MULTIPLICITY OF SOLUTIONS FOR SOME ...

MULTIPLICITY OF SOLUTIONS FOR SOME SEMILINEAR PROBLEMS INVOLVING NONLINEARITIES WITH ZEROS ´ AND LEONELO ITURRIAGA JORGE GARCÍA-MELIAN Abstract. In this paper we consider the semilinear elliptic problem −∆u = λf (u) in Ω u=0 on ∂Ω where f is a nonnegative, locally Lipschitz continuous function with r positive zeros, Ω is a smooth bounded domain and λ > 0 is a parameter. We show that for large enough λ there exist 2r positive solutions, irrespective of the behavior of f at zero or infinity, provided only that f verifies a suitable non integrability condition near each of its zeros, thereby generalizing previous known results. The construction of the solutions rely on the sub and supersolutions method and topological degree arguments, together with the use of a new Liouville theorem which is an extension of recent results to this type of nonlinearities.

1. Introduction and results In this paper, we deal with the question of multiplicity of positive (classical) solutions of the problem −∆u = λf (u) in Ω (1.1) u=0 on ∂Ω where f is a nonnegative, locally Lipschitz function defined in [0, +∞), Ω is a smooth bounded domain of RN (N ≥ 3) and λ > 0 is a parameter. It is well known that, when f is positive, its behavior at zero and infinity is relevant for the existence of solutions of (1.1). For example, when f (t) = tp , +2 existence of solutions in some domains depends on whether 1 < p < N N −2 N +2 or p ≥ N −2 . However, this situation changes drastically when f possesses some zero. In this case, problem (1.1) was considered in [8]. Later, the p-Laplacian version of (1.1) was analyzed in [6], [7], and the extension to Pucci’s operators setting in [2]. Assuming that f has a positive zero and under several additional hypotheses, it was shown in these works that there exist two positive solutions for large λ. The purpose of the present paper is two-fold. On one hand, we allow the nonlinear term f to possess several positive zeros while on the other hand, the behavior of f near its zeros is slightly more general than in previous works (cf. for instance [6]). It is to be remarked that only the behavior of f near its zeros is relevant to construct solutions for large λ, independently of the growth near zero or infinity. Let us next state our main result. We mention in passing that, although we are assuming N ≥ 3 throughout, the case N = 2 can also be dealt with, with some obvious modifications. 1

2

´ AND L. ITURRIAGA J. GARCÍA-MELIAN

Theorem 1. Assume N ≥ 3, and let f be a nonnegative locally Lipschitz function with a finite number of positive zeros α1 , . . . , αr . Suppose there exists δ > 0 such that Z αk +δ f (t) dt = +∞ (1.2) 2(N−1) αk (t − αk ) N−2 for k = 1, . . . , r. Then there exists λ0 > 0 such that, for λ > λ0 , problem (1.1) has 2r positive solutions uk,λ , k = 1, . . . , 2r, verifying ku2k−1,λ k∞ < αk < ku2k,λ k∞ < ku2k+1,λ k∞ < αk+1 , for k = 1, . . . , r and λ > λ0 . Remark 1. As a byproduct of the proof of Theorem 1, it is easy to see that the solutions constructed there actually verify lim ku2k−1,λ k∞ = lim ku2k,λ k∞ = αk .

λ→+∞

λ→+∞

The proof of this result is achieved in several steps: the solutions with supremum norm below each respective zero are obtained by means of sub and supersolutions. Those solutions whose maxima lie above each zero are constructed by means of a indirect argument. We first truncate the nonlinearity f above each zero to make it subcritical at infinity. This provides with a priori bounds for all solutions, allowing us to work with topological degree. We use a suitable parameterized version of the truncated problem which in particular needs no additional hypotheses on f to obtain solutions for every λ > 0. Finally, to show that the solutions of the truncated problem are indeed solutions of (1.1) for large λ, we use a Liouville theorem which is a slight generalization of a recent result in [1] where only positive nonlinearities were considered. It is at this point where condition (1.2) is required. The rest of the paper is organized as follows: in Section 2 we consider a Liouville theorem for nonlinearities with zeros, while Section 3 is devoted to the proof of Theorem 1. 2. A Liouville theorem In this section, we will obtain a nonexistence theorem for supersolutions of a semilinear problem related to (1.1) in RN , which will be instrumental in our proofs in the next section. It is concerned with (weak) positive supersolutions of the problem (2.1)

