EXISTENCE OF COMPACT SUPPORT SOLUTIONS ... - Project Euclid

Differential and Integral Equations

Volume 25, Numbers 7-8 (2012) , 629–656

EXISTENCE OF COMPACT SUPPORT SOLUTIONS FOR A QUASILINEAR AND SINGULAR PROBLEM Habib Maagli Departement de Mathématiques, Faculté des Sciences de Tunis Campus Universitaire, 2092 Tunis, Tunisia Jacques Giacomoni and Paul Sauvy LMAP (UMR 5142) Bat. Ipra, Université de Pau et des Pays de l’Adour Avenue de l’Université, 64013 Cedex Pau, France (Submitted by: Jesus Ildefonso Diaz) Abstract. Let Ω be a C 2 bounded domain of RN , N ≥ 2. We consider the following quasilinear elliptic problem:  −∆p u = K(x)(λuq − ur ), in Ω, (Pλ ) u = 0 on ∂Ω, u ≥ 0 in Ω, ` ´ def where p > 1 and ∆p u = div |∇u|p−2 ∇u denotes the p-Laplacian operator. In this paper, λ > 0 is a real parameter, the exponents q and r satisfy −1 < r < q < p − 1, and K : Ω −→ R is a positive function having a singular behaviour near the boundary ∂Ω. Precisely, K(x) = d(x)−k L(d(x)) in Ω, with 0 < k < p, L a positive perturbation function, and d(x) the distance of x ∈ Ω to ∂Ω. By using a sub- and supersolution technique, we discuss the existence of positive solutions or compact support solutions of (Pλ ) in respect to the blow-up rate k. Precisely, we prove that if k < 1 + r, (Pλ ) has at least one positive solution for λ > 0 large enough, whereas it has only compact support solutions if k ≥ 1 + r.

1. Introduction Let Ω be a C 2 bounded domain of RN , N ≥ 2. We discuss the existence of weak solutions in W01,p (Ω) ∩ L∞ (Ω) to −∆p u = K(x)(λuq − ur ) in Ω, (Pλ ) u = 0 on ∂Ω, u ≥ 0 in Ω. Accepted for publication: December 2011. AMS Subject Classifications: 35J20, 35J25, 35J70. 629

630

ˆ agli, and Paul Sauvy Jacques Giacomoni, Habib Ma

u ∈ W01,p (Ω) ∩ L∞ (Ω) is a weak solution to (Pλ ) if for all test functions ϕ ∈ D(Ω), Z Z p−2 K(x) (λuq − ur ) ϕ dx. (1.1) |∇u| ∇u · ∇ϕ dx = Ω

Ω

In the equation in (Pλ ), λ > 0 is a positive parameter, −1 < r < q < p − 1, and K ∈ C(Ω) is a positive function having a singular behaviour near the boundary ∂Ω. Precisely, K(x) = d(x)−k L(d(x)) in Ω, with 0 < k < p and def

L ∈ C 2 ((0, 2D]) a positive function, with D = diam(Ω), defined as follows: Z 2D z(s) L(t) = exp ds , (1.2) s t with z ∈ C([0, 2D)) ∩ C 1 ((0, 2D]) and z(0) = 0. Let us note that (1.2) implies that tL0 (t) =0 (1.3) lim t→0+ L(t) and for all ε > 0, lim tε L(t) = 0 (1.4) t→0+

and lim t−ε L(t) = +∞.

t→0+

(1.5)

The above asymptotics of L force ∀ξ > 0 ,

lim

t→0+

L(ξt) = 1. L(t)

Then L belongs to the Karamata class [9]. Setting K the class of functions satisfying (1.2), we get the following properties: if L1 , L2 ∈ K and if p ∈ R, then L1 , L2 ∈ K and L1 p ∈ K. Example 1.1. Let m ∈ N∗ and A > 0 large enough. Let us define m Y µn , t ∈ (0, 2D] L(t) = logn At n=1 def

with logn = log ◦ · · · ◦ log (n times) and µn > 0. Then L ∈ K. In the present paper, we investigate first the following issues for the problem (Pλ ): existence of non-trivial weak solutions according to λ > 0 and H¨ older regularity of weak solutions. Next, we study further the properties of nontrivial solutions. Since the non-linearity in the right-hand side is a singular absorption term near the boundary, a nontrivial weak solution may

Quasilinear and Singular Equations

631

not be positive everywhere in Ω and compact support (nontrivial) weak solutions or compactons (solutions with zero normal derivative at the boundary) may exist for stronger singularities, that is for large k > 0, whereas for small k > 0 any nontrivial weak solution is positive. Then, the natural question is to determine the borderline condition on the parameter k, which gives the strength of the singular potential K, between existence of positive weak solutions and existence of free-boundary weak solutions. The existence of compact support solutions is important in the applications, in particular in biology models (population dynamics and epidemiology models for instance) and was investigated quite intensely for nonlinear reaction diffusion equations with absorption in the last decades. In particular, concerning the case where the equation involves a quasilinear and degenerate operator, we can refer to the result in V´ azquez [13] where under a suitable condition about the behaviour of the non-linearity near the origin, a strong maximum principle is proved and thereby the positivity of solutions. The given condition is sharp in the sense that for different situations where this condition is not satisfied, the existence of free-boundary solutions is shown. In Alvarez-D´ıaz [2] (see also D´ıaz [4] for relted results on the subject), the authors consider a class of nonhomogeneous reaction-diffusion equations with strong absorption and study the behaviour of the solution near the free boundary. In particular, a nondegeneracy property (the solution grows faster than some function of the distance to the free boundary) is obtained when the growth of the reaction term near the boundary satisfies some estimate below. In Il’yasov-Egorov [8], the authors consider a semi-linear equation with a similar (and nonsingular) conflicting nonlinearity as in the equation in (Pλ ), and the existence of compactons is proved using the fibering method. An interesting feature of this result is that the Hopf lemma is violated for such equations. In the present work, we consider further the case where the equation involves a p-Laplace operator and a singular potential in the right-hand side and show that a more complex situation occurs in respect to the nonsingular case. In the next section, we give the main results proved in this paper. These results extend a previous work due to Haitao [7] in the semi-linear case (p = 2) and which involves a smaller class of nonlinearities.

2. Main results The main results of our paper concerning the problem (Pλ ) are stated below:

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Theorem 2.1. When k < 1 + r, there exists a constant Λ1 > 0 such that the following hold: (1) For λ > Λ1 , (Pλ ) admits a positive weak solution. 1,β (2) Any weak solution of (Pλ ) is C Ω for some β ∈ (0, 1). (3) For λ < Λ1 , (Pλ ) has no positive solution. Theorem 2.2. Let r > 0 and one of the two following conditions be satisfied: h (p − 1)(r + 1) 1 + r > q and k ∈ 1 + r, 1 + , (2.1) p−q+r 1+r ≥q

and

k ∈ [1 + r, 2 + r).

(2.2)

Then, there exists Λ2 > 0 such that the following hold: (1) For λ > Λ2 , (Pλ ) has a compact support weak solution uλ . (2) Any weak solution of (Pλ ) is C 1,β Ω for some β ∈ (0, 1). (3) For λ < Λ2 , (Pλ ) has no nontrivial solution. The outline of the paper is as follows. Before giving the proofs of those theorems, we establish some useful preliminary results in the next section. The proof of Theorem 2.1 is given in Section 4, and the proof of Theorem 2.2 is given in Section 5. The technical results stated in Section 3 are finally proved in Appendices A and B. The related regularity results are a consequence of the general regularity results stated in Appendix C. Throughout this paper, we will use the following notation: def

(1) To p ∈ (1, +∞) we associate p0 =

p p−1 .

def

(2) For x ∈ Ω, d(x) = dist(x, Ω) = inf d(x, y). y∈Ω

def

(3) D = diam(Ω) = sup d(x, y). x,y∈Ω

(4) Let ω be a nonempty set of Ω and f, g : ω −→ [0, +∞]. We write f (x) ∼ g(x) in ω if there exist two positive constants C1 and C2 such that ∀x ∈ ω,

C1 f (x) ≤ g(x) ≤ C2 f (x).

