On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers

Abstract

We prove that the semilinear elliptic equation −Δ𝑢=𝑓(𝑢), in Ω, 𝑢=0, on 𝜕Ω has a positive solution when the nonlinearity 𝑓 belongs to a class which satisfies 𝜇𝑡^𝑞≤𝑓(𝑡)≤𝐶𝑡^𝑝 at infinity and behaves like 𝑡^𝑞 near the origin, where 1<𝑞<(𝑁+2)/(𝑁−2) if 𝑁≥3 and 1<𝑞<+∞ if 𝑁=1,2. In our approach, we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity does not satisfy any hypotheses such those required by the blowup method. Furthermore, we do not impose any restriction on the growth of 𝑝.

1. Introduction

Since the 70s, several authors have been studying existence of solutions for the semilinear elliptic Dirichlet problem−Δ𝑢=𝑓(𝑢)inΩ,𝑢=0on𝜕Ω, (P)where Ω is a bounded domain in ℝ^𝑁, 𝑁≥2. Most part of these results were done under some hypotheses on the nonlinearity 𝑓 in order to make variational methods work. The most used hypothesis is the Ambrosetti-Rabinowitz condition [1] which makes the Euler-Lagrange functional associated to subcritical problem (P) satisfies the Palais-Smale condition. Also, the blowup method, due to Gidas and Spruck [2], is used to get existence of solutions and, to work, it needs an asymptotical behaviour like 𝑡^𝑝 for the subcritical nonlinearity 𝑓 (this is, for the case 1<𝑝<(𝑁+2)/(𝑁−2), when 𝑁≥3 or 1<𝑝<∞, when 𝑁=2). Most recently, two existence results related to the subject have been published: de Figueiredo and Yang [3] considered problem (P), where variational techniques were applied together with Morse's index, in Azizieh and Clément [4] using also variational techniques and the blowup method; problem (P) was studied for the 𝑚-Laplacean operator, with 𝑚>1. In both papers [3, 4], the existence of solutions depends on the following condition:𝜇𝑡^𝑝≤𝑓(𝑡)≤𝐶𝑡^𝑝forlarge𝑡. (1.1)

Our results complete those above in the sense that this paper treats this subject for a class of nonlinearities that lies between two different powers 𝑡^𝑞 and 𝑡^𝑝,1<𝑞<𝑝, with no restrictions on 𝑝. A multiplicity of solutions result was done by Li and Liu [5] with critical growth restriction on 𝑝. Here, we will suppose that 𝑓∶ℝ₊→ℝ₊ is a locally Hölder continuous function with the following decomposition 𝑓(𝑠)=𝑧(𝑠)+𝜆𝑔(𝑠). The hypotheses on the functions 𝑧 and 𝑔 are the following.

(H₁)lim_𝑠→0(𝑧(𝑠)/𝑠)=0.(H₂)There exists 𝜃>2 such that0<𝜃𝑍(𝑠)≤𝑧(𝑠)𝑠for𝑠≥0, (1.2)where 𝑍(𝑠)=∫^𝑠₀𝑧(𝑡)𝑑𝑡.(H₃)If 𝑁≥3, there exists 𝑞∈(1,(𝑁+2)/(𝑁−2)), and 𝑞∈(1,+∞) if 𝑁=1,2 such that|𝑧(𝑠)|≤|𝑠|^𝑞∀𝑠∈ℝ. (1.3)(H₄)There is a positive real sequence {𝑀_𝑛} satisfying lim_𝑛→+∞𝑀_𝑛=+∞,𝑔(𝑀_𝑛)>0, and𝑔(𝑀_𝑛)𝑀^𝑞_𝑛≥𝑔(𝑠)𝑠^𝑞∀𝑠∈[0,𝑀_𝑛],∀𝑛∈ℕ. (1.4)

For convenience, we rewrite problem (P) in the following way:−Δ𝑢=𝑧(𝑢)+𝜆𝑔(𝑢)inΩ,𝑢=0on𝜕Ω. (P)

Observe that conditions (H₁),(H₃), and (H₄) imply𝑙𝑖𝑚𝑠𝑢𝑝_𝑠→0+𝑓(𝑠)𝑠<𝜆₁, (1.5)where 𝜆₁ is the first eigenvalue of (−Δ;𝐻¹_𝑜(Ω)). Moreover, since 𝑓(𝑠)≥𝑧(𝑠), for all 𝑠>0, a necessary condition for the existence of solution for problem (𝑃) is that the nonlinearity 𝑓 crosses the line 𝜆₁𝑠.

