AN INDEFINITE NONLINEAR DIFFUSION PROBLEM IN ... - CiteSeerX

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 27, Number 2, June 2010

doi:10.3934/dcds.2010.27.617 pp. 617–641

AN INDEFINITE NONLINEAR DIFFUSION PROBLEM IN POPULATION GENETICS, I: EXISTENCE AND LIMITING PROFILES

Kimie Nakashima Tokyo University of Marine Science and Technology 4-5-7 Konan, Minato-ku, Tokyo 108-8477, Japan

Wei-Ming Ni and Linlin Su School of Mathematics University of Minnesota Minneapolis, MN 55455, USA Abstract. We study the following Neumann problem  d∆u + g(x)u2 (1 − u) = 0 in Ω, = 0 on ∂Ω, 0 ≤ u ≤ 1 in Ω and ∂u ∂ν where ∆ is the Laplace operator, Ω is a bounded smooth domain in RN with ν as its unit outward normal on the boundary ∂Ω, and g changes sign in Ω. This equation models the “complete dominance” case in population genetics R of two alleles. We show that the diffusion rate d and the integral Ω g dx play important roles for the existence of stable nontrivial solutions, and the sign of g(x) determines the limiting profile of solutions as d tends to 0. In particular, a conjecture of Nagylaki and Lou has been largely resolved. Our results and methods cover a much wider class of nonlinearities than u2 (1 − u), and similar results have been obtained for Dirichlet and Robin boundary value problems as well.

1. Introduction. Nonlinear reaction-diffusion equations have arisen in numerous applications in population genetics. One such model dealing with two types of genes (alleles) A1 , A2 is as follows  ut = d∆u + g(x)u(1 − u)[hu + (1 − h)(1 − u)] in Ω × (0, ∞), (1) ∂u on ∂Ω × (0, ∞), ∂ν = 0 2

∂ where Ω is a bounded smooth domain in RN (N ≥ 1), ∆ = ΣN i=1 ∂x2i is the Laplace operator, ν is the unit outward normal to ∂Ω, d is a positive parameter, g(x) changes sign in Ω, and 0 ≤ h ≤ 1 is a constant. Here u(x, t) represents the frequency of the allele A1 at time t and place x in the habitat Ω; therefore, only solutions with 0 ≤ u ≤ 1 are in consideration. The term ∆u represents the effect of population dispersal. d > 0 is essentially the ratio of the migration rate to the intensity of selection s0 . The zero Neumann condition at the boundary ∂Ω means that there is

2000 Mathematics Subject Classification. Primary: 35K57; Secondary: 35B25. Key words and phrases. Diffusion equations, indefinite nonlinearity, variational method, limiting behavior. Research supported in part by NSF.

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KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

no flux of genes across the boundary ∂Ω. If we denote the local selection coefficient of the genotype Ai Aj by rij (x), (1) may be derived by setting (see [3], [9], and [15]) r11 (x) = 1, r12 (x) = 1 − hs0 g(x), r22 (x) = 1 − s0 g(x),

where 0 ≤ h ≤ 1 specifies the degree of dominance (assumed to be independent of location). Moreover, the selection coefficient of Ai , denoted by ri (x), turns out to be r1 (x) = u + (1 − hs0 g(x))(1 − u), r2 (x) = (1 − hs0 g(x))u + (1 − s0 g(x))(1 − u)

respectively. It is clear that, for 0 < u < 1, r1 > r2 when g(x) > 0 and r1 < r2 when g(x) < 0. Therefore, the gene A1 has selective advantage at position x with g(x) > 0 and selective disadvantage at position x with g(x) < 0. The degree of dominance h plays an important role in the study of (1). To this date, all the mathematical results in the literature require at least that 0 < h < 1. For instance, the existence of nontrivial steady states has been obtained in [3], [14] for 0 < h < 1. The uniqueness of nontrivial steady states requires even more stringent restrictions, namely, 1/3 ≤ h ≤ 2/3. (See e.g., [1], [5], and [6].) The case h = 1 is, however, both biologically important and mathematically interesting. (The case h = 0 is similar.) Since in the case the heterozygote A1 A2 has the same fitness as the homozygote A2 A2 , i.e., r12 = r22 , we say that A2 is completely dominant to A1 , or A1 is recessive. Mathematically, in this case, problem (1) reduces to  ut = d∆u + g(x)u2 (1 − u) in Ω × (0, ∞), (2) ∂u on ∂Ω × (0, ∞). ∂ν = 0 Existing methods do not seem to apply to (2) presumably due to the “degeneracy” of the nonlinearity u2 (1 − u) at u = 0. Indeed, for (2), the following conjecture has been around for a long time (see Nagylaki and Lou [11, p. 152]): R (a) If Ω g dx = 0, then for every d > 0 problem (2) has a unique nontrivial equilibrium which is asymptotically stable. R (b) If Ω g dx > 0, then there exists d0 > 0 such that for every d ∈ (0, d0 ) problem (2) hasR a unique nontrivial equilibrium which is asymptotically stable. (c) If Ω g dx < 0, then there exists d˜0 > 0 such that for every d ∈ (0, d˜0 ) problem (2) has exactly two nontrivial equilibria, one is asymptotically stable and the other one is unstable. We should remark that the continuous-time but discrete-space version of this conjecture has been studied by Nagylaki [10] recently. However, the above conjecture, which is continuous-time and continuous-space, has eluded mathematicians’ effort thus far. In this paper, Part I of our study, we will establish the existence of stable nontrivial steady states of (2) and study their limiting profiles as d → 0 or ∞. In particular, Theorem R1.1 below guarantees that (2) has a stable nontrivial equilibrium for every R d > 0 if Ω g dx = 0, and, for every d > 0 small if Ω g dx 6= 0. Moreover, the asymptotic behaviors of those nontrivial steady states are obtained in Theorems 1.3 and 1.4 in this paper. In Part II of our study [7], the stability/instability properties of the trivial steady states of (2) will be investigated, and theRexistence of a second nontrivial steady state of (2), for every d > 0 small, in case Ω g dx < 0, which is

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unstable, will follow as a consequence. This way, except for the uniqueness in (a), (b), and the exact multiplicity in (c), the above conjecture is resolved. In fact, we shall study the steady states of a much more general problem than (1), namely,   d∆u + g(x)f (u) = 0 in Ω, 0≤u≤1 in Ω, (3)  ∂u = 0 on ∂Ω, ∂ν where g ∈ C α (Ω) (0 < α < 1), and f is Lipschitz continuous and assumed to satisfy f (0) = f (1) = 0, and f > 0 in (0, 1).

(4)

Note that under the assumption (4), f is allowed to be degenerate of any order at 0 and/or 1. It is easy to see that for (3) to have a nontrivial solution that g changes sign in Ω (5) R is necessary. It turns out that the integral Ω g dx and the diffusion rate d are crucial in the existence of nontrivial solutions for (3). To formulate our results, we introduce the following functional Z d Jd (u) := [ |Du|2 − g(x)F (u)] dx, u ∈ H 1 (Ω), (6) Ω 2 where  if u < 0,  0R u f (s) ds if 0 ≤ u ≤ 1, F (u) = 0  R1 f (s) ds if u ≥ 1. 0 Our first main result concerns the existence of stable solutions of (3). Theorem 1.1. Assume that g satisfies (5). Then the following statements hold. R (i) If Ω g dx = 0, then for every d > 0, problem (3) has a stable nontrivial solution ud which R is a global minimizer of Jd (·). (ii) If Ω g dx 6= 0, then there exists d∗ > 0 such that for every d ∈ (0, d∗ ), problem (3) has a stable nontrivial solution ud which is a global minimizer of Jd (·). To complement the existence result we have the following nonexistence result. R Theorem 1.2. (i) Suppose that Ω g dx 6= 0. If f satisfies that

f (s) = a1 > 0 for some k1 ≥ 1, and (7) sk1 f (s) lim− = a2 > 0 for some k2 ≥ 1, (8) s→1 (1 − s)k2 then (3) has no nontrivial solution for d > 0 large. R (ii) Suppose that Ω g dx > 0. If (8) holds and f ′ (s) ≥ 0 in (0, δ0 ) for some δ0 > 0, then (3) has no nontrivial solution for d > 0 large. R (iii) Suppose that Ω g dx < 0. If (7) holds and f ′ (s) ≤ 0 in (1 − δ0 , 1) for some δ0 > 0, then (3) has no nontrivial solution for d > 0 large. lim

s→0+

In particular, Theorem 1.2 applies to the degenerate cases h = 0 and h = 1 in our model problem (1). From both mathematical and biological point of view, it seems important to understand the shape of solutions. The solution ud guaranteed by Theorem 1.1 has the following property for d small.