(5) Let ω ⊂ RN ; LN (ω) denotes the N -dimensional Lebegue measure of ω. def (6) Let ε > 0; we define Ωε = {x ∈ Ω : d(x) < ε}. (7) ν : ∂Ω −→ RN denotes the outward normal associated to Ω. 1 R def (8) For v ∈ W01,p (Ω), we write kvk = k∇vkLp (Ω) = Ω |∇v|p dx p .


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(9) The function ϕ1 ∈ W01,p (Ω) denotes the positive and renormalized (i.e., kϕ1 kLp (Ω) = 1) eigenfunction corresponding to the first eigenvalue of −∆p , Z nZ o def 1,p p |v|p dx = 1 . |∇v| dx, v ∈ W0 (Ω), λ1 = inf Ω

Ω

It is a weak solution of the following eigenvalue problem: −∆p u = λ1 up−1 in Ω, u = 0 on ∂Ω, u ≥ 0 in Ω. Using Moser iterations and the regularity result in Lieberman [10], ϕ1 ∈ C 1,α Ω for some α ∈ (0, 1). Moreover, the strong maximum and boundary principles from Vázquez [13] guarantee that ϕ1 satisfies these two properties: (a) There exist two positive constants K1 and K2 depending only on p and Ω such that ∀x ∈ Ω,

K1 d(x) ≤ ϕ1 (x) ≤ K2 d(x).

(2.3)

(b) There exist ε∗ > 0 and δ ∗ > 0 depending only on p and Ω such that ∀x ∈ Ωδ∗ ,

|∇ϕ1 (x)| > ε∗ .

(2.4)

3. Preliminary results 3.1. A nonexistence lemma. Lemma 3.1. When k < 1 + r, there exists λ∗ > 0 such that (Pλ ) has no nontrivial solution for λ ≤ λ∗ . Proof. Let us define def

λ1,K =

p Ω |∇v| dx . K(x)|v|p dx

R inf

v∈W01,p (Ω) v6=0

R Ω

From Hardy’s inequality, there exists a constant C > 0 depending only on Ω and p such that for all v ∈ W01,p (Ω), Z Z |v|p dx ≤ C |∇v|p dx. p Ω Ω d(x)

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Since k < p, λ1,K > 0. Let u ∈ W01,p (Ω) ∩ L∞ (Ω) be a nontrivial solution of (Pλ ); then from (1.1) taking u ∈ W01,p (Ω) as a test function we get Z Z Z K(x) λuq+1 − ur+1 dx. (3.1) |∇u|p dx = K(x)up dx ≤ 0 < λ1,K Ω

Ω

Ω

def

This inequality cannot hold for λ ≤ λ∗ = min{1, λ1,K }. Indeed, - if u(x) ≤ 1, λuq+1 − ur+1 ≤ 0 whenever λ ≤ 1, - if u(x) > 1, K(x) λuq+1 − ur+1 < λ1,K K(x)uq+1 whenever λ ≤ λ1,K . Then, either LN ({x ∈ Ω : u(x) > 1}) = 0 and we get Z K(x)up dx ≤ 0, 0 < λ1,K Ω

or

Z λ1,K

1{u>1} K(x)up dx ≤ λ1,K

Z

1{u>1} K(x)uq+1 dx,

Ω

Ω

which contradicts q < p − 1.

3.2. Construction of a subsolution for (Pλ ). Lemma 3.2. When k < 1 + r, there exist M > 0, λ∗ > 0, and τ > 1 such def that uλ = M ϕ1 τ is a subsolution of (Pλ ) in Ω, provided that λ ≥ λ∗ . Proof. Let M > 0 and τ > 1; then we define uλ = M ϕ1 τ in Ω. A straightforward computation yields h i −∆p uλ = − (M τ )p−1 (p − 1)(τ − 1)|∇ϕ1 |p ϕ1 (τ −1)(p−1)−1 − λ1 ϕ1 τ (p−1) and K(x) (λuλ q − uλ r ) = −K(x) (M r ϕ1 τ r − λM q ϕ1 τ q ) . By properties (2.3) and (2.4) of the function ϕ1 , if we let n 1 o 1 ε∗ (τ − 1)(p − 1) p1 1 τ (q−r) def δ0 = min δ ∗ , , , K2 2λ1 K2 2λM q−r both of the above expressions are negative on Ωδ0 . Moreover, ∆p uλ (x) ∼ (M τ )p−1 (τ − 1)d(x)(τ −1)(p−1)−1 in Ωδ0 and K(x) (uλ r − λuλ q ) ∼ M r L(d(x))d(x)τ r−k in Ωδ0 . Since k < 1 + r, we can choose a constant τ > 1 satisfying (τ − 1)(p − 1) − 1 < τ r − k.


635

Hence, for M > 0 large enough we get −∆p uλ ≤ K(x) λuqλ − urλ in Ωδ0 . In Ω \ Ωδ0 , K, ϕ1 , and |∇ϕ1 | are bounded; therefore there exists λ∗ > 0 q ∗ r such that for λ ≥ λ , −∆p uλ ≤ K(x) λuλ − uλ in Ω \ Ωδ0 . Thus, uλ is a subsolution of (Pλ ) in Ω for M large enough and λ ≥ λ∗ . 3.3. Construction of a supersolution for (Pλ ). We consider the following problem: −∆p v = K(x)v q in Ω, (Q) v = 0 on ∂Ω, v > 0 in Ω, with q, p, and K satisfying the above assumptions. Lemma 3.3. (1) If k ∈ (0, 1 + q), (Q) has a unique solution v ∈ W01,p (Ω) ∩ C Ω satisfying v(x) ∼ d(x) in Ω. (2) If k = 1+q, (Q) has a unique solution v ∈ W01,p (Ω)∩C Ω satisfying Z 2d L(t) 1 p−k v(x) ∼ d(x) dt in Ω. t d(x) ), (Q) has a unique solution v ∈ W01,p (Ω)∩ (3) If k ∈ (1+q, 1+q+ p−(1+q) p C Ω satisfying v(x) ∼ d(x)

p−k p−(1+q)

L (d(x))

1 p−(1+q)

in Ω.

h 1,p (4) If k ∈ 1 + q + p−(1+q) , p , (Q) has a unique solution v ∈ Wloc (Ω) ∩ p C0 Ω satisfying 1 p−k p−(1+q) v(x) ∼ d(x) p−(1+q) L (d(x)) in Ω. (5) If k = p and if L satisfies the following condition, Z 2D 1 t−1 L(t) p−1 dt < +∞, 0

1,p (Q) has a unique solution v ∈ Wloc (Ω) ∩ C0 Ω satisfying Z d(x) p−1 1 p−(1+q) t−1 L(t) p−1 dt v(x) ∼ in Ω. 0

(3.2)

636


Proof. See Appendix A. From a solution of (Q), we can easily construct a supersolution of (Pλ ). 1,p Indeed, let us consider v ∈ Wloc (Ω) ∩ C0 Ω the solution of (Q) given by def

Lemma 3.3. Then, uλ = M v is a supersolution of (Pλ ) in Ω whenever 1 p−(1+q)

M ≥ λ . In particular, when k < 1 + r and λ ≥ λ∗ , choosing M sufficiently large uλ ∈ W01,p (Ω) ∩ C Ω and is a supersolution of (Pλ ) in Ω satisfying uλ ≤ uλ and uλ (x) ∼ d(x) in Ω. Now let us state a nonexistence result for the problem (Q). Proposition 3.1. Let v ∈ W01,p (Ω) ∩ C Ω be a positive subsolution of (Q) in Ω and assume that there exists ε > 0 such that Z K(x)ϕ1 p−1+ε dx = +∞. (3.3) Ω

1,p (Ω) ∩ C0 Ω such Then, for any η > 0, (Q) has no weak solution v ∈ Wloc that v ≥ ηv in Ω. Proof. See Appendix B.

