Nontrivial solutions of a class of nonlocal problems via local linking theory

Abstract

The existence of nontrivial solutions of Kirchhoff type equations is an important nonlocal quasilinear problem. In this paper, by using the local linking theory, we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems as follows:

$$\{\begin{array}{cc}-(a+b{\int }_{\Omega }{\left|\nabla u\right|}^2)\Delta u=f(x,u)\text{,}\hfill & \text{in }\Omega \text{;}\hfill \\ u=0\text{,}\hfill & \text{on }\partial \Omega \text{.}\hfill \end{array}$$