Infinitely many solutions for quasilinear elliptic equations without Ambrosetti-Rabinowitz condition and lack of symmetry

Abstract

In this paper, we consider the existence of solutions for the quasilinear elliptic problem:(P1){−div(A(x,u)∇u)+12At(x,u)|∇u|2=g(x,u)+h(x),inΩ,u=0,on∂Ω, where Ω⊂RN is an open bounded domain, N≥3, the real term A(x,t), At(x,t)=∂A∂t(x,t) and g(x,t) satisfy Carathéodory condition on Ω×R and h:Ω→R is a given measurable function. The intention of the article is to get new results of the existence of infinitely many weak solutions of the problem by weaken the Ambrosetti and Rabinowitz condition. We use a variant of perturbation techniques introduced by Rabinowitz (1982) to overcome the lack of symmetry. This extends the previous results.