Corollary 3.1. If k > p, there is no nontrivial weak solution of (Q). 4. Proof of Theorem 2.1 4.1. Existence of a C 1,β positive solution when λ ≥ λ∗ . Proposition 4.1. When k < 1 + r, provided λ ≥ λ∗ , (Pλ ) has a weak solution uλ ∈ W01,p (Ω) ∩ L∞ (Ω). Furthermore, there exists β ∈ (0, 1) such that uλ ∈ C 1,β Ω . def

Proof. In the equation of (Pλ ), the expression hλ (x, v) = K(x)(λv q − v r ) involves a singular term K(x) that blows up as d(x) → 0, which prevents the direct application of the sub- and supersolution method. To overcome this difficulty, we apply this method in a sequence of subdomains of Ω. Precisely, let us introduce (Ωk )k∈N∗ ⊂ Ω an increasing sequence of smooth subdomains of Ω such that Ωk −→ Ω in the Hausdorff topology with k→∞

∀k ∈ N∗ ,

1 k+1

< dist(∂Ω, ∂Ωk ) < k1 .

Then, for all k ∈ N∗ we consider the following problem: −∆p uk = K(x)(λuk q − uk r ) in Ωk , (Pk ) uk = uλ on ∂Ωk , uk ≥ 0 in Ωk .


637

By definition of Ωk , there exists Ck > 0 such that ∂h h i def λ ∀v ∈ Ik = min uλ , max uλ , sup (x, v) ≤ Ck . Ωk Ωk x∈Ωk ∂v As a consequence, there exists µk > 0 such that for all x ∈ Ωk , the function v 7−→ hλ (x, v) + µk v p−1 is increasing on Ik . Therefore by the sub- and supersolution method, (Pk ) has a solution uk ∈ W1,p (Ωk ). Indeed, we can construct the following iterative monotone scheme: for all n ∈ N∗ , let uk,n ∈ W1,p (Ωk ) be the weak solution of −∆p uk,n + µk (uk,n )p−1 = hλ (x, uk,n−1 ) + µk (uk,n−1 )p−1 in Ωk , (Pk,n ) uk,n = uλ on ∂Ωk , uk,n ≥ 0 in Ωk with uk,0 = uλ in Ωk . By induction on n ∈ N, (Pk,n ) has a unique solution uk,n ∈ W1,p (Ωk ). From the weak comparison principle, (uk,n )n∈N satisfies uλ ≤ uk,n ≤ uk,n+1 ≤ uλ in Ωk . Consequently, for all n ∈ N∗ , p−1 − (uk,n )p−1 ∈ L∞ (Ωk ) hλ (x, uk,n−1 ) + µk (uk,n−1 ) and since uλ is smooth in Ω, we can state by a regularity result due to Lieberman [10] (see Theorem 1) that (uk,n )n∈N is bounded in C 1,γ Ωk for some γ ∈ (0, 1). Precisely, there exists a constant C > 0 depending only on γ, Ωk , kuλ kL∞ (Ωk ) , and kuλ kL∞ (Ωk ) such that kuk,n kC 1,γ (Ωk ) ≤ C. From the Ascoli-Arzel` a theorem, there exist uk ∈ C 1 (Ωk ) and a subsequence (uk,m )m∈N such that uk,m −→ uk in C 1 Ωk . Passing to the limit when n → +∞ in m→∞

(Pk,n ), uk is a weak solution of (Pk ). def

For all k ∈ N, we define u ˜k = 1Ωk · uk in order to extend uk on Ω by zero. We prove that (˜ uk )k∈N is an increasing sequence in Ω. Indeed, since Ωk ⊂ Ωk+1 , if we compare uk+1 with every term of (uk,n )n∈N in Ωk , using the weak comparison principle we get ∀n ∈ N,

uk,n ≤ u ˜k+1 in Ωk . Passing to the limit in the above inequality, u ˜k (x) k∈N is nondecreasing for all x ∈ Ω. Therefore there exists uλ ∈ L∞ (Ω) such that u ˜k −→ uλ almost k→∞

everywhere in Ω and uλ ≤ uλ ≤ uλ in Ω.

(4.1)

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It follows that u ˜k −→ uλ in D0 (Ω) and uλ satisfies (1.1). Using inequalk→∞

0

ity (4.1) and Hardy’s inequality, K(x) [λ(uλ )q − (uλ )r ] ∈ W−1,p (Ω), from which it follows that uλ ∈ W01,p (Ω). Finally, applying Proposition C.1 of Appendix C, we get the C 1,β Ω regularity of uλ . 4.2. Existence of Λ1 . Let us define def

Λ1 = inf {λ > 0 : (Pλ ) has a positive solution} . By Lemma 3.1 and Proposition 4.1, λ∗ ≤ Λ1 ≤ λ∗ < +∞. By definition of Λ1 , for any λ > Λ1 there exists µ ∈ (Λ1 , λ) such that (Pµ ) has a positive solution uµ ∈ W01,p (Ω) ∩ L∞ (Ω). Moreover, using Proposition C.1, uµ ∈ C 1,β Ω . Since uµ is a subsolution to (Pλ ), we prove that uµ ≤ uλ in Ω. Indeed, K(x) > 0 in Ω, so there exists δ0 > 0 such that −∆p uµ ≤ 0 ≤ −∆p (C0 ϕ1 ) in Ωδ0 , with C0 > 0 large enough to satisfy uµ ≤ C0 ϕ1 on ∂Ωδ0 . By the weak comparison principle, uµ ≤ C0 ϕ1 in Ωδ0 . Moreover, uµ and ϕ1 are bounded in Ω \ Ωδ0 ; thus uµ ≤ Cϕ1 in Ω for some constant C > 0. Therefore choosing M sufficiently large in the definition of uλ , we get uµ ≤ uλ in Ω. Finally, applying again sub and supersolution technique as in Step 1, we get a solution uλ ∈ C 1,β Ω of (Pλ ). Proof of Theorem 2.1. The proof follows from Proposition 4.1 and Subsection 4.2. 5. Proof of Theorem 2.2 5.1. Existence of a solution under condition (2.1) or (2.2). Proposition 5.1. Let k ∈ [1 + r, 1 + q + p−(1+q) ). Then, under the condition p Z K(x) (uλ )r+1 dx < +∞, (5.1) Ω

λ∗∗

there exists > 0 such that the problem (Pλ ) has a nontrivial weak solution uλ ∈ W01,p (Ω) ∩ L∞ (Ω) for λ > λ∗∗ . uλ ∈ Lp (Ω). So Remark 5.1. Since uλ ∈ W01,p (Ω), by Hardy’s inequality d(x) using H¨ older’s inequality, assumption (5.1) in Theorem 5.1 is satisfied if 0