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KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

Theorem 1.3. Assume that (5) holds. Then as d approaches zero, ud → 1 uniformly on any compact subset of Ω+ \ (∂Ω+ ∩ Ω), and ud → 0 uniformly on any compact subset of Ω− \ (∂Ω− ∩ Ω), where Ω+ = {x ∈ Ω : g(x) > 0} and Ω− = {x ∈ Ω : g(x) < 0}. In case N = 1, i.e., when Ω is an interval, we have further information of the behavior of the solution ud on the zero set of g. The interesting part is the behavior of ud at a point c where g changes sign. We will show that when g changes sign “cleanly” at c (i.e., g(x) = γ|x − c|k−1 (x − c) + o(|x − c|k ) for some γ 6= 0 and k > 0), ud (c) tends to 1/2 as d → 0 provided that f (s) = f (1 − s), and for our model problem (1) where f (u) = u(1 − u)[hu + (1 − h)(1 − u)], 0 ≤ h ≤ 1, ud (c) must converge to a limit l, where   < 1/2 if 0 ≤ h < 1/2, = 1/2 if h = 1/2, l  > 1/2 if 1/2 < h ≤ 1.

In particular, for the important “completely dominant” case, namely, h = 1, ud (c) → l > 1/2 as d → 0. In general, the limit of ud (c), as d → 0, is determined by the value of the unique solution of a suitably scaled problem on the entire real line. (See Theorem 3.7, Examples 3.8 R and 3.9, and the remarks in Section 3.) On the other hand, when Ω g dx = 0, any nontrivial solution of (3) has the following limiting behavior as d tends to infinity. R Theorem 1.4. Assume that (5) holds and Ω g dx = 0. Then, as d → ∞, any sequence of nontrivial solutions of (3) has a subsequence converging in C 2 (Ω) to a constant c ∈ [0, 1]. Moreover, if k is the least positive integer such that f (k) (c) 6= 0, then c ∈ (0, 1) and k must be even; in other words, f takes on an extremum at c. The above two results seem to have nice biological explanations for the genetic model (1). Since d is proportional to the migration rate, in the limit d = 0 there is no dispersal. Therefore, it seems reasonable that as d approaches zero, nontrivial steady state tends to 1 in Ω+ where A1 is favored, and to 0 in Ω− where A2 is favored. On the other hand, the unique extreme point of f (u) = u(1 − u)[hu + (1 − h)(1 − u)] (with 0 ≤ h ≤ 1) in √ (0, 1) is the maximum point c0 , namely, c0 = 1/2 if h = 1/2 and c0 = (2 − 3h − 1 − 3h + 3h2 )/(3 − 6h) if h 6= 1/2. Therefore, Theorem 1.4 implies that any nontrivial steady state converges to c0 as d tends to infinity; in particular, c0 = 2/3 in the “completely dominant” case h = 1. This result seems reasonable because when d is large, the dispersal is rapid and we may think that the population acts as Ra single unitR in the habitat Ω where A1 and A2 have “equal advantage” because of Ω+ g dx = Ω− (−g) dx. Thus, in the limit d = ∞ nontrivial solutions tend to an intermediate constant state. Our approach applies equally well to the equation in (3) under Dirichlet or Robin boundary conditions, which will be discussed in Section 4. This paper is organized as follows. In Section 2, we prove Theorems 1.1 and 1.2 — the existence and non-existence results. In Section 3, we establish the behavior of a global minimizer ud as d tends to zero (Theorem 1.3, Propositions 3.4 and 3.5, and Theorem 3.7), and the behavior of any nontrivial solution of (3) as d tends to infinity (Theorem 1.4). Finally, the corresponding results for the Dirichlet and Robin problems are included in Section 4.

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2. Existence of a stable nontrivial global minimizer. Proof of Theorem 1.1. Fix d > 0. We first show that Jd (·) (defined as in (6)) has a global minimizer ud in H 1 (Ω) with 0 ≤ ud ≤ 1. Since F (·) is bounded, we see that β := inf{Jd (u) : u ∈ H 1 (Ω)} > −∞. Let {un }∞ n=1 be a minimizing sequence, i.e., Jd (un ) decreases to β. We may assume 0 ≤ un ≤ 1 for each n, as otherwise we simply replace un by  if un > 1,  1 un if 0 ≤ un ≤ 1, u ˜n :=  0 if un < 0,

which is still in H 1 (Ω) and satisfies Jd (˜ un ) ≤ Jd (un ). Hence {un }∞ n=1 is bounded 1 in H -norm. Therefore, we may assume, passing to a subsequence if necessary, un ⇀ u weakly in H 1 (Ω),

(9)

un → u strongly in L2 (Ω).

(10)

and

From (9) we see that Z

Ω

|Du|2 dx ≤ lim inf

Z

Ω

|Dun |2 dx.

(11)

From (10) it follows that un → u almost everywhere by passing to a subsequence if necessary, and therefore, Z Z gF (un ) dx → gF (u) dx Ω

Ω

by Lebesgue Dominated Convergence Theorem. This together with (11) implies that Z d Jd (u) = [ |Du|2 − g(x)F (u)] dx Ω 2 Z d ≤ lim inf [ |Dun |2 − g(x)F (un )] dx = β. Ω 2 Thus u is a global minimizer and 0 ≤ u ≤ 1. Now, standard calculus of variations and elliptic regularity theory guarantee that u is a classical solution of (3). This shows that for any d > 0 there always exists a solution 0 ≤ ud ≤ 1 of (3) which is a global minimizer of Jd (·). Therefore, to establish the existence of a nontrivial solution, it suffices to show that neither u ≡ 0 nor u ≡ 1 Ris a global minimizer. To establish Part (i), we assume that (5) holds and Ω g dx = 0. So we have Jd (0) = JdR(1) = 0. Fix some constant c ∈ (0, 1), and choose some 0 < φ ∈ C 1 (Ω) such that Ω gφ dx > 0. Let u = c + εφ, where ε > 0 is a small number. Then for

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KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

any d > 0 fixed,

= = =

Jd (u) = Jd (c + εφ) Z d { ε2 |Dφ|2 − g(x)[F (c) + F ′ (c)εφ + o(|εφ|)]} dx 2 ZΩ d 2 [ ε |Dφ|2 − εf (c)gφ + o(|εφ|)] dx Ω 2 Z Z Z d o(|εφ|) ε|Dφ|2 dx − f (c) gφ dx + dx] < 0, ε[ 2 ε Ω Ω Ω

for ε sufficiently small. This implies that for every d > 0 problem (3) has a nontrivial solution. R For Part (ii), under the assumption (5), we have Ω+ gF (1) dx := δ1 > 0 and R − Ω− gF (1) dx := δ2 > 0. Then Jd (0) = 0 and Jd (1) = −δ1 + δ2 . Since Z Z − gF (χ{g>0} ) dx = − gF (1) dx = −δ1 , Ω

Ω+

we can find a function u˜ ∈ H 1 (Ω) such that Z δ1 δ2 − gF (˜ u) dx < min{− , −δ1 + }. 2 2 Ω Then we choose d0 > 0 small such that Z d0 δ1 δ2 |D˜ u|2 dx < min{ , }. 2 2 2 Ω Hence, for any 0 < d ≤ d0 ,

δ2 δ1 δ2 δ1 , −δ1 + } + min{ , } 2 2 2 2 ≤ min{0, −δ1 + δ2 } = min{Jd (0), Jd (1)}.

