Double phase anisotropic variational problems and combined effects of reaction and absorption terms

Abstract

This paper deals with the existence of multiple solutions for the quasilinear equation −divA(x,∇u)+V(x)|u|α(x)−2u=f(x,u) in RN, which involves a general variable exponent elliptic operator in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like |ξ|q(x)−2ξ for small |ξ| and like |ξ|p(x)−2ξ for large |ξ|, where 1<α(⋅)≤p(⋅)<q(⋅)<N. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works A. Azzollini et al. (2014) [4] and N. Chorfi and V. Rădulescu (2016) [11] from cases where the exponents p and q are constant, to the case where p(⋅) and q(⋅) are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the Cerami compactness condition.