L(d(x))d(x)r+1−k ∈ Lα (Ω),


where α =

p r+1

639

> 1. And this last condition is satisfied if

p − (1 + r) . (5.2) p So (5.2) implies (5.1), but this condition is not sharp and can be weakened by using the precise behaviour of uλ given in Lemma 3.3. Indeed, (1) if k ∈ [1 + r, 1 + q), uλ (x) ∼ d(x) in Ω. Therefore condition (5.1) is satisfied if k < 2 + r. (5.3) (2) if k = 1 + q, condition (5.1) is also satisfied if k < 2 + r. p−k 1 p−(1+q) L (d(x)) p−(1+q) in , u ∼ d(x) (3) if k ∈ 1 + q, 1 + q + p−(1+q) λ p Ω. Therefore, condition (5.1) is satisfied if k q, (5.3) is always true for k ∈ [1 + r, 1 + q], and since k q, (5.5) p−q+r condition (2.1) implies (5.1). Similarly, if 1 + r ≤ q, by equivalence (5.5), (5.4) is never satisfied for k ∈ 1 + q, 1 + q + p−(1+q) and condition (2.2) p implies (5.1). We can easily check that both conditions (2.1) and (2.2) are weaker than (5.2). Moreover, if one of the following conditions holds, (p − 1)(r + 1) p − (1 + q) 1 + r > q and k ∈ 1 + ,1 + q + , p−q+r p p − (1 + q) 1 + r ≥ q and k ∈ 2 + r, 1 + q + , p then, using Lemma 3.3 again, condition (5.1) is not satisfied, which shows the “sharpness” of conditions (2.1) and (2.2). 1+q 0 such that   Cp |x − y|p if p ≥ 2, p−2 p−2 |x − y|2 h|x| x − |y| y, x − yi ≥ if 1 < p < 2.  Cp (|x| + |y|)2−p

640


Proof. See Lemma 4.2 in Lindqvist [11].

Proof of Proposition 5.1. Let us introduce the functional Z Z Z 1 1 λ p r+1 Iλ (v) = |∇v| dx + K(x)|v| dx − K(x)|v|q+1 dx, p Ω r+1 Ω q+1 Ω with v ∈ W01,p (Ω). Let ϕ0 6= 0 ∈ D(Ω) be a nonnegative function. Therefore, there exists λ∗∗ > 0 such that Iλ (ϕ0 ) < 0 for λ > λ∗∗ . Let us fix a constant M > 1 such that M uλ ≥ ϕ0 in Ω and introduce the cut-off function fλ defined in Ω × R by  if v > M uλ (x),  K(x) [λ(M uλ )q − (M uλ )r ] q − |v|r ] K(x) [λ|v| if v ∈ [0, M uλ (x)] , fλ (x, v) =  0 if v < 0. It is easy to see that the function v 7−→ fλ (x,Rv) is a Carathéodory function. v For (x, v) ∈ Ω × R, let us set Fλ (x, v) = 0 fλ (x, t) dt and consider the functional Eλ defined as follows: Z Z 1 Eλ (v) = ∀v ∈ W01,p (Ω), |∇v|p dx − Fλ (x, v(x)) dx. p Ω Ω A straightforward computation yields Z λ 1 1 |∇v|p dx − A(v, q) + A(v, r) Eλ (v) = p Ω q+1 r+1 (5.6) r q −λB(v, q) + B(v, r) − C(r) + λ C(q), r+1 q+1 with Z def 1{0≤v≤M uλ } K(x)|v|s+1 dx, A(v, s) = ZΩ def B(v, s) = 1{v≥M uλ } K(x) (M uλ )s v dx Ω

and def

Z

C(s) =

Ω

1{v≥M uλ } K(x)(M uλ )s+1 dx.

Let ε > 0 and v ∈ W01,p (Ω); then we split the integral A(v, q) in Ω \ Ωε and Ωε : Z Z q+1 A(v, q) = 1{0≤v≤M uλ } K(x)|v| dx + 1{0≤v≤M uλ } K(x)|v|q+1 dx Ω\Ωε

def

= AΩ\Ωε (v, q) + AΩε (v, q).

Ωε


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Since in Ω \ Ωε , K is bounded, from the embedding W01,p (Ω) ,→ Lq+1 (Ω), there exists a constant C1 such that AΩ\Ωε (v, q) ≤ C1 kvkq+1 .

(5.7)

Furthermore, by H¨ older’s inequality we have Z 1−τ τ 1{0≤v≤M uλ } K(x)|v|p dx AΩε (v, q) ≤ AΩε (v, r) , Ωε

with τ = p−(1+q) p−(1+r) < 1. Using (1.4) and Hardy’s inequality, we finally obtain, for ε small enough, Z 1−τ 1 |v|p (p−k)(1−τ ) τ 2 dx AΩε (v, r) AΩε (v, q) ≤ C2 ε p Ωε d(x) 1

≤ C2 ε 2 (p−k)(1−τ ) (τ A(v, r) + C3 (1 − τ )kvkp ) . From the above arguments and since Z Z q B(v, q) = 1{v≥M uλ } K(x) (M uλ ) v dx + Ω\Ωε

Ωε

(5.8)

1{v≥M uλ } K(x) (M uλ )q v dx

def

= BΩ\Ωε (v, q) + BΩε (v, q),

we also get BΩ\Ωε (v, q) ≤ C4 kvk

(5.9)

and 1

BΩε (v, q) ≤ C5 ε 2 (p−k)(1−τ ) (τ B(v, r) + C6 (1 − τ )kvkp ) .

(5.10)

Using inequalities (5.7) to (5.10), Eλ (v) ≥

C1 1 1 kvkp − λ kvkq+1 − λC4 kvk + B(v, r) 2p q+1 2 1 r q + A(v, r) − C(r) + λ C(q), 2(r + 1) r+1 q+1

(5.11)

for ε > 0 sufficiently small. Together with (5.1), (5.11) implies that Eλ is coercive and bounded from below on W01,p (Ω). So let us define def

cλ =

inf

v∈W01,p (Ω)

Eλ (v)

and let (vn )n∈N ⊂ W01,p (Ω) be a minimizing sequence of Eλ , that is to say Eλ (vn ) −→ cλ . Eλ (vn ) n∈N is bounded, and then (vn )n∈N is bounded in n→∞

W01,p (Ω). Thus, there exist uλ ∈ W01,p (Ω) and a subsequence (vm )m∈N such

642


that vm −→ uλ weakly in W01,p (Ω), strongly in Lq+1 (Ω) and in L1 (Ω) and m→∞ almost everywhere in Ω. Then we get kuλ k ≤ lim inf kvm k.

(5.12)

m→+∞

Using Fatou’s lemma and inequality (5.1), it follows that 1 1 A(uλ , r) + B(uλ , r) ≤ lim inf A(vm , r) + B(vm , r) < +∞. (5.13) m→+∞ r r Again from Fatou’s lemma and inequalities (5.8), (5.10), and (5.12), λ AΩ (uλ , q) + λBΩε (uλ , q) (5.14) q+1 ε λ 1 ≤ lim inf AΩε (vm , q) + λBΩε (vm , q) ≤ C7 ε 2 (p−k)(1−τ ) . m→+∞ q + 1 Since vm −→ uλ in Lq+1 (Ω) and in L1 (Ω), m→∞

AΩ\Ωε (vm , q) −→ AΩ\Ωε (uλ , q) m→∞

and BΩ\Ωε (vm , q) −→ BΩ\Ωε (uλ , q). m→∞

(5.15) Gathering the estimates (5.12) to (5.15) and using (5.6), we obtain 1

1

cλ = lim inf E(vm ) ≥ Eλ (uλ ) − C7 ε 2 (p−k)(1−τ ) ≥ cλ − C7 ε 2 (p−k)(1−τ ) . m→+∞

Passing to the limit as ε → 0, we finally get Eλ (uλ ) = cλ . By definition of cλ , uλ satisfies Eλ (uλ ) = min Eλ (v) v∈W01,p (Ω)

and since Eλ is Gˆ ateaux differentiable, uλ satisfies the Euler-Lagrange equation associated to Eλ , Z Z 1,p p−2 |∇uλ | ∇uλ · ∇v dx = fλ (x, uλ )v dx. ∀v ∈ W0 (Ω), Ω

Ω

W01,p (Ω),

In particular, setting v = (uλ )− ∈ by the weak maximum principle it follows that uλ ≥ 0 almost everywhere in Ω. Moreover, since M uλ is a supersolution of (Pλ ), for all nonnegative v ∈ W01,p (Ω), Z Z K(x) [λ(M uλ )q − (M uλ )r ] v dx. |∇(M uλ )|p−2 ∇(M uλ ) · ∇v dx ≥ Ω

Ω

)+

W01,p (Ω),

Setting v = (uλ − M uλ ∈ we obtain Z 0= fλ (x, uλ ) − K(x) [λ(M uλ )q − (M uλ )r ] (uλ − M uλ )+ dx Ω


≥

Z

p−2

|∇uλ |

p−2

∇uλ − |∇(M uλ )|

643

+ ∇(M uλ ) · ∇ (uλ − M uλ ) dx.

