Positive solutions of nonlinear elliptic equations involving supercritical Sobolev exponents without Ambrosetti and Rabi
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Calculus of Variations
Positive solutions of nonlinear elliptic equations involving supercritical Sobolev exponents without Ambrosetti and Rabinowitz condition Anderson L. A. de Araujo1 · Luiz F. O. Faria2 · Jéssyca L. F. Melo Gurjão1 Received: 2 October 2019 / Accepted: 30 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The purpose of the present paper is to study a class of semilinear elliptic Dirichlet boundary value problems in the ball, where the nonlinearities involve the sum of a sublinear variable exponent and a superlinear (may be supercritical) variable exponents of the form 0 ≤ f (r , u) ≤ a1 |u| p(r )−1 , if u ≥ 0, where r = |x|, p(r ) = 2∗ + r α , with α > 0, and 2∗ = 2N /(N − 2) is the critical Sobolev embedding exponent. We do not impose the Ambrosetti–Rabinowitz condition on the nonlinearity (or some additional conditions) to obtain Palais–Smale or Cerami compactness condition. We employ techniques based on the Galerkin approximations scheme, combining with a Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces (due to do Ó et al. in Calc Var Partial Differ Equ 55:83, 2016), to establish the existence result. Mathematics Subject Classification 35J62 · 37L65 · 35B33
Communicated by P.Rabinowitz. A.L.A de Araujo and J. L. F. Melo Gurjão were partially supported by FAPEMIG/FORTIS, L. F.O Faria was partially supported by FAPEMIG CEX APQ 02374/17.
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Anderson L. A. de Araujo [email protected] Luiz F. O. Faria [email protected] Jéssyca L. F. Melo Gurjão [email protected]
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Departamento de Matemática, UFV, CCE, Viçosa, MG 36570-900, Brazil
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Departamento de Matemática, UFJF, ICE, Juiz de Fora, MG 36036-330, Brazil 0123456789().: V,-vol
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A. L. A. de Araujo et al.
1 Introduction and main results Consider the following boundary value problem −Δu = |u| p−2 u in Ω u=0 on ∂Ω,
(1)
where Ω is a smooth bounded domain of R N (N ≥ 3), and p > 1. This problem plays a very important role in Nonlinear Analysis and has been the main subject of investigation in many works in the last thirty years. If 1 < p < 2N /(N − 2), problem (1) can be studied by using variational methods, thanks to the compactness of Sobolev embedding. When p = 2∗ = 2N /(N − 2), this problem has no positive solution provided the domain is star-shaped. However, in the seminal paper of Brézis and Nirenberg [6], the authors showed that with small linear perturbations the problem can provide positive solutions. A typical ∗ example considered in [6] is −Δu = λu + |u|2 −2 u. They proved that a nontrivial bounded positive solution only exists for 0 < λ < λ1 , N > 3, and 0 < λ∗ < λ < λ1 , N = 3, still by variational arguments. After that, in [3], Ambrosetti, Brézis and Cerami proved some results on existence and multiplicity of solutions for a sublinear perturbation of the critical power (see also [4]). More precisely, they studied the existence of positive solutions for the following problem −Δu = λ|u|q−2 u + |u| p−2 u in Ω (2) u=0 on ∂Ω, where 1
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