Existence of solutions for a quasilinear elliptic system with local nonlinearity on ${\mathbb{R}}^N$

Abstract

In this paper, we investigate the existence of solutions for a class of quasilinear elliptic system \begin{eqnarray*} \begin{cases}{ccc} -\mbox{div}(ϕ_1(|\nabla u|)\nabla u)+V_1(x)ϕ_1(|u|)u=λF_u(x, u,v), \ \ x\in \mathbb R^N, -\mbox{div}(ϕ_2(|\nabla v|)\nabla v)+V_2(x)ϕ_2(|v|)v=λF_v(x, u,v), \ \ x\in \mathbb R^N, u\in W^{1,Φ_1}(\mathbb R^N), v\in W^{1,Φ_2}(\mathbb R^N), \end{cases} \end{eqnarray*} where $N\ge 2$, ${\inf }_{{\mathbb{R}}^N}{V}_i(x)>0,i=1,2$, and $\lambda >0$. We obtain that when the nonlinear term $F$ satisfies some growth conditions only in a circle with center $0$ and radius $4$, system has a nontrivial solution $(u_{\lambda },v_{\lambda })$ with $\Vert (u_{\lambda },v_{\lambda }){\Vert }_{\infty }\le 2$ for every $\lambda$ large enough, and the families of solutions $\{(u_{\lambda },v_{\lambda })\}$ satisfy that $\Vert (u_{\lambda },v_{\lambda })\Vert \to 0$ as $\lambda \to \infty$. Moreover, a corresponding result for a quasilinear elliptic equation is also obtained, which is better than the result for the elliptic system.

References in corpus (1)

• Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces