The Existence of Nontrivial Solutions to a Class of Quasilinear Equations

Abstract

In this paper, we study the following quasilinear equation: $-\mathrm{div}\left(\varphi \left(\left|\nabla u\right|\right)\nabla u\right)+\varphi \left(\left|u\right|\right)u=f\left(u\right)\text{in}{ℝ}^N$ , where $\varphi \in C^1\left({ℝ}^{+},{ℝ}^{+}\right)$ and $\Phi \left(t\right)={\int }_0^{t}s\varphi \left(\mid s\mid \right)ds$ . In the Orlicz-Sobolev space, by variational methods and a minimax theorem, we prove the equation has a nontrivial solution.