Jd (˜ u) < min{−

Now, set d∗ := sup{d > 0 : ∃ u ∈ H 1 (Ω) such that Jd (u) < min{Jd (0), Jd (1)}}. The above arguments show that d∗ > 0 and for any d ∈ (0, d∗ ) we obtain a nontrivial global minimizer solution of (3). Finally, the existence of a stable nontrivial global minimizer for every d ∈ (0, d∗ ) follows from some standard arguments, which we include in Appendix B for completeness. Proof of Theorem 1.2. We shall show that (3) has no nontrivial solution for d large under conditions in (i), (ii) and (iii) respectively. For the sake of simplicity, we write k · k for k · kL∞ (Ω) throughout. Suppose, for a contradiction, that there ∞ exist {dn }∞ n=1 and {un }n=1 with dn → ∞ such that for each n,  1  ∆un + dn g(x)f (un ) = 0 in Ω, 0 0, let Ω+,h = {x ∈ Ω+ : g(x) ≤ h}. It is easy to see that limh→0 |Ω+,h | = 0 and there exists h0 > 0 such that 0 < |Ω+,h0 | ≤ ε by the continuity of g(x) and the assumption (5). Then we have Z Z g(x) dx ≥ g(x) dx := δ1 > 0. {x∈Ω+ :1−udn ≥ε}

Ω+ ,h0

Therefore, Jdn (udn ) ≥ −

Z

Ω+

g(x)F (1) dx + δ2 ,

(13)

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where δ2 := [F (1) − F (1 − ε)]δ1 > 0. On the other hand, we can choose some u in H 1 (Ω) such that Z Z δ2 − g(x)F (1) dx + , g(x)F (u) dx < − 2 Ω Ω+ and choose d0 > 0 small such that Z d0 δ2 |Du|2 dx < . 2 2 Ω

Then for any dn with dn ≤ d0 , we have Z Jdn (u) < − g(x)F (1) dx + δ2 . Ω+

In view of (13), this implies that udn is not a global minimizer of Jdn (·), which is a contradiction. Lemma 3.2. Assume that (5) holds. Then as d tends to 0, ud → 1 uniformly on any compact subset of Ω+ , and ud → 0 uniformly on any compact subset of Ω− . Proof. Let D+ be a compact subset of Ω+ . For any x0 in D+ , we may find a number ρ > 0 such that B2ρ (x0 ) ⊂ Ω+ . We claim that for any ε > 0 there exists d(ε) > 0 such that inf ud (x) ≥ 1 − ε, for all 0 < d < d(ε). Bρ (x0 )

Suppose not, then there exist some ε > 0 and a sequence {dn }∞ n=1 with dn → 0 as n → ∞ such that inf udn (x) < 1 − ε, for all n ≥ 1. Bρ (x0 )

For each n, assume that udn achieves its infimum in Bρ (x0 ) at yn ∈ Bρ (x0 ). Since Bρ (yn ) ⊂ B2ρ (x0 ) ⊂ Ω+ , we have 1 g(x)f (udn ) < 0 in Bρ (yn ). dn By the mean value inequality for superharmonic functions, we get Z 1 ud dx ≤ udn (yn ) < 1 − ε. |Bρ (yn )| Bρ (yn ) n ∆udn = −

Hence, for each n, Z

udn dx

=

B2ρ (x0 )

Z

B2ρ (x0 )\Bρ (yn )

udn dx +

Z

udn dx

Bρ (yn )

< |B2ρ (x0 ) \ Bρ (yn )| + (1 − ε)|Bρ (yn )| = |B2ρ (x0 )| − ε|Bρ (yn )|.

(14)

On the other hand, udn → 1 in measure in B2ρ (x0 ) as n → ∞ by Lemma 3.1, from which it follows that Z udn dx → |B2ρ (x0 )| as n → ∞. B2ρ (x0 )

But this contradicts (14). Hence ud → 1 uniformly in Bρ (x0 ) as d → 0. By standard compactness argument we get ud → 1 uniformly on D+ . Similarly, ud → 0 uniformly on any compact subset of Ω− .

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KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

In the following lemma, we study the asymptotic behavior of ud near a boundary point P . Lemma 3.3. (i) Let P ∈ Ω− ∩ ∂Ω with a neighborhood U of P such that (U ∩ Ω) ⊂ Ω− . Then there exists a neighborhood U ′ of P such that ud → 0 uniformly on U ′ ∩ Ω− as d → 0. (ii) Let P ∈ Ω+ ∩ ∂Ω with a neighborhood U of P such that (U ∩ Ω) ⊂ Ω+ . Then there exists a neighborhood U ′ of P such that ud → 1 uniformly on U ′ ∩ Ω+ as d → 0.

Proof. (i) We first straighten a boundary portion near P as in [12]. Through translation and rotation of the coordinate system, we may assume that P is the origin with the inner normal to ∂Ω at P pointing in the direction of the positive xN -axis. Then there exists a smooth function ψ(x′ ), x′ = (x1 , ..., xN −1 ), defined for |x′ | < δ0 , such that (A) ψ(0) = 0 and Dψ(0) = 0; (B) ∂Ω ∩ N = {(x′ , xN ) : xN = ψ(x′ )} and Ω∩N = {(x′ , xN ) : xN > ψ(x′ )}, where N ⊂ U is a neighborhood of P . For y ∈ RN with |y| sufficiently small, we define a mapping x = Φ(y) = (Φ1 (y), ..., ΦN (y)) by ( ∂ψ (y ′ ) for j = 1, 2, ..., N − 1, yj − yN ∂x j (15) Φj (y) = yN + ψ(y ′ ) for j = N .

As DΦ(0) = Id by virtue of Dψ(0) = 0, Φ has the inverse mapping y = Φ−1 (x) for |x| < δ1 (< δ0 ). We write Ψ(x) = (Ψ1 (x), ..., ΨN (x)) for Φ−1 (x). The diffeomorphism y = Ψ(x) straightens a boundary portion near P . We may assume that Φ is defined in an open set containing the closure of B2ρ := B2ρ (0) ⊂ RN (ρ > 0) such that Φ(B2ρ ) ⊂ {|x| < δ1 } ∩ N . We put + , vd (y) := ud (Φ(y)), for y ∈ B2ρ

+ where B2ρ = {y ∈ B2ρ : yN > 0}. So vd (y) satisfies

d

N X

aij (y)(vd )yi yj (y) + d

i,j=1

N X

bi (y)(vd )yi (y) + g(Φ(y))f (vd (y)) = 0

i=1

PN + on B2ρ , where aij (y) = l=1 extend vd to B2ρ by reflection: v˜d (y) :=

∂Ψj ∂Ψi ∂xl (Φ(y)) ∂xl

(

(Φ(y)), bi = ∆Ψi (Φ(y)). Next, we

+ vd (y) if y ∈ B2ρ , − ′ vd (y , −yN ) if y ∈ B2ρ ,

− where B2ρ = {y ∈ B2ρ : yN < 0}, y ′ = (y1 , ..., yN −1 ), and put  aij (y) if yN ≥ 0, a ˜ij (y) := (−1)δiN +δjN aij (y ′ , −yN ) if yN < 0,  bi (y) if yN ≥ 0, ˜bi (y) := (−1)δiN bi (y ′ , −yN ) if yN < 0, and  g(Φ(y)) if yN ≥ 0, g˜(y) := g(Φ(y ′ , −yN ))) if yN < 0, where δiN is the Kronecker symbol. Then v˜d (y) satisfies

d

N X

i,j=1

a ˜ij (y)(˜ vd )yi yj (y) + d

N X i=1

˜bi (y)(˜ vd )yi (y) + g˜(y)f (˜ vd (y)) = 0

(16)

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in B2ρ \ {yN = 0}. Here are several observations which will be used later: ∂vd (a) ∂y |{yN =0} = 0. This is due to N ∂vd |{y =0} ∂yN N

= =

∂ud ∂Φj · |{y =0} ∂xj ∂yN N N −1 X j=1

=

∂ud ∂ψ ∂ud · (− )+ ∂xj ∂xj ∂xN

∂ud · |(−Dx′ ψ, 1)|, ∂ν

by virtue of (15). (b) a ˜ij , a ˜N N , ˜bi with 1 ≤ i, j ≤ N − 1 are Lipschitz continuous on B2ρ . For 1 ≤ i ≤ N −1, a ˜iN belongs to C 1 (B2ρ ), because aiN (y ′ , 0) = 0. ˜bN is not continuous, but it is a bounded measurable function. + (c) DΨ(Φ(y)) is nonsingular at each y ∈ B2ρ . Therefore, the matrix A(y) := t ˜ (aij (y)) = (DΨ)(DΨ) (Φ(y)) is positive definite, and so is the matrix A(y) := (˜ aij (y)) for each y ∈ B2ρ . Moreover, owing to the continuity of a ˜ij , there exist constants 0 < λ, Λ < ∞ such that 0 < λ|ξ|2 ≤

N X

i,j=1

a ˜ij ξi ξj ≤ Λ|ξ|2 , for all y ∈ B2ρ , ξ ∈ RN . ∂vd ∂yN |{yN =0}

Since v˜d ∈ W 1,2 (B2ρ ) and solution satisfying L˜ vd :=

N X

[˜ aij (y)(˜ vd )yj (y)]yi +

i,j=1

N X j=1

= 0, we may view v˜d as a weak lower

[˜bj (y) − (˜ aij )yi (y)](˜ vd )yj (y) ≥ 0,

which is an elliptic operator with bounded measurable coefficients in the domain B2ρ , noting that the generalized first order derivatives of a ˜ij are bounded and measurable functions. So we can apply the local estimate for weak lower solutions [4, Theorem 8.17] and obtain that ˜ sup v˜d ≤ ck˜ vd kLp (B2ρ ) , p > 1, c = c(N, Λ/λ, γρ, p), Bρ

˜ and γ are constants such that for all y ∈ B2ρ where Λ N X

i,j=1

˜ 2 , λ−2 |˜ aij (y)|2 ≤ Λ

N X

i,j=1

|˜bj (y) − (˜ aij )yi (y)|2 ≤ γ 2 .