As an example, we may consider 𝑧(𝑠)=𝑠^𝑞,𝑔(𝑠)=𝑠^𝑝(1+cos𝑠),𝜃=𝑞+1,𝑀_𝑛=2𝑛𝜋. (1.6) The function 𝑓(𝑠)=𝜆𝑠^𝑝(1+cos𝑠)+𝑠^𝑞 satisfies the above conditions but does not satisfy Ambrosetti-Rabinowitz conditions. Moreover, 𝑓 also does not belong to any class of solutions contained in the references [1–3] or [4].

Our main result is the following.Theorem 1.1. There exists 𝜆_∗>0 such that problem (P) has a positive solution for all 𝜆∈(0,𝜆_∗].In our proof, we adapt an idea explored by Chabrowski and Yang in [6]. Comparing our approach to others cited in the literature, it is important to stress that ours does not use Morse Index method and consequently we do not need derivatives of the nonlinearity as in [3], and we do not have any restriction 1<𝑝≤𝑁/(𝑁−2) as required in [4], for the case 𝑚=2.

2. Proof of Theorem 1.1

Hereafter, let us denote by ‖𝑣‖ the usual norm of 𝐻¹_𝑜(Ω), that is,‖𝑣‖=(_Ω|∇𝑣|²𝑑𝑥)^1/2, (2.1)and by 𝑐_𝑞 the minimax level obtained by the mountain-pass theorem of Ambrosetti and Rabinowitz applied to the functional𝐽_𝑞1(𝑣)=2_Ω|∇𝑣|²𝑑𝑥−_Ω𝑍(𝑣₊)𝑑𝑥, (2.2)where 𝑣₊(𝑥)=max{0,𝑣(𝑥)}.

The proposition below establishes an estimate involving the 𝐿^∞-norm of a solution related to a subcritical problem. This estimate is an important point in our approach, and its proof is an immediate consequence of bootstrap arguments. The constant 𝜆_∗ that appears in Theorem 1.1 depends only on the constant 𝑘 below and on the sequence (𝑀_𝑛) given by (H₄), as we can see in this section.

Proposition 2.1. Let 𝑣 be a solution of the problem−Δ𝑣=ℎ(𝑣)inΩ,𝑣=0on𝜕Ω, (2.3) where ℎ∶ℝ→ℝ is a continuous functions verifying |ℎ(𝑠)|≤2|𝑠|^𝑞, for all 𝑠>0. Then, for all 𝐶>0, there exists a constant 𝑘=𝑘(Ω,𝑞,𝐶)>0 such that if‖𝑣‖²≤𝐶,‖𝑣‖_∞≤𝑘. (2.4)

Applying Proposition 2.1, for the constant 𝐶=2𝑐_𝑞(1/2−1/(𝑞+1))⁻¹ there exists a constant 𝑘=𝑘(Ω,𝑞) such that (2.4) holds.

Let us fix 𝑛∈ℕ such that 𝑀_𝑛>𝑘=𝑘(Ω,𝑞), and let 𝜆_∗>0 satisfy𝜆_∗𝑔(𝑀_𝑛)𝑀^𝑞_𝑛≤1,𝜆_∗(1𝜃+1𝑞+1)𝑔(𝑀_𝑛)𝑀_𝑛|Ω|≤𝑐_𝑞, (2.5)where |Ω| denotes the Lebesgue's measure of the set Ω.