Ω

Using Lemma 5.1, ∇ (uλ − M uλ )+ = 0 almost everywhere in Ω and by Poincaré’s inequality uλ ≤ M uλ almost everywhere in Ω. Finally, Iλ (uλ ) = Eλ (uλ ) =

min

v∈W01,p (Ω)

Eλ (v) ≤ Eλ (ϕ0 ) = Iλ (ϕ0 ) < 0,

therefore uλ is a nontrivial weak solution of (Pλ ).

5.2. Compact support of the solution. In this section we define 1 r q−r def def gλ (t) = tr − λtq , t ∈ [0, +∞) and a∗ = (5.16) λq in such a way that gλ is positive and increasing on the interval (0, a∗ ). We first state a result which guarantees the existence of an appropriate supersolution of (Pλ ) near the boundary. Lemma 5.2. Let uλ ∈ W01,p (Ω) ∩ L∞ (Ω) be a weak solution of (Pλ ). Then uλ ∈ C Ω and there exist δ∗ > 0, M > 0 and α ∈ (1, p0 ) such that uλ (x) ≤ M ϕ1 (x)α

in Ωδ∗ .

In the proof of this lemma, we will use the following weak comparison principle: Proposition 5.2. Let us consider the Dirichlet problems −∆p u − b(x, u) = f in Ω, u = f0 on ∂Ω

(5.17)

and

−∆p v − b(x, v) = g v = g0 0

in Ω, on ∂Ω. 1

(5.18)

,p

Assume that f ≤ g in Lp (Ω), f 0 ≤ g 0 in W p0 (∂Ω), u, v ∈ W1,p (Ω) are any weak solutions of the Dirichlet problems (5.17) and (5.18), respectively, and b(x, ·) : R → R is nonincreasing for almost every x ∈ Ω. Then u ≤ v in Ω. Proof. See Proposition 2.3 in Cuesta-Takáˇc [3].

def

Proof of Lemma 5.2. According to the previous notation, the set ω ∗ = {x ∈ Ω : uλ (x) ≤ a∗ } contains a neighbourhood of ∂Ω and there exists δ0 > 0

644


such that Ωδ0 ⊂ ω ∗ . Since uλ is bounded, there exists C ∗ > 0 large enough such that uλ ≤ C ∗ ϕ1 on ∂Ωδ0 . Hence, uλ and C ∗ ϕ1 satisfy −∆p uλ ≤ −∆p (C ∗ ϕ1 ) in Ωδ0 , (5.19) uλ ≤ C ∗ ϕ1 on ∂Ωδ0 . Therefore, by the weak comparison principle uλ ≤ C ∗ ϕ1 in Ωδ0 . From this estimate and the interior regularity result of Serrin [12], uλ ∈ C Ω . Let M > 0 and α ∈ (1, p0 ); we want to construct a supersolution v to (Pλ ) def

near the boundary such that v = M ϕ1 α . From the proof of Lemma 3.2, there exists a δ1 > 0 depending only on Ω, p, M , and α such that ∆p v ∼ (M α)p−1 (α − 1)(p − 1)d(x)(α−1)(p−1)−1

in Ωδ1

(5.20)

in Ωδ1 .

(5.21)

and K(x)(v r − λv q ) ∼ M r L(d(x))d(x)αr−k Precisely, 1 1 1o n ε∗ (α − 1)(p − 1) p1 1 1 α(q−r) α def , , δ1 = min δ ∗ , K2 2λ1 K2 2λ M where ε∗ and δ ∗ are defined in (2.3). By the definition of δ1 , choosing α > 1 1 ε∗ (α−1)(p−1) p small enough, δ1 = K and we can impose 2λ 2 1

M≤

inf L(δ)δ −(α(p−r−1)−(p−k)) i h δ≤δ 1

αp−1 (α − 1)

1 p−(1+r)

.

(5.22)

Then, by estimates (5.20) and (5.21), v is a supersolution of (Pλ ) in Ωδ1 . Moreover, if we set p−(1+r) n a∗ a∗ αp−1 (α − 1) p−k o def , δ2 = min δ0 , ∗ , C K2 C1 inf L(δ) δ h(a∗ ) for x ∈ Ωδε provided ε is sufficiently small. Indeed, h p−(r+1) p−(r+1) i p−(r+1) h(a∗ ) ≤ C1 ε p − (a∗ ) p < −C2 ε αp ≤ j(x), for x ∈ Ωδε , with C1 and C2 two positive constants independent of ε. With all this notation, we finally define the function w in Ωδε by def

w(x) = h−1 (J(x)) ,

for x ∈ Ωδε .

646


In other words, ε

Z

Gλ (s)

− p1

for x ∈ Ωδε .

ds = J(x),

w(x)

Using the last relation, w is nonnegative in Ωδε and w ≤ a∗ in Ωδε . Moreover, w vanishes when d(x) is small. Indeed, for s ∈ (0, a∗ ), Z s s s gλ (t) dt > gλ . gλ (s)s > Gλ (s) > s 2 2 2

Then for ε ∈ (0, a∗ ), Z ε

− p1

Gλ (s)

Z ds

p

2 2

ds

ds,

0

for ε > 0 small. Then, from the definitions of J and w, w has compact support in Ω. To complete the proof, it is enough to show that uλ ≤ w in Ωδε . Since w has compact support, J ∈ W1,p (Ωδε ) and w = h−1 ◦ J ∈ W1,p (Ωδε ) and satisfies 1 ∇w = −Gλ (w) p ∇j in D0 (Ωδε ) . Then, 1 1 ∆p w + Gλ (w) p0 ∆p j = 0 gλ (w)|∇j|p in D0 (Ωδε ) . p Provided ε is sufficiently small, we have h ϕ i−1 1 1 1 p p ϕ1 |∇j| = |∇ϕ | g 6 K(x) in Ωδε 1 λ 0 0 p p 2 2 and ϕ i− 10 1 ϕ ϕ h 1 p 1 1 1 p ϕ1 0 ϕ1 gλ gλ (gλ ) ∆p j = 0 |∇ϕ1 | + p 2 2 2 2 4 2 1 ϕ i 0 h p 1 p−1 ϕ1 gλ ≥0 in Ωδε . +λ1 ϕ1 2 2 Hence, ∆p w ≤ K(x)gλ (w) in Ωδε . Moreover, since gλ ≥ 0 on ∂Ωδε , we have uλ (x) ≤ ε ≤ w(x) on ∂Ωδε . Therefore, from Proposition 5.2, uλ (x) ≤ w(x) in Ωδε .