+ By Lemma 3.1, ud → 0 as d → 0 in measure on Φ(B2ρ ) ⊂ Ω− , so Z Z vdp | det DΦ(y)| dy = upd dx → 0 as d → 0. + B2ρ

+ Φ(B2ρ )

This implies Z

+ B2ρ

vdp dy → 0 as d → 0,

(17)

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KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

+ because | det DΦ(y)| ≥ c1 on B2ρ for some positive constant c1 . Hence, Z Z v˜dp dy = 2 vdp dy → 0 as d → 0. + B2ρ

B2ρ

From (17) it follows that Z sup v˜d ≤ c( Bρ

B2ρ

v˜dp dy)1/p → 0 as d → 0,

which implies that v˜d → 0 uniformly on Bρ as d → 0. Therefore, ud → 0 uniformly on U ′ ∩ Ω− as d → 0, where U ′ = Φ(Bρ ). To establish Part (ii), we may apply exactly the same arguments as above to (1 − ud ). Now, Theorem 1.3 follows from Lemmas 3.2 and 3.3 via standard compactness arguments. Next, we investigate the behavior of ud on the zero set of g when the space dimension N = 1. Let Ω = (a, b), a < b, and Ω+ = {x ∈ (a, b) : g(x) > 0},

Ω− = {x ∈ (a, b) : g(x) < 0},

Ω0 = {x ∈ [a, b] : g(x) = 0}. A connected component of Ω0 is either a closed interval or a single point. Since u′′d = 0 on Ω0 , ud is a straight line on each connected interval component I0 of Ω0 . Hence, to see the behavior of ud on I0 , we only need to understand the behavior of ud at boundary points of I0 . When g changes sign infinitely many times in Ω (e.g., g takes on values oscillating around zero), the situation becomes quite involved, and we will leave it for future consideration. Here, to make it simple, we assume that both Ω+ and Ω− have finitely many connected components.

(18)

We first describe the behavior of ud on Ω0 \ (Ω+ ∩ Ω− ) in the following two simple propositions. Proposition 3.4. Assume that (18) holds. Then for c ∈ ∂Ω0 ∩ Ω, we have the following conclusions: (i) limd→0 ud (c) = 0, if c ∈ Ω− and c 6∈ Ω+ . (ii) limd→0 ud (c) = 1, if c ∈ Ω+ and c 6∈ Ω− . Proposition 3.5. Assume that (18) holds. Then for c ∈ ∂Ω0 ∩ {a, b}, we have the following conclusions: (i) limd→0 ud (c) = 0 if c ∈ Ω− , and limd→0 ud (c) = 1 if c ∈ Ω+ . (ii) limd→0 ud (c) = 0 if p ∈ Ω− , and limd→0 ud (c) = 1 if p ∈ Ω+ , when c and p are the two endpoints of a connected component of Ω0 . To prove Propositions 3.4 and 3.5, we need the following lemma. Lemma 3.6. Let I+ (I− ) denote a connected component of Ω+ (Ω− ). Assume that c ∈ ∂I+ (∂I− ). Then ud (c) 9 1 (0) as d → 0 implies that {u′d (c)} is unbounded.

Proof. Assume that c ∈ ∂I+ and ud (c) 9 1. Then there exists {dk }∞ k=1 with dk → 0 as k → ∞, and udk (c) → β for some 0 ≤ β < 1. For any integer n, let h = (1 − β)/(4n). We may assume without loss of generality that c is the left endpoint of I+ . Pick some x ∈ I+ such that 0 < x − c < h. Since udk (x) → 1 as k → ∞, there exists some k0 such that udk (x) ≥ (3 + β)/4 for all k ≥ k0 . Since

EXISTENCE AND LIMITING PROFILES

629

udk (c) → β as k → ∞, there exists some k1 such that udk (c) ≤ (1 + β)/2 for all k ≥ k1 . Hence, for k ≥ max{k0 , k1 } we have udk (x) − udk (c) (3 + β)/4 − (1 + β)/2 ≥ = n. x−c (1 − β)/4n

As u′′d < 0 in I+ , we see that u′dk (c) ≥ (udk (x) − udk (c))/(x − c). Therefore, u′dk (c) ≥ n for k ≥ max{k0 , k1 }. This proves that {u′d(c)} is unbounded. The proof of the dual statement is similar. Proof of Proposition 3.4. (i) Near c we have the following three cases I− c

I−

I0

,

(A)

c

I−

, and

I− c

(B)

I0

.

(C)

For case (A): We pick two points x1 < c, x2 > c, x1 , x2 ∈ I− . Since u′′d (x) ≥ 0 on [x1 , x2 ], ud (c) ≤ max{ud(x1 ), ud (x2 )}. Hence, 0 ≤ lim ud (c) ≤ max{ lim ud (x1 ), lim ud (x2 )} = 0. d→0

d→0

d→0

For case (B) and case (C): Let p be the other endpoint of I0 . Then |u′d (c)| ≤ 1/|p−c|. Hence, limd→0 ud (c) = 0 by Lemma 3.6. The proof of Part (ii) is similar. Proof of Proposition 3.5. (i) When c ∈ Ω− , or c ∈ Ω+ , the conclusions follow from Lemma 3.6. (ii) When c and p are the two endpoints of a connected component of Ω0 , we have u′d (p) = u′d (c) = 0 in view of the Neumann boundary condition. Therefore, limd→0 ud (p) = 0 if p ∈ Ω− , and limd→0 ud (p) = 1 if p ∈ Ω+ by Lemma 3.6. Then the conclusion follows from ud (c) = ud (p). The more interesting and delicate case is when c belongs to Ω+ ∩ Ω− (which is not covered by Proposition 3.4). As d goes to zero, ud tends to 0 on one side of c and to 1 on the other side. Hence, ud has a sharp transition layer at c when d is small. Without loss of generality, we assume that c = 0 and that g changes sign from negative to positive when x passes through 0. We obtain the following result of the limiting behavior of ud (0). Theorem 3.7. Suppose that, for some γi , ki > 0 (i = 1, 2), and ε > 0,  γ1 xk1 + o(|x|k1 ) in [0, ε), g(x) = −γ2 |x|k2 + o(|x|k2 ) in (−ε, 0). Then

  0 1 lim ud (0) = d→0  Φ(0)

if k1 > k2 , if k1 < k2 , if k1 = k2 ,

where Φ is the unique solution of  ′′ Φ + g˜(y)f (Φ) = 0, 0 ≤ Φ ≤ 1 Φ(−∞) = 0, Φ(∞) = 1, and g˜(y) =



γ1 y k1 −γ2 |y|k1

in (−∞, ∞),

in [0, ∞), in (−∞, 0).