For that 𝑛∈ℕ previously fixed , let us consider the function ℎ_𝑛∶ℝ→ℝ given by ℎ_𝑛(𝑠)=0,if𝑠≤0,𝑔(𝑠),if0≤𝑠≤𝑀_𝑛,𝑔(𝑀_𝑛)𝑀^𝑞_𝑛𝑠^𝑞,if𝑠≥𝑀_𝑛. (2.6)From condition (H₄),|ℎ_𝑛(𝑠)|≤𝑔(𝑀_𝑛)𝑀^𝑞_𝑛|𝑠|^𝑞,∀𝑠, (2.7)and then 𝑓_𝑛(𝑠)=𝑧(𝑠)+𝜆ℎ_𝑛(𝑠) satisfies the inequality 𝑓_𝑛(𝑠)≤2|𝑠|^𝑞, since 𝜆≤𝜆_∗.

It is easy to check that the truncated function 𝑓_𝑛 satisfies Ambrosetti-Rabinowitz condition in the interval [𝑀_𝑛,+∞), that is,𝜃𝐹_𝑛(𝑠)−𝑠𝑓_𝑛(𝑠)≤0,∀𝑠≥𝑀_𝑛, (2.8)where 𝐹_𝑛(𝑠)=∫^𝑠_𝑜𝑓_𝑛(𝑡)𝑑𝑡. In fact, denoting 𝐻_𝑛(𝑠)=∫^𝑠_𝑜ℎ_𝑛(𝑡)𝑑𝑡 we have𝐻_𝑛(𝑠)=^𝑀𝑛_𝑜𝑔(𝑡)𝑑𝑡+^𝑠_𝑀𝑛𝑔(𝑀_𝑛)𝑀^𝑞_𝑛𝑡^𝑞𝑑𝑡, (2.9)for 𝑠>𝑀_𝑛, which implies𝐻_𝑛(𝑠)=^𝑀𝑛_𝑜1𝑔(𝑡)𝑑𝑡+𝑞+1𝑔(𝑀_𝑛)𝑀^𝑞_𝑛[𝑠^𝑞+1−𝑀_𝑛^𝑞+1]. (2.10)Integrating inequality (H₄), from 0 to 𝑀_𝑛, we have^𝑀𝑛_𝑜1𝑔(𝑡)𝑑𝑡≤𝑞+1𝑔(𝑀_𝑛)𝑀^𝑞_𝑛𝑀_𝑛^𝑞+1. (2.11)Consequently, from (2.10) and (2.11), the inequality𝐻_𝑛1(𝑠)≤𝑞+1𝑔(𝑀_𝑛)𝑀^𝑞_𝑛𝑠^𝑞+1 (2.12)holds. Thus,(𝑞+1)𝐻_𝑛(𝑠)−𝑠ℎ_𝑛(𝑠)≤0∀𝑠>𝑀_𝑛, (2.13)which, together with (H₂), implies𝜃𝐹_𝑛(𝑠)−𝑠𝑓_𝑛(𝑠)≤0∀𝑠≥𝑀_𝑛, (2.14)as we wanted to check (observe that (H₂) and (H₃) imply that 𝜃≤𝑞+1).

Now, consider the following truncated problem:−Δ𝑢=𝑧(𝑢)+𝜆ℎ_𝑛(𝑢)inΩ,𝑢>0inΩ,𝑢=0on𝜕Ω. (Pn)

Let us recall that 𝜆≤𝜆_∗ and 𝑀_𝑛>𝑘. It is well known that problem (𝑃_𝑛) has a positive solutions 𝑢∈𝑊_𝑜^1,𝑝(Ω). This occurs since the energy functional related to (𝑃_𝑛) given by𝐽_𝑛1(𝑣)=2_Ω|∇𝑣|²𝑑𝑥−_Ω𝑍(𝑣₊)𝑑𝑥−𝜆_Ω𝐻_𝑛(𝑣)𝑑𝑥 (2.15)or𝐽_𝑛(𝑣)=𝐽_𝑞(𝑣)−𝜆_Ω𝐻_𝑛(𝑣)𝑑𝑥 (2.16)satisfies the hypotheses of the mountain pass theorem [1]. Moreover, from the inequality 𝐽_𝑛(𝑣)≤𝐽_𝑞(𝑣), for all 𝑣∈𝐻¹_𝑜(Ω), the minimax level 𝑏_𝑛 of functional 𝐽_𝑛 satisfies 𝑏_𝑛≤𝑐_𝑞.

