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

TL;DR

This paper proves the existence of solutions for a class of quasilinear elliptic systems on  space under specific growth conditions, showing solutions vanish as a parameter increases, with an improved result for related equations.

Contribution

It introduces new existence results for quasilinear elliptic systems with localized nonlinearities, extending previous work and providing better results for related equations.

Findings

Existence of nontrivial solutions for large mbda

Solutions tend to zero as mbda  

Improved results for related elliptic equations

Abstract

In this paper, we investigate the existence of solutions for a class of quasilinear elliptic system \begin{eqnarray*} \begin{cases}{ccc} -\mbox{div}(\phi_1(|\nabla u|)\nabla u)+V_1(x)\phi_1(|u|)u=\lambda F_u(x, u,v), \ \ x\in \mathbb R^N, -\mbox{div}(\phi_2(|\nabla v|)\nabla v)+V_2(x)\phi_2(|v|)v=\lambda F_v(x, u,v), \ \ x\in \mathbb R^N, u\in W^{1,\Phi_1}(\mathbb R^N), v\in W^{1,\Phi_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 $\|(u_{\lambda},v_{\lambda})\|_{\infty}\le 2$ for every $\lambda$ large enough, and the families of solutions $\{(u_\lambda,v_\lambda)\}$ satisfy that $\|(u_\lambda,v_\lambda)\|\to 0$ 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.