(19)

(20)

630

KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

Proof. The fact that problem (19) has a unique solution follows from Theorem A.1 proved in Appendix A by using the method of Fife and Peletier [2]. We may assume that r1 r2 g(x) ≥ xk1 in (0, ε), g(x) ≤ − |x|k2 in (−ε, 0), (21) 2 2 otherwise, take a smaller ε such that (21) holds. For each d > 0 small, the global minimizer ud satisfies  du′′d + g(x)f (ud ) = 0 in (−ε, ε), 0 < ud < 1 in (−ε, ε). Now, let x = λy, where λ > 0 is a scaling constant, and define vd (y) := ud (λy). Then vd satisfies d

ε ε vd′′ + g(λy)f (vd ) = 0, y ∈ (− , ), λ2 λ λ

i.e., d d

vd′′ o(|x|k1 ) k1 k1 ε + [λk1 γ1 y k1 + λ y ]f (vd ) = 0, y ∈ [0, ), 2 λ |x|k1 λ

vd′′ o(|x|k2 ) k2 k2 ε k2 k2 + [−λ γ |y| + λ |y| ]f (vd ) = 0, y ∈ (− , 0). 2 2 k 2 λ |x| λ

We first consider the case that k1 = k2 . Setting λ = d1/(k1 +2) , we have vd′′ + [˜ g (y) +

ε ε o(|x|k1 ) k1 |y| ]f (vd ) = 0, y ∈ (− 1/(k +2) , 1/(k +2) ). k 1 1 1 |x| d d

(22)

Thus, for any R > 0 fixed, (22) holds in (−R, R) for any 0 < d < dR , where dR = k1 ) ( Rε )k1 +2 . As d → 0, k[˜ g(y) + o(|x| |y|k1 ]f (vd )kL∞ ((−R,R)) is uniformly bounded, |x|k1 and so is kvd kC 2,α ((−R,R)) . Therefore, there exists a sequence {vdn }∞ n=1 with vdn → v in C 2 ((−R, R)) as dn → 0, where v satisfies  ′′ v + g˜(y)f (v) = 0 y ∈ (−R, R), 0≤v≤1 y ∈ (−R, R). Since R can be arbitrarily large, by a diagonal process, we obtain a subsequence of ∞ 2 {vdn }∞ n=1 (still denoted by {vdn }n=1 ) such that vdn → v in C -norm on any compact subset of R. So v satisfies  ′′ v + g˜(y)f (v) = 0 y ∈ (−∞, ∞), (23) 0≤v≤1 y ∈ (−∞, ∞). Since v is convex for y < 0 and concave for y > 0 and f satisfies (4), we see that the solutions to (23) must be either v ≡ 0, v ≡ 1, or v = Φ — the unique solution of (19). Next, we will show that, in fact, v ≡ 0 and v ≡ 1 cannot happen. Suppose that v ≡ 1. This implies that, for any R > 0, vdn → 1 in C 2 ((−R, R)) as dn → 0. In particular, vdn (0) → 1, vd′ n (0) → 0 as dn → 0.

Moreover, we have

lim vdn (−

dn →0

ε 1/(k +2) dn 1

) = lim udn (−ε) = 0 dn →0

EXISTENCE AND LIMITING PROFILES

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by Theorem 1.3. Therefore, for some fixed α ∈ (0, 1/2), there exists d0 > 0 such that for any dn ∈ (0, d0 ), we have vd′ n (0) < α,

vdn (0) > 2α,

and vdn (−

ε

1/(k +2) dn 1

) < α.

(24)

Now, for each dn ∈ (0, d0 ), let yn be the unique minimum point of vdn on [− 1/(kε1 +2) , 0]. The uniqueness of yn is due to the strict convexity of vdn for y < 0. dn

Then we have vd′ n (yn ) ≥ 0, and both vdn and vd′ n are strictly increasing in (yn , 0). Moreover, there exists a (unique) point y˜n in (yn , 0) such that vdn (˜ yn ) = α. We have, for any y ∈ [˜ yn , y˜n + 1], that Z y vd′ n (z) dz ≤ α + vd′ n (0) < 2α < vdn (0), vdn (y) = vdn (˜ yn ) + y˜n

which is due to (24) and the fact that vd′ n (0) is the maximum of vd′ n (y) for all y in the domain of vd′ n . This implies that y˜n + 1 < 0,

i.e.,

y˜n < −1.

Thus, vd′ n (0) ≥ ≥ =

yn ) vd′ n (yn ) + vd′ n (˜ yn + 1) − vd′ n (˜ yn ) vd′ n (˜ yn + 1) − vd′ n (˜ Z y˜n +1 vd′′n (y) dy y˜n y˜n +1

≥

Z

γ2 k1 |y| f (vdn (y)) dy 2

≥

Z

γ2 k1 |y| δ dy, 2

y˜n y˜n +1

y˜n

where δ = min f (s) > 0. α≤s≤2α

Therefore, vd′ n (0) ≥

γ2 δ 2

vd′ n (0) ε

Z

0

1

y k1 dy =

γ2 δ > 0. 2(k1 + 1)

But this contradicts that → 0 as dn → 0. Thus, v 6≡ 1. Similarly, using the equation of vdn on [0, 1/(k1 +2) ], we can rule out the possibility of v ≡ 0. Hence, dn v = Φ and, in fact, the above argument shows that any sequence of vd has a subsequence converging in C 2 -norm to Φ on any compact subset of R. Therefore, vd converges in C 2 -norm to Φ on any compact subset of R. In particular, we have limd→0 ud (0) = limd→0 vd (0) = Φ(0). Next, we consider the case that k1 > k2 . Setting λ = d1/(k2 +2) , we have vd′′ + [γ1 λk1 −k2 y k1 +

o(|x|k1 ) k1 −k2 k1 ε λ |y| ]f (vd ) = 0, y ∈ (0, 1/(k +2) ), k 1 2 |x| d

(25)

o(|x|k2 ) k2 ε |y| ]f (vd ) = 0, y ∈ (− 1/(k +2) , 0). k 2 2 |x| d

(26)

vd′′ + [−γ2 |y|k2 +

632

KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

As before, we get a sequence of vdn such that, for any R > 0, vdn → v in C 2 ((−R, R)) as dn → 0, where v satisfies  ′′ y ∈ [0, ∞),  v =0 v ′′ − γ2 |y|k2 f (v) = 0 y ∈ (−∞, 0), (27)  0≤v≤1 y ∈ (−∞, ∞).

It is easy to see that v ≡ 0 and v ≡ 1 are the only possible solutions to (27). In light of the second equation in (27), we can rule out the possibility of v ≡ 1 as before. Therefore, vd → 0 in C 2 -norm on any compact subset of R. In particular, we have limd→0 ud (0) = limd→0 vd (0) = 0. Similarly, in case k1 < k2 , we set λ = d1/(k1 +2) and get vd → 1 in C 2 -norm on any compact subset of R. In particular, we have limd→0 ud (0) = limd→0 vd (0) = 1. This completes the proof of the theorem. Theorem 3.7 suggests that it is both interesting and important to know the value of Φ at 0. For simplicity, in the following discussion, we will assume that g˜(y) = γ|y|k−1 y in R for some γ > 0 and k > 0 in equation (19). We observe that if f is symmetric, namely, f (s) = f (1 − s) for s ∈ [0, 1], then Φ must satisfy Φ(y) = 1 − Φ(−y) by the uniqueness property of (19), which implies that Φ(0) = 12 . On the other hand, it is not hard to construct examples with Φ(0) 6= 12 when f is not symmetric. Example 3.8. Let f1 (s) = s(1 − s), s ∈ [0, 1], and  f1 (s) 0 ≤ s ≤ 12 , f2 (s) = 1 f˜ 2 ≤ s ≤ 1, where f˜ ∈ C 1 ([ 12 , 1]) satisfies that f˜( 12 ) = 14 , f˜(1) = 0, f˜(s) > f1 (s) for all s ∈ ( 12 , 1). Let Φi (i = 1, 2) be the unique solution of (19) with f = fi . We observe that Φ1 is a lower solution of (19) with f = f2 , since for y > 0, 12 < Φ1 (y) < 1 and hence Φ′′1 + γ|y|k−1 yf2 (Φ1 ) = γy k [−Φ1 (1 − Φ1 ) + f˜(Φ1 )] > 0, whereas, for y ≤ 0, Φ1 is an exact solution. Next, we construct an upper solution of (19) with f = f2 by the method used in Appendix A. Let w(y) be the solution of  ′′ w − γw(1 − w) = 0 in (−∞, −1), w(−∞) = 0, w(−1) = 1, and set ¯2 = Φ



w(y) 1

for y < −1, for y ≥ −1.