Thus, from equality 𝑏_𝑛=𝐽_𝑛(𝑢)=𝐽_𝑛(𝑢)−(1/𝜃)𝐽^′_𝑛(𝑢)𝑢, we have(12−1𝜃)‖𝑢‖²+_Ω(1𝜃𝑢𝑓_𝑛(𝑢)−𝐹_𝑛(𝑢))𝑑𝑥=𝑏_𝑛 (2.17)and from (2.8) we have(12−1𝜃)‖𝑢‖²≤𝜆_{{𝑢≤𝑀𝑛}}|1𝜃𝑢ℎ_𝑛(𝑢)−𝐻_𝑛(𝑢)|𝑑𝑥+𝑏_𝑛. (2.18)From inequality (2.7), we have |(1/𝜃)𝑢ℎ_𝑛(𝑢)−𝐻_𝑛(𝑢)|≤(1/𝜃+1/(𝑞+1))𝑔(𝑀_𝑛)𝑀_𝑛 in {𝑥∈Ω∶𝑢(𝑥)≤𝑀_𝑛}, hence(12−1𝜃)‖𝑢‖²1≤𝜆(𝜃+1𝑞+1)𝑔(𝑀_𝑛)𝑀_𝑛|Ω|+𝑐_𝑞. (2.19)From the choice of 𝜆_∗, we get ‖𝑢‖²≤2𝑐_𝑞(12−1)𝑞+1⁻¹. (2.20)Using the constant 𝐶 obtained in Proposition 2.1 and from the last inequality, we have that 𝑢(𝑥)≤𝑘, for all 𝑥∈Ω. But 𝑀_𝑛>𝑘 implies that 𝑢(𝑥)≤𝑀_𝑛 and thus 𝑢 is a solution for (P). The proof is done.Remark 2.2. Hypothesis (H₄) can be replaced by
(H₄)^'There is a positive real 𝑀>𝑘=𝑘(Ω,𝑞) satisfying𝑔(𝑀)𝑀^𝑞≥𝑔(𝑠)𝑠^𝑞∀𝑠∈[0,𝑀]. (2.21)

3. Final Comments

The methods applied in this paper with few modifications can be used to establish existence of positive solutions for the problem −Δ_𝑝𝑢=𝑧(𝑢)+𝜆𝑔(𝑢)inΩ,𝑢=0on𝜕Ω, (3.1)with 𝑁>𝑝>1 and 𝑧,𝑔 verifying

(H₅)lim_𝑠→0(𝑧(𝑠)/|𝑠|^𝑝−2𝑠)=0.(H₆) There exists 𝜃>𝑝 such that0<𝜃𝑍(𝑠)≤𝑧(𝑠)𝑠,for𝑠≥0, (3.2)where 𝑍(𝑠)=∫^𝑠₀𝑧(𝑡)𝑑𝑡.(H₇) There exists 𝑞∈(𝑝−1,(𝑁(𝑝−1)+𝑝)/(𝑁−𝑝)) such that|𝑧(𝑠)|≤|𝑠|^𝑞∀𝑠∈ℝ. (3.3)(H₈) There is a positive real sequence {𝑀_𝑛} satisfying lim_𝑛→+∞𝑀_𝑛=+∞,𝑔(𝑀_𝑛)>0, and𝑔(𝑀_𝑛)𝑀^𝑞_𝑛≥𝑔(𝑠)𝑠^𝑞∀𝑠∈[0,𝑀_𝑛],∀𝑛∈ℕ. (3.4)

The ideas developed in this paper may be used when we deal with the p-Laplacian operator. The main difference is concerned with the proof of Proposition 2.1, where we should replace bootstrap arguments by Moser's iteration methods, and then repeat the same approach explored by Chabrowski and Yang [6].

Acknowledgments

This work is partially sponsored by CNPq-Milenium-AGIMB. The second author is supported by CNPq - Brazil.