− ∆u ≥ f (u) in RN ,

where N ≥ 3 and f is a continuous, nonnegative, given nonlinearity (observe that the case N = 2 is immediate, since the only nonnegative superharmonic functions in R2 are the constant functions). The result we are going to prove is an extension of Theorem 1 in [1], where only positive nonlinearities were considered, to nonnegative nonlinearities. We refer the interested reader to [1] and [3] for more references on Liouville theorems.

NONLINEARITIES WITH ZEROS

3

Theorem 2. Assume N ≥ 3 and let f : [0, ∞) → R be continuous and nonnegative. Suppose in addition that for every zero α of f there exists δ > 0 such that Z α+δ f (t) dt = +∞. (2.2) 2(N−1) α (t − α) N−2 Let u be a positive solution of (2.1). Then u ≡ β in RN , for some β with f (β) = 0. Proof. Assume u is a positive, nonconstant solution of (2.1) and denote m(R) = inf |x|=R u(x). Thanks to the maximum principle, we have that m(R) = inf |x|≤R u(x), so that it is a strictly decreasing function. Take an increasing sequence Rk → +∞ and consider the problem   −∆v = f (v) in R1 < |x| < Rk v = m(R1 ) on |x| = R1 (2.3)  v = m(Rk ) on |x| = Rk .

Since v¯ = u is a supersolution of this problem while v = m(Rk ) is a subsolution with v¯ ≥ v, we deduce the existence of a minimal, radially symmetric solution vk of (2.3) (cf. for instance the Appendix in [1]), verifying 0 < vk ≤ u in R1 < |x| < Rk . In particular, the sequence {vk }∞ k=1 is uniformly bounded in compact sets of RN \ BR1 , and it is standard to obtain a N 1 subsequence –denoted again {vk }∞ k=1 – such that vk → v in Cloc (R \ BR1 ). Passing to the limit in (2.3), we achieve −∆v = f (v) in |x| > R1 with v|∂BR1 = m(R1 ), so that v is nontrivial and the strong maximum principle gives v > 0. Next observe that each function vk is radially symmetric, so that v also is. Thus v(x) ≤ m(|x|), which ensures that v is not constant, since m is strictly decreasing and v = m(R1 ) on ∂BR1 . It is also well-known that v is decreasing, so we may define l := limr→∞ v(r) and w = v − l. We have a radially symmetric solution of −∆w = f (w + l) =: g(w) in |x| > R1 , with limr→+∞ w(r) = 0. The function w verifies −w′′ −

N −1 ′ w = g(w) r

in r > R1 , 2−N

so that, with the change of variables s = rN −2 , z(s) = w(r), we have a positive solution of the Cauchy problem −z ′′ = as−γ g(z) in (0, s0 ) (2.4) z(0) = 0 −1) for some a > 0 and some small positive s0 , where γ = 2(N N −2 . Now we observe that when g(0) > 0, that is, f (l) > 0, then for small positive δ: Z δ g(τ ) (2.5) dτ = +∞, τγ 0

since γ > 2, while for g(0) = f (l) = 0, condition (2.5) translates into Z l+δ f (τ ) dτ = +∞, (τ − l)γ l


4

which is precisely condition (2.2). Hence we have a contradiction with Theorem 6 in [1]. This contradiction shows that u has to be a constant β, hence f (β) = 0. For the benefit of the reader, we include a short sketch of the proof of Theorem 6 in [1]. Observe that z ′ ≥ 0, z ′′ ≤ 0, so that by the mean value theorem z(s) = z ′ (ξ)s ≥ z ′ (s)s (for some ξ ∈ (0, s)). Hence 0 < z ′ (s0 ) ≤ z ′ (s) ≤ z(s)/s in (0, s0 ). Multiply the equation in (2.4) by z ′ and integrate in (s, s0 ) to obtain Z s0 g(z(t)) z(t) γ ′ ′ 2 (2.6) z (s) ≥ 2a z (t)dt. z(t)γ t s for every s ∈ (0, s0 ). Hence ′

2

′

γ

z (s) ≥ 2az (s0 ) (2.7)

= 2az ′ (s0 )γ

Z Z

s0 s

g(z(t)) ′ z (t)dt z(t)γ

z(s0 ) z(s)

g(τ ) dτ τγ

in (0, s0 ).

This implies in particular that z ′ (s) → +∞ as s → 0+, so that, diminishing s0 if necessary we may assume z ′ (s0 ) ≥ 1. Denote by H(z(s)) the righthand side of the last line in (2.7) with z ′ (s0 ) replaced by one. Then z ′ (s) ≥ 1 H(z(s)) 2 . Taking this inequality again in (2.6) we have Z s0 Z z(s0 ) γ γ g(z(t)) g(τ ) ′ ′ 2 2 z (s) ≥ 2a H(z(t)) z (t)dt = 2a H(τ ) 2 dτ γ γ z(t) τ s z(s) γ Z z(s0 ) +1 γ H(z(s)) 2 . H ′ (τ )H(τ ) 2 dτ = =− γ z(s) 2 +1 ∞ Iterating this procedure we get sequences {ak }∞ k=1 and {bk }k=1 given by γ 2 a1 = 1, ak = γ2 ak−1 + 1, b1 = 1, bk = bk−1 ak such that

z ′ (s)2 ≥

H(z(s))ak bk

in (0, s0 ) for every k. k−1

(γ ) It can be checked that ak ≥ ( γ2 )k−1 and bk ≤ C1 2 for some C1 > 1. Finally, choose δ ∈ (0, s0 ) such that H(z(s)) > 2C1 in (0, δ). We obtain γ k−1 in (0, δ) for every k. Letting k → +∞, we arrive at a z ′ (s)2 ≥ 2( 2 ) contradiction, since γ > 2. The proof is concluded. 3. Proof of Theorem 1 This section will be devoted to prove Theorem 1. The proof proceeds in several steps, so that it will be split in a series of lemmas. Lemma 3. Assume f is a locally Lipschitz function with an isolated zero α > 0. Then for every δ > 0 there exists λ0 > 0 such that problem (1.1) admits a positive solution uλ for λ ≥ λ0 , which verifies in addition α − δ < kuλ k∞ < α.


5

Proof. Choose a point x0 ∈ Ω and positive numbers R2 > R1 > 0 such that A = A(R1 , R2 ) = {x ∈ RN : R1 < |x − x0 | < R2 } ⊂⊂ Ω. Our first intention it to look for a positive, radially symmetric subsolution of the problem −∆v = λ0 f (v) in A (3.1) v=0 on ∂A for some fixed value λ0 > 0. Such a subsolution has to verify v(x) = w(r), where r = |x − x0 | and    −w′′ − N − 1 w′ ≤ λ0 f (w) in R1 < r < R2 r (3.2)   w(R ) = w(R ) = 0. 1 2

Now we perform the change of variables w(r) = z(s), where s = r 2−N /(N − 2). Then (3.2) is equivalent to −z ′′ ≤ λ0 g(s)f (z) in a < s < b z(a) = z(b) = 0, for some b > a > 0, where g(s) = r 2(N −1) . Thus it suffices to have 2(N −1) −z ′′ = λ0 R1 f (z) in a < s < b z(a) = z(b) = 0. It is rather standard to solve this problem: there exists a solution, which is symmetric with respect to the midpoint of the interval [a, b] and is given implicitly by Z z(s) a+b ds p = (2λ0 )1/2 R1N −1 (s − a) in a < s < 2 F (¯ z ) − F (s) 0 Rs where F (s) = 0 f (t)dt, and z¯ (the maximum of z) is a solution of the equation Z z¯ ds b−a p (3.3) . = (2λ0 )1/2 R1N −1 2 F (¯ z ) − F (s) 0 To solve this equation it suffices to take an arbitrary value of z¯ ∈ (α − δ, α), and then define λ0 by means of (3.3). Next, define v(x) x ∈ A u(x) = 0 x 6∈ A.

Then u is a (weak) subsolution of (1.1) for λ ≥ λ0 . Since u = α is a supersolution and u ≤ u , we obtain a solution uλ of (1.1) for every λ ≥ λ0 , which verifies α − δ < z¯ ≤ kuλ k∞ ≤ α. Finally, the strong maximum principle gives kuλ k∞ < α for every λ ≥ λ0 . To proceed further, choose K ∈ (αi , αi+1 ) and define a truncated function as follows:  0≤t≤K  f (t), ˜ (3.4) f (t) =  f (K) tp , t > K. Kp


6

for some p < (3.5)

N +2 N −2 .

We consider the truncated problem −∆u = λf˜(u) in Ω u=0 on ∂Ω

and observe that a solution u of (3.5) is a solution of the original problem (1.1) if kuk∞ < K. We next consider the existence of solutions of (3.5). Lemma 4. For every λ > 0, problem (3.5) admits a positive solution uλ with kuλ k∞ > αi . The proof of this lemma follows with a degree theoretic argument. For this sake, we choose a locally Lipschitz function h such that h ≥ 0, h(0) = 1 and h = 0 in [ α21 , +∞), say. For σ ∈ [0, 1], τ ≥ 0, consider the following parameterized version of (3.5): −∆u = λf˜(u) + σh(u) + τ in Ω (3.6) u=0 on ∂Ω. We first show the existence of a priori bounds for all positive solutions of (3.6). Lemma 5. For every Λ > 0, there exist positive constants M , τ0 such that if u is a positive solution of (3.6) for some σ ∈ [0, 1], τ ≥ 0, λ ≥ Λ, then kuk∞ < M

and

τ < τ0 .

Proof. Notice first that if u is a positive solution of (3.6), then −∆u ≥ τ in Ω. Thus if we multiply by the positive eigenfunction φ associated to the principal eigenvalue λ1 of −∆ in Ω and integrate in Ω, we arrive at Z Z φ. uφ > τ λ1 Ω

Ω

Thus τ < λ1 kuk∞ . The rest of the proof is rather standard, following the lines of Theorem 1 in [5]. We are not giving full details. Assume there exists a sequence of positive solutions of (3.6) {un }∞ n=1 cor∞ ⊂ [0, +∞) and {λ }∞ ⊂ [Λ, +∞), ⊂ [0, 1], {τ } responding to {σn }∞ n n=1 n n=1 n=1 such that Mn := kun k∞ → +∞. Choose points {xn }∞ n=1 ⊂ Ω with un (xn ) = −1/2

− p−1 2

Mn and define vn (y) = u(xn + µn y)/Mn , where µn = λn Mn These functions verify  µ2  −∆vn = n (λn f˜(Mn vn ) + σn h(Mn vn ) + τn ) in Ω Mn  vn = 0 on ∂Ω

→ 0.

where Ωn = {y ∈ RN : xn + µn y ∈ Ω}. We have two possibilities: either 1/2 1/2 d(xn )µn → +∞ or d(xn )µn → d > 0, where d(xn ) stands for the distance from xn to ∂Ω. Consider the first possibility. Notice that τn /Mn < λ1 , so that, after 1 (RN ), where v is a choosing an appropriate subsequence, vn → v in Cloc positive solution of f (K) p v in RN −∆v = Kp with v(0) = 1. This contradicts the Liouville theorem in [4].