¯ ′′2 + γ|y|k−1 yf2 (Φ ¯ 2 ) < 0 for y < −1 and Φ ¯ ′′2 + γ|y|k−1 yf2 (Φ ¯ 2) = 0 We observe that Φ ¯ for y > −1, whereas for y near −1, Φ2 is the minimum of two regular upper solutions. ¯ 2 is a weak upper solution of (19) with f = f2 . Therefore, Φ ¯ 2 , since Φ ¯2 Furthermore, by the uniqueness (Theorem A.1), we see that Φ1 ≤ Φ may serve as the upper solution in a pair of order-related lower and upper solutions in constructing the solution of (19) with f = f1 . (See the Appendix A.) Therefore, ¯ 2 . Moreover, the inequalities are strict by the maximum principle. In Φ1 ≤ Φ2 ≤ Φ particular, Φ2 (0) > Φ1 (0) = 12 .

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Example 3.9. In the genetic model (1), f (u) = u(1 − u)[hu + (1 − h)(1 − u)] := fh , where 0 ≤ h ≤ 1. We denote the unique solution of (19) with f = fh by Φh . For each 0 ≤ h ≤ 1, we introduce  fh (s) if 0 ≤ s ≤ 21 , ˜ fh (s) := fh (1 − s) if 12 < s ≤ 1. ˜ h , satisfies Then f˜h is symmetric, and the solution of (19) with f = f˜h , denoted by Φ 1 ˜ Φh (0) = 2 . It is straightforward to compute that f (s) − f (1 − s) = (2h − 1)(2s − 1)s(1 − s).

Then, for h = 21 , f (s) = f (1 − s), and therefore, fh (s) > f˜h (s) if 21 < s < 1. Therefore, the same

Φh (0) = 12 . For any h ∈ ( 21 , 1], computation as in Example 3.8 ˜ guarantees that Φh is a lower solution of (19) with f = fh , and  w(y) for y < −1, ¯h = Φ 1 for y ≥ −1, where w(y) is the solution of  ′′ w − γ f˜h (w) = 0 in (−∞, −1), w(−∞) = 0, w(−1) = 1,

˜h ≤ Φ ¯ h . Therefore, by the is an upper solution of (19) with f = fh satisfying Φ ˜ ¯ uniqueness property of problem (19), we have Φh ≤ Φh ≤ Φh ; in particular, Φh (0) > ˜ h (0) = 1 by the maximum principle. For any h ∈ [0, 1 ), we have that fh (s) < f˜h (s) Φ 2 2 ˜ h is an upper solution of if 12 < s < 1. Similar, one may check that, in this case, Φ (19) with f = fh , and  v(y) for y > 1, Φh = 0 for y ≤ 1, where v(y) is the solution of  ′′ v + γfh (v) = 0 in (1, ∞), v(1) = 0, v(∞) = 1,

˜ h . Therefore, by the is a lower solution of (19) with f = fh satisfying Φh ≤ Φ ˜ h ; in particular, Φh (0) < uniqueness property of problem (19), we have Φh ≤ Φh ≤ Φ ˜ h (0) = 1 by the maximum principle. Φ 2 In view of Theorem 3.7, this indicates that with f (u) = u(1−u)[hu+(1−h)(1−u)], at a point c where g changes sign, namely, g(x) = γ|x − c|k−1 (x − c) + o(|x − c|k ) for some γ > 0 and k > 0, we have ud (c) → l, where   < 1/2 if 0 ≤ h < 1/2, = 1/2 if h = 1/2, l  > 1/2 if 1/2 < h ≤ 1.

Finally, we conclude this section by proving Theorem 1.4 about the behavior of any nontrivial solutions as d tends to infinity.

Proof of Theorem 1.4. Suppose the assumptions in Theorem 1.4 hold. Let {dn }∞ n=1 be any sequence such that dn → ∞ as n → ∞, and let un be any nontrivial solution of (3) with d = dn . Then there exists a subsequence {unk }∞ k=1 such that unk → u in C 2 (Ω), where u satisfies  ∆u = 0, 0 ≤ u ≤ 1 in Ω, ∂u on ∂Ω. ∂ν = 0

634

KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

Thus u ≡ c for some constant c ∈ [0, 1]. Now, assume that k is the least positive integer such that f (k) (c) 6= 0. Dividing both sides of the equation of un by f (un ) and integrating, we obtain Z |Dun |2 · f ′ (un ) dx = 0. 2 Ω f (un )

This implies that f ′ (c) = 0 and f ′ must change sign in a neighborhood of c in [0, 1]. Therefore, c ∈ (0, 1) and k is even. 4. The Dirichlet and Robin problems. This the Dirichlet boundary value problem   d∆u + g(x)f (u) = 0 0≤u≤1  u=0

section is devoted to the study of in Ω, in Ω, on ∂Ω,

and the Robin boundary value problem   d∆u + g(x)f (u) = 0 0≤u≤1  ∂u ∂ν + β(x)u = 0, β(x) ≥ 0, 6≡ 0

in Ω, in Ω, on ∂Ω,

(28)

(29)

where g ∈ C α (Ω) (0 < α < 1), f is Lipschitz continuous and satisfies (4), and β ∈ C 1,α (Ω). It is easy to see that for (28) and (29) to have a nontrivial solution, it is necessary that g(x) is positive somewhere in Ω. (30) Problems (28) and (29) have the following parallel properties to the Neumann problem (3). Theorem 4.1. Assume that g satisfies (30). Then the following statements hold. (i) There exists d∗ > 0 such that for every d ∈ (0, d∗ ), problem (28) has a stable nontrivial solution ud which is a global minimizer of Jd,D (·), where Z d Jd,D (u) := [ |Du|2 − g(x)F (u)] dx, u ∈ H01 (Ω). Ω 2 (ii) There exists d∗ > 0 such that for every d ∈ (0, d∗ ), problem (29) has a stable nontrivial solution ud which is a global minimizer of Jd,R (·), where Z Z d d Jd,R (u) := [ |Du|2 − g(x)F (u)] dx + β(x)u2 dsx , u ∈ H 1 (Ω). Ω 2 ∂Ω 2 (iii) Neither (28) nor (29) has a nontrivial solution for d > 0 large. Proof. We shall only show the proofs of the assertions for (29), since those of (28) are similar and simpler. As in the proof of Theorem 1, in order to find a nontrivial solution of (29), it suffices to show that the trivial solution R u ≡ 0 is not a global minimizer of Jd,R (·). Under the assumption (30), we have Ω+ gF (1) dx := δ1 > 0, R and hence −R Ω gF (χ{g>0} ) dx = −δ1 < 0. Thus we can find a function u ∈ H 1 (Ω) such that − Ω gF (u) dx < −δ1 /2. Then we choose d0 > 0 small such that Z Z d0 d0 δ1 2 |Du| dx + β(x)u2 dsx < . 2 2 2 Ω ∂Ω Thus, for any 0 < d ≤ d0 , Jd,R (u) < 0 = Jd,R (0). Now, set d∗ := sup{d > 0 : ∃ u ∈ H 1 (Ω) such that Jd (u) < Jd (0)}.

EXISTENCE AND LIMITING PROFILES

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The above arguments show that d∗ > 0 and for any d ∈ (0, d∗ ) we obtain a nontrivial global minimizer solution of (29). Then, for every d ∈ (0, d∗ ) the existence of a stable global minimizer follows from some standard arguments as in Appendix B. To prove that (29) has no nontrivial solution for d > 0 large, we suppose, for contradiction, that there exists a sequence {dn }∞ n=1 with dn → ∞ as n → ∞ such that (29) has a solution 0 < un < 1 for each d = dn . Since un satisfies  1  ∆un + dn g(x)f (un ) = 0 0 < un < 1  ∂un ∂ν + β(x)un = 0

in Ω, in Ω, on ∂Ω,

we may assume that un → u in C 2 (Ω), passing to a subsequence if necessary, where u satisfies  ∆u = 0 in Ω, ∂u + β(x)u = 0 on ∂Ω. ∂ν Therefore, u ≡ 0. Setting u ˜n = un /kun k, we have 

∆˜ un + d1n g(x)f (un )/kun k = 0 ∂u ˜n un = 0 ∂ν + β(x)˜

in Ω, on ∂Ω,

where k · k denotes the L∞ -norm. Since f is Lipschitz continuous, k d1n g(x)f (un )k/ kun k is uniformly bounded, and so is k˜ un kC 2,α (Ω) . Hence u ˜n → u˜ in C 2 (Ω) by passing to a subsequence if necessary, where u ˜ satisfies  ∆˜ u=0 in Ω, ∂u ˜ + β(x)˜ u = 0 on ∂Ω. ∂ν Thus u ˜ ≡ 0, a contradiction, since k˜ uk = limn→∞ k˜ un k = 1. The solution ud guaranteed by Theorem 4.1 has the following limiting behavior as d → 0. Theorem 4.2. Assume that (30) holds. Then as d approaches zero, ud → 0 uniformly on any compact subset of Ω− \(∂Ω− ∩ Ω) (if Ω− 6= ∅), and ud → 1 uniformly on any compact subset of Ω+ . Proof. We shall show that under the assumption (30) and Dirichlet or Robin boundary conditions, the conclusions of Lemmas 3.1, 3.2, and Part (i) of Lemma 3.3 still hold, from which Theorem 4.2 follows. Recall in the proof of Lemma 3.1, we define Ω+,h = {x ∈ Ω+ : g(x) ≤ h},

h > 0.