647

Proof of Theorem 2.2. Since uλ is compactly supported in Ω, inequality (3.1) is also satisfied when k ≥ 1 + r, which implies the existence of a critical parameter λ∗∗ > 0 such that (Pλ ) has no nontrivial solution for λ < λ∗∗ . Thanks to Propositions 5.1 and 5.3 and Remark 5.1, from the regularity result of Lieberman [10] and the same arguments as in paragraph 4.2, Theorem 2.2 follows. Appendix A. Proof of Lemma 3.3 def

A.1. When 0 ≤ k < 1 + q. By (1.5), v = mϕ1 is a subsolution of (Q) in Ω for m > 0 small enough. Now let us define def

f (x) = M d(x)−(k−q) L (d(x))

in Ω,

with M > 0. Let (k − q)+ < ε < 1; therefore 0 < f (x) ≤ C1 d(x)−ε in Ω. Thus if we consider the problem −∆p v = f in Ω, Q v = 0 on ∂Ω, v > 0 in Ω, by a result of Giacomoni, Schindler, and Takáˇc [6] Q has a unique solution v ∈ C 1,α Ω , with α ∈ (0, 1) and v ∼ d(x) in Ω. Therefore, −∆p v ≥ K(x)v q in Ω for M > 0 sufficiently large. Hence we get that both the sub- and supersolution of problem Q , namely v and v, behave like the distance function d(x) near ∂Ω. Using the sub- and supersolution method as in 1,p Section 4, we get a solution v ∈ W0 (Ω) ∩ C Ω satisfying v(x) ∼ d(x) in Ω. Now let w ∈ W01,p (Ω) ∩ C Ω be a solution to Q satisfying w(x) ∼ d(x) in Ω. Then we can define def

C ∗ = sup {C > 0 : Cw ≤ v in Ω} ∈ R. It is easy to see that C ∗ > 0 and C ∗ w ≤ v in Ω, so for all x ∈ Ω we get q −∆p (C ∗ ) p−1 w(x) = K(x) (C ∗ w(x))q ≤ K(x)v(x)q = −∆p (v(x)). q

If we suppose C ∗ < 1, the weak maximum principle implies (C ∗ ) p−1 w ≤ v q in Ω. But (C ∗ ) p−1 > C ∗ because C ∗ < 1 and q < p − 1; therefore, q

C ∗ w < (C ∗ ) p−1 w 6 v in Ω, which contradicts the definition of C ∗ . So C ∗ ≥ 1 and w ≤ C ∗ w ≤ v in Ω. Interchanging the role of w and v, we finally get that w = v, and this proves

648


the uniqueness of the solution of Q in the convex set n o def Λ1 = v ∈ W01,p (Ω) ∩ C Ω : v(x) ∼ d(x) in Ω . A.2. When 1 + q ≤ k ≤ p. For t ∈ (0, D] we define Z 2D y(s) def Θ(t) = exp ds , s t with y ∈ C ([0, 2D]) ∩ C 1 ((0, 2D]) such that y(0) = 0 and limt→0+ Then, tΘ00 (t) tΘ0 (t) = 0 and lim = −1. lim t→0+ Θ 0 (t) t→0+ Θ(t)

ty 0 (t) y(t)

= 0.

(A.1)

def

Let β ∈ [0, 1]; for x ∈ Ω we also define w(x) = ϕ1 (x)β Θ (ϕ1 (x)) in Ω. Then, p−1 (β−1)(p−1)−1 ϕ1 Θ0 (ϕ1 ) ϕ1 Θ0 (ϕ1 ) p−2h β+ λ1 ϕ1 p −∆p w = Θ(ϕ1 ) ϕ1 β+ Θ(ϕ1 ) Θ(ϕ1 ) ϕ1 Θ0 (ϕ1 ) ϕ21 Θ00 (ϕ1 ) i + (p − 1)|∇ϕ1 |p β(1 − β) − 2β . − Θ(ϕ1 ) Θ(ϕ1 ) We now distinguish the following cases: A.2.1. Case 1 : 0 < β < 1. There exists ε > 0 sufficiently small such that for x ∈ Ωε , β ϕ1 (x)Θ0 (ϕ1 (x)) 3β ≤β+ ≤ 2 Θ(ϕ1 (x)) 2 and β(1 − β) ϕ1 (x)Θ0 (ϕ1 (x)) ϕ1 (x)2 Θ00 (ϕ1 (x)) 3 ≤ β(1 − β) − 2β − ≤ β(1 − β). 2 Θ(ϕ1 (x)) Θ(ϕ1 (x)) 2 Therefore, we get −∆p w(x) ∼ Θ(ϕ1 (x))p−1 ϕ1 (x)(β−1)(p−1)−1 in Ω, which implies −∆p w(x) w(x)−q ∼ Θ(ϕ1 (x))p−(1+q) ϕ1 (x)(β−1)(p−1)−1−qβ in Ω. p−k When 1 + q < k < p, if we choose β = p−(1+q) ∈ (0, 1) and y(t) = for t ∈ [0, 2d], w satisfies ( −∆p w(x) w(x)−q ∼ K(x) in Ω, w = 0 on ∂Ω, w > 0 in Ω.

z(t) p−(1+q)


649

Therefore, there exist C1 , C2 > 0 such that C1 w and C2 w are respectively sub- and supersolutions of the problem (Q). Thus, (Q) has a solution v ∈ 1,p Wloc (Ω) ∩ C0 Ω satisfying p−k

1

v(x) ∼ d(x) p−(1+q) L (d(x)) p−(1+q)

in Ω.

(A.2)

Using the same arguments as Section A.1, we get the uniqueness of the solution in the set n o p−k 1 def 1,p (Ω) ∩ C0 (Ω) : v(x) ∼ d(x) p−(1+q) L(d(x)) p−(1+q) in Ω . Λ2 = v ∈ Wloc Moreover, uλ ∈ W01,p (Ω) if and only if the right-hand term in the equation 0 of problem (Q) is W−1,p (Ω), i.e., if and only if there exists a constant C > 0 such that Z 1,p q ∀v ∈ W0 (Ω), K(x)uλ (x) v(x) dx ≤ Ckvk. Ω

Using estimate (A.2), Hardy’s and Hölder’s inequalities, and property (1.4), this condition is satisfied if k < 1 + q + p−(1+q) . Moreover, by (A.2), we have p Z K(x) (uλ (x))q+1 dx < +∞ Ω

only if k ≤ 1 + q +

p−(1+q) . p

Then, for k > 1 + q +

p−(1+q) , p

uλ ∈ / W01,p (Ω).

A.2.2. Case 2 : β = 1. The computation of −∆p w gives p−2 ϕ1 Θ0 (ϕ1 ) p−2 0 −∆p w = Θ (ϕ1 ) Θ(ϕ1 ) 1+ × Θ(ϕ1 ) h ϕ1 Θ00 (ϕ1 ) i Θ(ϕ1 ) p p λ ϕ + (p − 1)|∇ϕ | − 2 − . 1+ 1 1 1 ϕ1 Θ0 (ϕ1 ) Θ0 (ϕ1 ) We choose Θ such that C1 ϕ1 p ≤ −

Θ(ϕ1 )ϕ1 p−1 ≤ C2 ϕ1 p−1 Θ0 (ϕ1 )

near the boundary, which is equivalent to requiring C1 t ≤ −

Θ(t) ≤ C2 , for t > 0 small enough. Θ0 (t)

Hence, −∆p w(x) w(x)−q ∼ −Θ0 (ϕ1 (x)) Θ (ϕ1 (x))p−2−q ϕ1 (x)−q in Ω.