7

1 (RN ) ∩ If the second possibility holds, we similarly have vn → v in Cloc + C(RN ), where v verifies + (K) p in RN −∆v = fK p v + v=0 on ∂RN + N : x and v(deN ) = 1, where RN N > 0}, contradicting the + = {x ∈ R Liouville theorem in [5]. This contradiction shows the existence of a positive constant M such that kuk∞ ≤ C for every positive solution u of (3.6) with σ ∈ [0, 1] and τ ≥ 0. Observe finally that, since τ < λ1 kuk∞ , we then obtain τ < λ1 C =: τ0 for every solution of (3.6). This concludes the proof.

Proof of Lemma 4. Choose L > 0 such that the function λf˜(u) + σh(u) + Lu + τ is increasing for u ∈ [0, M ], σ ∈ [0, 1] and τ ≥ 0, where M is given by Lemma 5. Denote by Sf the unique solution of the problem −∆u + Lu = f in Ω u=0 on ∂Ω. It is well-known that S : C(Ω) → C01 (Ω) is a compact, strongly positive linear operator. If F (σ, τ, ·) denotes the Nemytskii operator of the function λf˜(u)+σh(u)+Lu+τ , then F : [0, 1]×[0, τ0 ]×C01 (Ω) → C(Ω) is continuous and increasing in the third variable, and problem (3.6) is equivalent to the fixed point equation u = T (σ, τ, u) 1 in C0 (Ω), where T (σ, τ, ·) = S(F (σ, τ, ·)) is a compact operator, increasing in the third variable. Now consider the open set ∂u 0, it follows by Lemma 5 and the strong maximum principle that there are no fixed points of T (σ, τ, ·) on ∂O for σ ∈ (0, 1], τ ∈ [0, τ0 ]. Thus the Leray-Schauder degree deg(I − T (σ, τ, ·), O, 0) makes sense and does not depend on σ nor on τ , as long as σ > 0. Since there are no solutions of (3.5) in O for τ = τ0 , by Lemma 5, we obtain deg(I − T (σ, 0, ·), O, 0) = deg(I − T (σ, τ0 , ·), O, 0) = 0 for every σ ∈ (0, 1]. Next, define the set ∂u 0, so that the degree deg(I − T (σ, 0, ·), Q, 0) is also well defined. But observe that u = 0 is a

8


subsolution of (3.6), while u = αi is a supersolution, due to f˜(αi ) = h(αi ) = 0. Since T (σ, 0, ·) is increasing, it leaves Q invariant, hence it is standard that deg(I − T (σ, 0, ·), Q, 0) = 1 (a proof of this fact follows easily by considering the homotopy H(σ, ·, t) = I − tT (σ, 0, ·) − (1 − t)θ for t ∈ [0, 1], where θ ∈ (0, αi ) is arbitrary). Thanks to the excision property of the degree: deg(I − T (σ, 0, ·), O \ Q, 0) = −1 thus there exists a fixed point uλ,σ of T (σ, 0, ·) with kuλ,σ k∞ > αi . This fixed point is a positive solution of (3.6) with τ = 0. Next, using again the bounds given by Lemma 5, which are independent of σ, we can choose an arbitrary sequence σn → 0 to obtain that (by passing to a subsequence) uλ,σn → uλ in C01 (Ω). Thus uλ is a nonnegative weak solution of (3.5) which verifies kuλ k∞ ≥ αi . By the strong maximum principle, uλ > 0 in Ω and kuλ k∞ > αi . This concludes the proof. We are finally ready to put all pieces together and proceed to the proof of Theorem 1. Proof of Theorem 1. Using Lemma 3 we obtain the existence of λ0 > 0 such that (1.1) admits a positive solution u1,λ for every λ > λ0 with ku1,λ k < α1 . Now choose K ∈ (α1 , α2 ) and truncate f as in (3.4). By means of Lemma 4, for every λ > 0, there exists a solution u2,λ of (3.5) with ku2,λ k∞ > α1 . We claim that u2,λ is actually a solution of (1.1) when λ is large enough. We will in fact prove that ku2,λ k∞ → α1 as λ → +∞. For this aim, take a point xλ ∈ Ω such that u2,λ (xλ ) = ku2,λ k∞ and define vλ (y) = u(xλ + λ−1/2 y),