Under the assumption (5) there, we conclude that there exists h0 > 0 such that 0 < |Ω+,h0 | ≤ ε. Now under the assumption (30), it may also happen that there exists h1 > 0 such that |Ω+,h1 | > ε, whereas |Ω+,h | = 0 for all 0 < h < h1 . In this case, we have Z g(x) dx ≥ εh1 . {x∈Ω+ :1−udn ≥ε}

636

KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

Therefore, Jdn (udn )

≥ − Z

Z

g(x)F (1) dx +

Ω+

{x∈Ω+ :1−udn ≥ε}

≥ −

Z

g(x)[F (1) − F (1 − ε)] dx

g(x)F (1) dx + δ˜2 ,

Ω+

where δ˜2 := [F (1) − F (1 − ε)]εh1 . Then the rest of the proof is exactly the same as that of Lemma 3.1. Next, the conclusions of Lemma 3.2 follow from that of Lemma 3.1 as before. Finally, under Dirichlet or Robin boundary conditions we have, as in ∂vd |{yN =0} ≤ 0, which still guarantees that v˜d is a weak the proof of Lemma 3.3 (i), ∂y N lower solution of L in B2ρ , and therefore, Part (i) of Lemma 3.3 still holds. When N = 1, Proposition 3.4 and Theorem 3.7 work well for problems (28) and (29), since they have nothing to do with the boundary of Ω. Under the same assumption as in Proposition 3.5, with Dirichlet boundary conditions we have limd→0 ud (c) = 0 trivially, and with Robin boundary conditions we have the following property by slightly modifying the proof of Proposition 3.5. Proposition 4.3. Assume that (18) holds. Then under Robin boundary conditions, for c ∈ ∂Ω0 ∩ {a, b}, we have the following conclusions: (i) limd→0 ud (c) = 0 if c ∈ Ω− , and limd→0 ud (c) = 1 if c ∈ Ω+ . (ii) limd→0 ud (c) = 0 if p ∈ Ω− , and limd→0 ud (c) = 1/[1 + β(c)|c − p|] if p ∈ Ω+ , when c and p are the two endpoints of a connected component of Ω0 . Acknowledgments. We wish to thank Professor Yuan Lou at the Ohio State University for suggesting this problem and many helpful comments. Appendix A. In this Appendix, we prove the existence and uniqueness of solutions of the following problem:  ′′ in (−∞, ∞),  Φ + g(y)f (Φ) = 0 0≤Φ≤1 in (−∞, ∞), (31)  Φ(−∞) = 0, Φ(∞) = 1, where f is Lipschitz continuous on [0, 1] and satisfies (4), g is continuous in R and assumed to satisfy: (g1) g is nondecreasing in R and strictly increasing in an open interval I ⊂ R. (g2) There exists a number N > 0 such that g(N ) > 0 and g(−N ) < 0. Problems of type (31) have been studied by Fife and Peletier [2]. Although their results do not quite cover (19) where  γ1 y k1 in [0, ∞), g(y) = −γ2 |y|k1 in (−∞, 0), for some γ1 , γ2 , k1 > 0 (as they require that g be bounded, see the assumption A1 in [2]), their methods of proof do work well here. For the sake of completeness, we include the proof here. We will follow the arguments in [2] closely. Theorem A.1. Problem (31) has a unique solution Φ. Moreover, 0 < Φ(y) < 1 and Φ′ (y) > 0 for all y ∈ R.

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Proof. We first establish the existence of a solution of (31) via lower- and uppersolution method. A lower solution can be constructed as follows. Let v(y) be defined by the relation Z v(y) Z 1 1 y−N = {2 g(N )f (r)dr}− 2 ds, y ≥ N. (32) 0

s

By virtue of f > 0 in (0, 1) and g(N ) > 0, we see that the function Z h Z 1 1 H(h) := {2 g(N )f (r)dr}− 2 ds 0

s

is strictly increasing in h ∈ [0, 1). Since f is Lipschitz continuous and f (1) = 0, there exists L > 0 such that 0 ≤ f (r) ≤ L(1 − r) for all r ∈ [0, 1]. Hence Z 1 Z 1 2 g(N )f (r) dr ≤ 2Lg(N ) (1 − r) dr = Lg(N )(1 − s)2 . s

s

Therefore,

h

Z

0

{2

Z

1

1

g(N )f (r)dr}− 2 ds

s 1

Z

h

≥

(Lg(N ))− 2

=

(Lg(N ))− 2 [− ln(1 − h)] → ∞ as h → 1− .

0

1

(1 − s)−1 ds

This shows that v(y) is well-defined for all y ≥ N and satisfies that v(N ) = 0, 0 < v(y) < 1 for all y > N , and v(∞) = 1. Since H(h) is twice differentiable and H ′ (h) > 0 for all 0 ≤ h < 1, we see that v(y) is twice differentiable. Then it is straightforward to verify that v(y) is a solution of the problem  ′′ v (y) + g(N )f (v(y)) = 0 y > N, v(N ) = 0, v(∞) = 1. Next, define

Φ(y) = We observe that for y > N ,



0 for y ≤ N , v(y) for y > N .

Φ′′ + g(y)f (Φ) = v ′′ + g(y)f (v) ≥ v ′′ + g(N )f (v) = 0,

and for y < N , Φ(y) ≡ 0 is an exact solution. Then it is easy to see that Φ(y) is a weak lower solution of (31), since near y = N , Φ is the maximum of two lower solutions. A weak upper solution can be constructed similarly:  w(y) for y < −N , Φ(y) = 1 for y ≥ −N , where w is the solution of the problem  ′′ w (y) + g(−N )f (w(y)) = 0 w(−N ) = 1, w(−∞) = 0,

y < −N,

given by the relation

−y − N =

Z

1

w(y)

{2

Z

0

s

1

[−g(−N )f (r)]dr}− 2 ds, y ≤ −N.

Since Φ < Φ, there exists a solution Φ between them [13]. Moreover, it is easy to see that Φ(−∞) = 0 and Φ(∞) = 1. This proves that problem (31) has a solution.

638

KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

Next, we prove that Φ′ (y) > 0 for any y ∈ R and any solution Φ of (31). First, we have 0 < Φ(y) < 1 for any y ∈ R by the maximum principle. Then, by (g1) and (g2) there exists y0 ∈ R such that  ≤ 0 in (−∞, y0 ), g(y) ≥ 0 in (y0 , ∞).

Now suppose that there exists y1 ∈ R such that Φ′ (y1 ) = 0. If y1 ≥ y0 , then Φ′′ (y) ≤ 0 for y ≥ y1 . Hence, Φ′ (y) ≤ 0 for y ≥ y1 and Φ(∞) ≤ Φ(y1 ) < 1. This contradicts Φ(∞) = 1. If y1 ≤ y0 , then Φ′′ (y) ≥ 0 for y ≤ y1 . Hence, Φ′ (y) ≤ 0 for y ≤ y1 and Φ(−∞) ≥ Φ(y1 ) > 0, a contradiction, since Φ(−∞) = 0. Therefore, Φ′ (y) > 0 for any y ∈ R. Finally, we show that (31) has at most one solution. Suppose that Φ1 and Φ2 are two solutions of (31) and Φ1 < Φ2 on some interval (a, b). Assume that the interval is maximal, i.e., a and b are allowed to take the values −∞ and ∞, respectively. Therefore, Φ′1 (a) ≤ Φ′2 (a),

Φ1 (a) = Φ2 (a) = α,

Φ′1 (b)

Φ1 (b) = Φ2 (b) = β,

≥

(33)

Φ′2 (b).