(A.3)

650


To get −∆p w(x) w(x)−q ∼ ϕ1 (x)−k L (ϕ1 (x)) in Ω, we require t

−(1+q)

p−(q+1)

y(t)Θ(t)

∼t

−k

Z

2D

t

z(s) ds in (0, D]. s

This condition can be satisfied only if k = 1 + q. Then taking 1 Z 2D p−(1+q) s−1 L(s) ds Θ(t) = , 0 < t ≤ D, t

Θ satisfies conditions (A.1) and (A.3). Thus, if k = 1 + q and Z 2D 1 p−(1+q) w(x) = ϕ1 (x) t−1 L(t) dt in Ω, ϕ1 (x)

there exist C1 , C2 > 0 such that C1 w and C2 w are respectively sub- and supersolutions of (Q). Thus, there exists a solution v ∈ W01,p (Ω) ∩ C Ω of (Q) satisfying Z 2D 1 p−(1+q) v(x) ∼ d(x) t−1 L(t) dt in Ω. d(x)

Using the same argument as in Section A.1 we get the uniqueness of the solution in the set o Z 2D 1 n p−(1+q) def 1,p in Ω . Λ3 = v ∈ W0 (Ω) ∩ C(Ω) : v(x) ∼ d(x) t−1 L(t) dt d(x)

A.2.3. Case 3 : β = 0. In this case, we get p−1 h ϕ1 Θ00 (ϕ1 ) i −∆p w = ϕ(x)−1 Θ0 (ϕ1 ) λ1 ϕ1 p − (p − 1)|∇ϕ1 |p . Θ(ϕ1 ) Hence, −∆p w(x) w(x)−q ∼ ϕ1 (x)−1 Θ0 (ϕ1 (x))p−1 Θ(ϕ1 (x))−q in Ω. Similarly as the previous case, to get ϕ1 (x)−1 Θ0 (ϕ1 (x))p−1 Θ(ϕ1 (x))−q ∼ ϕ1 (x)−k L(ϕ(x)) in Ω, we require −p

t

p−(1+q)

Θ(t)

− y(t)

p−1

−k

∼t

exp

Z t

2D

z(s) dt in (0, t]. s


651

This condition can be satisfied only if k = p. Then if condition (3.2) holds and if we choose Z 2D y(s) Z t p−1 1 p−(1+q) s−1 L(s) p−1 ds Θ(t) = exp ds = C , 0 < t ≤ D, s t 0 we get that Θ satisfies conditions (A.1) and (A.3). Thus if k = p and Z ϕ1 (x) p−1 1 p−(1+q) −1 p−1 t L(t) dt w(x) = C , 0

there exist C1 , C2 > 0 such that C1 w and C2 w are respectively sub and 1,p supersolutions of (Q) and there exists a solution v ∈ Wloc (Ω) ∩ C0 Ω of (Q) satisfying Z d(x) p−1 1 p−(1+q) v(x) ∼ in Ω. s−1 L(s) p−1 ds 0

Using the same argument as in Section A.1, we get the uniqueness of the solution in the set Z d(x) o p−1 n 1 p−(1+q) def 1,p s−1 L(s) p−1 ds in Ω . Λ4 = v ∈ Wloc (Ω) ∩ C0 (Ω) : v(x) ∼ 0

Appendix B. Proof of Proposition 3.1 To prove this proposition, we need the following two lemmas: Lemma B.1. (Picone’s Identity) Let u, v ∈ C 1 Ω be two positive functions satisfying Hopf ’s lemma. Then u p−1 u p def |∇v|p − p |∇v|p−2 ∇v · ∇u L(u, v) = |∇u|p + (p − 1) v v satisfies L(u, v) ≥ 0 in Ω and L(u, v) = R(u, v) where up def R(u, v) = |∇u|p − |∇v|p−2 ∇v · ∇ p−1 . v Moreover, L(u, v) = 0 in Ω if and only if there exists C > 0 such that u = Cv in Ω. Proof. See Theorem 1.1 in Allegretto-Huang [1].

L∞ (Ω)

Lemma B.2. (D´ıaz-Saa inequality) For i = 1, 2 let w such that i ∈1 1 1,p ∞ wi ≥ 0 almost everywhere in Ω, wi p ∈ W (Ω), ∆p wi p ∈ L (Ω), and ∞ 1 w2 w1 = w2 on ∂Ω. Moreover if w w2 , w1 ∈ L (Ω), we have the inequality 1 1 Z −∆p (w1 p ) ∆p (w2 p ) + (w1 − w2 ) dx ≥ 0. (B.1) p−1 p−1 Ω w1 p w2 p

652


Futhermore, (B.1) becomes an equality if and only if there exists C > 0 such that w1 = Cw2 almost everywhere in Ω. Proof. See Lemma 2 in D´ıaz-Saa [5].

Proof of Proposition 3.1. We argue by contradiction. If Proposition 3.1 1,p does not hold, there exist v ∈ Wloc (Ω) ∩ C0 Ω weak solution of (Q), η > 0, and ε > 0 satisfying v ≥ ηu almost everywhere in Ω and (3.3) holds. B.1. Step 1: when q ≥ 0. We consider the following perturbed problem: −∆p v = Kn (x)v q , v > 0 in Ω, (Qn ) v = 0 on ∂Ω, where (Kn )n∈N ⊂ L∞ (Ω) is an increasing sequence satisfying Kn −→ K n→+∞

almost everywhere in Ω. We will prove there existsa unique solution of (Qn ) in W01,p (Ω) and show that this solution is C 1,α Ω for some α ∈ (0, 1). Let us consider the functional Z Z def def Kn (x)wq+1 dx = 1 . In (u) = |∇u|p dx, u ∈ V = w ∈ W01,p (Ω), Ω

Ω

W01,p (Ω)

,→ Lq+1 (Ω), there exists a nonFrom the compactness embedding 1,p negative and nontrivial vñ ∈ W0 (Ω) satisfying In (˜ vn ) = min In (u). u∈V

Therefore, from the Lagrange multiplier rule, there exists λn > 0 such that −∆p vñ = λn Kn (x) (˜ vn )q in Ω, vñ = 0 on ∂Ω. By homogeneity of the p-Laplacian operator, if we define def

1

vn = (λn ) p−(1+q) vñ ∈ W01,p (Ω), vn satisfies

−∆p vn = Kn (x)vn q , vn = 0 on ∂Ω.

in Ω,

Since q < p − 1 and Kn ∈ L∞ (Ω), using Moser iterations we prove that vn ∈ L∞ (Ω) and from the regularity result in Lieberman [10], vn ∈ C 1,α Ω for some α ∈ (0, 1). Then, from the strong maximum principle in Vázquez [13], vn > 0 in Ω and vn is a solution of (Qn ).


653

Now, let us prove the uniqueness of such a solution. We use for that the D´ıaz-Saa inequality (B.1). So let un ∈ C 1,α Ω be another solution of (Qn ); then Z ∆p vn −∆p un (B.2) + p−1 (un p − vn p ) dx ≥ 0, p−1 u vn n Ω which implies Z 1 1 Kn (x) − (un p − vn p ) dx = 0. p−(1+q) p−(1+q) u v n n Ω Then inequality (B.2) becomes an equality; therefore, by Lemma B.1 there exists C > 0 such that un = Cvn in Ω. Furthermore, by homogeneity arguments, −∆p (Cvn ) 6= Kn (x)(Cvn )q in Ω if C 6= 1, so un = vn in Ω and we get the uniqueness. Now, we will prove that for all n ∈ N vn ≤ v. For that, we apply the suband supersolution method in a compact subset of Ω. So let us fix n ∈ N and define (Ωm )m∈N∗ an increasing sequence of smooth subdomains of Ω such that Ωm −→ Ω in the Hausdorff topology with m→∞

