Multiple solutions for a Kirchhoff equation with critical growth

Abstract

We consider the problem

$$\begin{aligned} -m\left( \int \limits _{\varOmega } |\nabla u|^2\mathrm{d}x\right) \Delta u =\lambda f(x,u)+\mu |u|^{2^*-2}u, \quad x \in \varOmega ,\, u \in H_0^1(\varOmega ), \end{aligned}$$

where $\varOmega \subset {\mathbb {R}}^N$, $N \ge 3$, is a bounded smooth domain, $2^*=2N/(N-2)$, $\lambda ,\,\mu >0$ and m is an increasing positive function. The function f is odd in the second variable and has superlinear growth. In our first result we obtain, for each $k \in {\mathbb {N}}$, the existence of k pairs of nonzero solutions for all $\mu >0$ fixed and $\lambda$ large. Under weaker assumptions of f, we also obtain a similar result if $N=3$, $\lambda >0$ is fixed and $\mu$ is close to 0. In the proofs, we apply variational methods.