Infinitely Many Solutions for a Class of Quasi-linear Elliptic Problem

In this paper, we study the following quasi-linear elliptic equation

\(\left\{{\matrix{{- \,{\rm{div(}}\phi {\rm{(}}\left| {\nabla u} \right|{\rm{)}}\nabla u{\rm{) = \lambda}}\psi {\rm{(}}\left| u \right|{\rm{)}}u + \,\varphi {\rm{(}}\left| u \right|{\rm{)}}u,\,\,\,\,{\rm{in}}\,\,\,\Omega,\,\,\,} \cr {u = 0,\,\,\,\,\,\,\,{\rm{on}}\,\,\partial \Omega {\rm{,}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \cr}} \right.\) $$\{\begin{array}{c}-\mathrm{d}\mathrm{i}\mathrm{v}(\varphi (\mid \mathrm{\nabla }u\mid )\mathrm{\nabla }u)=\lambda \psi (\mid u\mid )u+\phi (\mid u\mid )u,\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=0,\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$$

where Ω ⊂ ℝ^N is a bounded domain, λ > 0 is a parameter. The function ψ(∣t∣)t is the subcritical term, and ϕ(∣t∣)t is the critical Orlicz-Sobolev growth term with respect to φ. Under appropriate conditions on φ, ψ and ϕ, we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation, for λ ∈ (0, λ₀), where λ₀ > 0 is a fixed constant.