∀m ∈ N∗ ,

1 m+1

< dist(∂Ω, ∂Ωm )
0 in Ωm , 1,p with v ∈ W01,p (Ω) ∩ C Ω the subsolution of (P). Since v ∈ Wloc (Ω) ∩ L∞ (Ω) and v ≥ ηv in Ω and using the same arguments as in the proof of Proposition 4.1, for all m ∈ N there exists vn,m ∈ W 1,p (Ωm ) ∩ C Ωm a unique solution of (Qn,m ). Moreover, vn,m satisfies ηv ≤ vn,m ≤ v in Ω. Now, setting vñ,m to be the extension of vn,m by 0 in Ω \ Ωm , the sequence (˜ vn,m )m∈N∗ is an increasing sequence which converges pointwise to an el 1,p (Ω) ∩ C0 Ω a solution of (Qn ), by similar arguments as in ement un ∈ Wloc the proof of Proposition 4.1. Then the uniqueness argument implies un = vn in Ω, and then ∀n ∈ N, vn ≤ v in Ω. B.2. Step 2: when q < 0. Let us define the following problem: q −∆p v = Kn (x) v + n1 , v > 0 in Ω, 0 (Qn ) v = 0 on ∂Ω. Using a method similar to that used in Step 1, we get the existence and the uniqueness of a sequence of weak solutions of (Q0n ) in W01,p (Ω).

654


B.3. Step 3: Applying Picone’s Identity with u = ϕ1 β ∈ C 1 Ω , where β = p−1+ε and v = un ∈ C 1 Ω , we get p Z up 0≤ |∇u|p − |∇v|p−2 ∇v · ∇ p−1 dx. (B.3) v Ω (1) When q ≥ 0 this inequality gives Z Z ∇ϕ1 β p dx |∇ϕ1 |p ϕ1 (β−1)p dx = βp Ω Ω Z Z ϕ βp ϕ1 βp 1 p−2 |∇vn | ∇vn · ∇ K (x) ≥ dx = dx. n vn p−1 vn p−(q+1) Ω Ω Therefore, passing to the limit as n → +∞, there exists C > 0 such that Z 1 inf K(x)ϕ1 p−1+ε dx ≤ C, ∀ω ⊂⊂ Ω. y∈ω v(y)p−(q+1) ω This inequality does not hold for ω close enough to Ω, i.e., when dist(Ω, ω) is sufficiently small, because Z K(x)ϕ1 p−1+ε dx = +∞, Ω

by assumption. (2) When q < 0 arguing similarly as in the first case, we get Z Z ϕ1 βp βp |∇ϕ1 |p ϕ1 (β−1)p dx ≥ Kn (x) p−(q+1) dx. Ω Ω vn + n1 Therefore, passing to the limit as n → +∞, Z 1 inf K(x)ϕ1 p−1+ε dx ≤ C, y∈ω (v(y) + 1)p−(q+1) ω and we conclude as above.

∀ω ⊂⊂ Ω,

Appendix C. C 1,β regularity We consider the following quasilinear elliptic boundary value problem: −div(a(x, ∇u)) = f (x) in Ω, (P ) u = 0 on Ω. In this equation, f ∈ L∞ loc (Ω) and N X ∂ div(a(x, ∇u)) = ai (x, ∇u(x)), for x ∈ Ω and u ∈ W01,p (Ω) ∂xi def

i=1

(C.1)


655

0

with values in W−1,p (Ω). Moreover, the components ai of the vector field a : Ω × RN → RN , a = (a1 , . . . , aN ), are functions of x and η ∈ RN , such ∂ai that for i, j ∈ {1, . . . , N }, ai ∈ C(Ω × RN ) and ∂η ∈ C(Ω × (RN \ {0})). We j assume that a satisfies the following ellipticity and growth conditions: (H1) There exist some constants κ ∈ [0, 1], γ, Γ ∈ (0, +∞), and α ∈ (0, 1), such that for all x, y ∈ Ω, all η ∈ RN \ {0}, and ξ ∈ RN , ai (x, 0) = 0,

for i = 1, . . . , N,

N X

∂ai (x, η)ξi ξj ≥ γ(κ + |η|)p−2 |ξ|2 , ∂ηj i,j=1 N X ∂ai ≤ Γ(κ + |η|)p−2 , (x, η) ∂ηj

(C.2) (C.3)

(C.4)

i,j=1

N X

|ai (x, η) − ai (y, η)| ≤ Γ(1 + |η|)p |x − y|α .

(C.5)

i=1

We remark that conditions (C.2) through (C.5) are motivated by the elliptic boundary value problem, −∆p u = f (x) in Ω, (P ) u = 0 on Ω. Finally, we impose the following growth condition on the function f : (H2) f ∈ L∞ loc (Ω) and there exist some constants c > 0 and ε ∈ (0, 1) such that, for almost all x ∈ Ω, |f (x)| ≤ cd(x)−ε .

(C.6)

Proposition C.1. Assume that a(x, η) satisfies the structural hypotheses (C.2) through (C.5) and f (x) satisfies the growth hypothesis (C.6). Let u ∈ W01,p (Ω) be a weak solution of the problem (P ). In addition assume 0 ≤ u(x) ≤ u ≤ Cd(x)

for almost all x ∈ Ω,

(C.7)

W01,p (Ω)

where C > 0 and u ∈ is a weak supersolution of −div(a(x, ∇u)) = |f (x)| in Ω, P u = 0 on Ω. Then there exist constants β ∈ (0, α) and M ≥ 0 depending only on Ω, N, p, on γ, Γ, α in (C.2) through (C.5), on the constants c, ε in (C.6) and on the constant C in (C.7), but not on κ ∈ [0, 1], such that u satisfies u ∈ C 1,β Ω and kukC 1,β (Ω) ≤ M.

656


Proof. The proof is similar to the proof of Theorem B.1 in Giacomoni, Schindler, and Tak´ aˇc [6]. In particular, condition (C.6) replacing the growth condition (B.8) implies the estimate (B.17) in [6]. The C 1,β regularity of uλ is directly proved applying this proposition with def

a(x, η) = |η|p−2 η

def

and f (x) = K(x) [λuλ (x)q − uλ (x)r ]

for x ∈ Ω and η ∈ RN . References [1] W. Allegretto and Y. Huang, A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal., 32 (1998), 819–830. [2] L. Alvarez and J.I. D´ıaz, On the behaviour of the free boundary of some nonhomogeneous elliptic problems, Positivity, 36 (1990), 131–144. [3] M. Cuesta and P. Tak´ aˇc, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations, 13 (2000), 721–746. [4] J.I. D´ıaz, “Nonlinear Partial Differential Equations and Free Boundaries,” Elliptic Equations, Research Notes in Mathematics, Vol. I, 106, Pitman, Londres, 1985. [5] J.I. D´ıaz and J.E. Sa´ a, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C.R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521–524. [6] J. Giacomoni, I. Schindler, and P. Tak´ aˇc, Sobolev versus H¨ older local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6 (2007), 117–158. [7] Y. Haitao, Positive versus compact support solutions to a singular elliptic problem, J. Math. Anal. Appl., 319 (2007), 830–840. [8] Y. Il’yasov and Y. Egorov, Hopf boundary maximum principle violation for semilinear elliptic equations, Nonlinear Anal., 72 (2010), 3346–3355. ¨ [9] J. Karamata, Uber die Hardy-Littlewoodsche Umkehrung des Abelschen St¨ atigkeitssatzes, Math. Zeitschrift, 32 (1930), 319–320. [10] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203–1219. [11] P. Lindqvist, On the equation div (|∇u|p−2 ∇u) + λ|u|p−2 u = 0, Proc. Amer. Math. Soc., 109 (1990), 157–164. [12] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247–302. [13] J.L. V´ azquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math & Opt., 12 (1984), 1992–2002.