Multiple solutions for critical quasilinear elliptic equations

Abstract

We study the existence of infinitely many solutions for the following quasilinear elliptic equations with critical growth: where $$ b_{ij}\in C^{1}(\mathbb {R},\mathbb {R})$$ satisfies the growth condition $$|b_{ij}(t)|\sim |t|^{2s-2}$$ at infinity, $$s\ge 1$$ , $$\Omega \subset \mathbb {R}^N$$ is an open bounded domain with smooth boundary, a is a constant. Here we use the notations: $$D_i=\frac{\partial }{\partial x_i}, b'_{ij}(t)=\frac{db_{ij}(t)}{dt}.$$ We will study the effect of the terms $$a|v|^{2s-2}v$$ and $$b_{ij}(v)$$ on the existence of an unbounded sequence of solutions for (P). Here, we do not assume the crucial global monotone condition. We overcome the difficulties caused by the lack of such monotone condition by performing various kinds of changes of variables.