Infinitely many solutions for quasilinear elliptic equations with lack of symmetry

In this paper we look for weak solutions of the quasilinear elliptic model problem −div(A(x,u)∇u)+12At(x,u)|∇u|2=g(x,u)+h(x)in Ω,u=0on ∂Ω,where Ω⊂RN is a bounded domain, N≥2, the real terms A(x,t), At(x,t)=∂A∂t(x,t) and g(x,t) are Carathéodory functions on Ω×R and h:Ω→R is a given measurable map. We prove that, even if At(x,t)≢0, under suitable assumptions infinitely many solutions exist in spite of the lack of symmetry. A suitable supercritical growth is allowed for the nonlinear term g(x,t). We use a variant of the variational perturbation techniques introduced by Rabinowitz in Rabinowitz (1982) but by means of a weak version of the Cerami–Palais–Smale condition.