y ∈ Ωλ ,

where Ωλ := {y ∈ RN : xλ + λ−1/2 y ∈ Ω}. Then −∆vλ = f (vλ ) in Ωλ , with vλ = 0 on ∂Ωλ . Observe that, according to Lemma 5, kvλ k∞ = kuλ k∞ < M , for some positive M , independent of λ. This in turn gives bounds for kvλ kC 1 . Now take an arbitrary sequence λn → +∞. As in the proof of Lemma 5, two situations are possible: either λn d(xλn ) → +∞ or λn d(xλn ) → d > 0. In the second case, since we may assume, by passing to a subsequence, 1 (RN ) ∩ C(RN ), we obtain that v is a positive solution of that vn → v in Cloc + + −∆v = f (v) in RN + v=0 on ∂RN + with v(deN ) = kvk∞ . However, according to Theorem 3.1 in [9], the solution v is strictly increasing in the direction of eN , and we obtain a contradiction. 1 (RN ), where Thus λn d(xλn ) → +∞, and we may assume vn → v in Cloc −∆v = f (v) in RN . By Theorem 2, we deduce v ≡ α1 , and in particular ku2,λn k∞ → α1 . Since the sequence λn → ∞ is arbitrary, it follows that limλ→+∞ ku2,λ k∞ = α1 . Thus there exists λ1 such that ku2,λ k∞ < K for λ > λ1 , and this implies that u2,λ is a positive solution of (1.1) with ku2,λ k∞ > α1 when λ > λ1 . Proceeding similarly, we obtain the existence of positive solutions uk,λ , k = 3, . . . , 2r, for large enough λ, such that ku2k−1,λ k∞ < αk , ku2k,λ k∞ > αk , for k = 2, . . . , r. The proof is concluded.


9

Acknowledgements. J. G-M was partially supported by Ministerio de Ciencia e Innovaci´ on under grant MTM2011-27998 (Spain). J. G-M and L. I. were partially supported by USM Grant No. 121211 and Fondecyt grant 1120842. References ´ n, J. Garc´ıa-Melia ´ n, A. Quaas, Optimal Liouville theorems for super[1] S. Alarco solutions of elliptic equations involving the Laplacian, submitted for publication. ´ n, L. Iturriaga, A. Quaas, Existence and multiplicity results for Pucci’s [2] S. Alarco operators involving nonlinearities with zeros. Calc. Var. Partial Differential Equations 45 (2012), no. 3-4, 443–454. [3] S. N. Armstrong, B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Part. Diff. Eqns. 36 (2011), 2011–2047. [4] B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525–598. [5] B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883–901. [6] L. Iturriaga, S. Lorca, E. Massa, Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 763–771. ´ nchez, P. Ubilla, Positive solutions of the p[7] L. Iturriaga, E. Massa, J. Sa Laplacian involving a superlinear nonlinearity with zeros, J. Diff. Eqns. 248 (2010), 309–327. [8] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (4) (1982), 441–467. [9] A. Quaas, B. Sirakov, Existence results for nonproper elliptic equations involving the Pucci’s Operator, Comm. Partial Differential Equations 31 (2006), 987–1003. ń J. Garc´ıa-Melia ´ lisis Matema ´ tico, Universidad de La Laguna. Departamento de Ana ´ nchez s/n, 38271 – La Laguna, SPAIN C/. Astrof´ısico Francisco Sa and ´ mica, Instituto Universitario de Estudios Avanzados (IUdEA) en F´ısica Ato ´ nica, Facultad de F´ısica, Universidad de La Laguna Molecular y Foto ´ nchez s/n, 38203 – La Laguna , SPAIN C/. Astrof´ısico Francisco Sa E-mail address: [email protected] L. Iturriaga ´ tica, Universidad T´ Departamento de Matema ecnica Federico Santa Mar´ıa ˜ a, 1680 – Valpara´ıso, CHILE. Casilla V-110, Avda. Espan E-mail address: [email protected]