Positive solutions of nonlinear elliptic equations involving supercritical Sobolev exponents without Ambrosetti and Rabinowitz condition

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 $$0\le f(r,u)\le a_1|u|^{p(r)-1}$$ , if $$u\ge 0$$ , where $$r = |x|$$ , $$p(r ) = 2^* + r^{\alpha }$$ , with $$\alpha > 0$$ , 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 O et al. in Calc Var Partial Differ Equ 55:83, 2016), to establish the existence result.