Since z = Φi (y) (i = 1, 2) is strictly increasing, we have the inverse function y = ′ Φ−1 i (z), z ∈ (0, 1). Multiplying the equation of Φi by 2Φi and integrating over (a, b), we have Z b [Φ′i (b)]2 − [Φ′i (a)]2 = −2 g(y)f (Φi (y))Φ′i (y) dy a

=

−2

Z

β

α

g(Φ−1 i (z))f (z) dz.

Subtracting the above equation for i = 1 from the one for i = 2 gives

=

{[Φ′2 (b)]2 − [Φ′1 (b)]2 } − {[Φ′2 (a)]2 − [Φ′1 (a)]2 } Z β −2 [g(Φ2−1 (z)) − g(Φ−1 1 (z))]f (z) dz.

(34)

α

In view of (33), the left hand side of (34) is not greater than zero, whereas the −1 right hand side of (34) is nonnegative because of Φ−1 2 (z) < Φ1 (z) for z ∈ (α, β) ′ and (g1). Hence both sides equal zero. In particular, Φ2 (a) = Φ′1 (a). But this can happen only when a = −∞ by the uniqueness property of initial value problems. Similarly, we have b = ∞. Therefore, (α, β) = (0, 1). That the right hand side of −1 (34) is zero indicates that g(Φ−1 2 (z)) = g(Φ1 (z)) for all z ∈ (0, 1). However, (g1) −1 −1 implies that g(Φ2 (z)) < g(Φ1 (z)) when Φ−1 2 (z) ∈ I. This gives a contradiction and completes the proof. We conclude this Appendix by the observation that the unique solution Φ of problem (31) has the following monotone property with respect to the function g. Let Φi (i = 1, 2) be the solution of (31) with g = gi , respectively, where g1 ≥ g2 , g1 6≡ g2 . We claim that Φ1 > Φ2 . To verify this we see that Φ′′1 + g2 (y)f (Φ1 ) ≤ Φ′′1 + g1 (y)f (Φ1 ) = 0 and the inequality is strict for some y, which shows that Φ1 is a strict upper solution of (31) with g = g2 . On the other hand, we take some y1 such that g2 (y1 ) > 0 and let v be the solution of  ′′ v (y) + g2 (y1 )f (v(y)) = 0 y > y1 , v(y1 ) = 0, v(∞) = 1.

EXISTENCE AND LIMITING PROFILES

Then Φ(y) :=



0 v(y)

639

for y ≤ y1 , for y > y1 ,

is a lower solution of (31) with g = g2 as in the proof of Theorem A.1. Moreover, for y > y1 , Φ′′ + g1 (y)f (Φ) ≥ Φ′′ + g2 (y)f (Φ) ≥ v ′′ + g2 (y1 )f (v) = 0,

and for y < y1 , Φ′′ + g1 (y)f (Φ) = 0. This implies that Φ is a lower solution of (31) with g = g1 as well. By the uniqueness property, we see that Φ ≤ Φ1 , and hence Φ ≤ Φ2 ≤ Φ1 . Moreover, the inequalities are strict by the maximum principle. Appendix B. For the sake of completeness, we include the following standard arguments for the existence of a stable global minimizer. More precisely, we want to show that if min{Jd (u) : u ∈ H 1 (Ω)} < min{Jd (0), Jd (1)}, then there exists at least one stable global minimizer solution, where Jd (·) is given by (6). Suppose, for a contradiction, that all the global minimizers are unstable. Let v0 > 0 be a global minimizer. Without loss of generality, we assume that v0 is unstable from above, then there exists a minimal solution above v0 , denoted by v1 [8, Theorem 3.1]. Clearly, v0 < v1 < 1 is a global minimizer which is stable from below. Set S = {v0 < u < 1 : u is a global minimizer which is stable from below}.

S 6= ∅ because of v1 ∈ S. The usual partial order “ ≤ ” in S is defined as u, v ∈ S, u ≤ v if u(x) ≤ v(x) ∀ x ∈ Ω.

By Zorn’s Lemma, there exists a maximal well-ordered subset of S denoted by W . We have the following two situations. (a) W has a greatest element w. Since w ∈ S, w must be unstable from above. Hence, there exists a minimal solution above w, denoted by w. ˜ Then w ˜ ∈ S and {w} ˜ ∪ W is a well-ordered subset of S which is greater than W . This contradicts the maximum of W . (b) W has no greatest element. We define w ¯ := sup u(x). u∈W

Then w ¯ 6∈ W . We observe that w ¯ ∈ S. For this, it is enough to find a sequence u1 ≤ u2 ≤ · · · ≤ uk ≤ · · · in S such that uk → w ¯ in C 2 (Ω). Let {xl : l = 1, 2, ...} (l) be a dense subset of Ω. For every l, there exists a sequence {uk }∞ k=1 in W such (l) that uk (xl ) increases to w(x ¯ l ) as k → ∞ by the definition of w. ¯ Noting that W is well-ordered, we set (l) uk := max{uk : 1 ≤ l ≤ k}. (l)

Then uk ∈ W and u1 ≤ u2 ≤ · · · ≤ uk ≤ · · · . Since uk (xl ) ≥ uk (xl ) when k ≥ l, we have (l) lim uk (xl ) ≥ lim uk (xl ) = w(x ¯ l ). k→∞

k→∞

Also, lim uk (xl ) ≤ w(x ¯ l)

k→∞

640

KIMIE NAKASHIMA, WEI-MING NI AND LINLIN SU

by the definition of w. ¯ This implies that lim uk (xl ) = w(x ¯ l ).

k→∞

Next, we will show that uk → w ¯ in C 2 (Ω) by passing to a subsequence if necessary. Since {uk } ⊂ S, {uk } is bounded in C 2,α (Ω) (for some 0 < α < 1). Therefore, uk → ϕ in C 2 (Ω) for some ϕ ∈ C 2 (Ω) by passing to a subsequence if necessary. We claim that ϕ = w. ¯ If not, then there exists x0 ∈ Ω such that ϕ(x0 ) < w(x ¯ 0 ) − ε for some ε > 0. Therefore, there exists a neighborhood O1 of x0 such that ϕ(x) < w(x ¯ 0 ) − ε for all x in O1 .

On the other hand, there exists u0 ∈ W such that

u0 (x0 ) > w(x ¯ 0 ) − ε,

which implies the existence of a neighborhood O2 of x0 such that u0 (x) > w(x ¯ 0 ) − ε for all x in O2 .

For any xl ∈ O1 ∩ O2 , we have ϕ(xl ) = limk→∞ uk (xl ) = w(x ¯ l ), but the above inequalities show that w(x ¯ l ) ≥ u0 (xl ) > w(x ¯ 0 ) − ε > ϕ(xl ),

which gives a contradiction. Therefore, ϕ = w. ¯ This completes the proof of w ¯ ∈ S. Hence, {w} ¯ ∪ W is a well-ordered subset of S which is greater than W . Again, this contradicts the maximum of W . Thus, at least one global minimizer must be stable. REFERENCES [1] K. J. Brown and P. Hess, Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Differential and Integral Equations, 3 (1990), 201–207. [2] P. C. Fife and L. A. Peletier, Nonlinear diffusion in polulation genetics, Arch. Rational Mech. Anal., 64 (1977), 93–109. [3] W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219–233. [4] D. Gilbarg and N. Trudinger, “Elliptic Partial Differential Equations of Second Order,” reprint of the 1998 edition, Springer-Verlag, Berlin, 2001. [5] D. Henry, “Geometric Theory of Semilinear Parabolic Equations,” Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin-New York, 1981. [6] Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population gentics, J. Differential Equations, 181 (2002), 388–418. [7] Y. Lou, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity, preprint. [8] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401–454. [9] T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595–615. [10] T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance, Theoret. Population Biol., preprint (2009). [11] T. Nagylaki and Y. Lou, The dynamics of migration-selection models, in “Tutorials in Mathematical Biosciences” (ed. A. Friedman), Lecture Notes in Math., 1922, Springer, Berlin, (2008), 117–170. [12] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819–851. [13] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979–1000.

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[14] S. Senn, On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics, Comm. Partial Differential Equations, 8 (1983), 1199–1228. [15] M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733–756.

Received October 2009; revised February 2010. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]