Existence and Multiplicity Results for Kirchhoff Problems

Abstract

We consider the following superlinear Kirchhoff problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\Omega }|\nabla u|^{2}\right) \triangle u=f(x,u)&{}\quad \hbox {in}\ \ \Omega ,\ \\ u=0&{}\quad \hbox {on}\ \partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $${\mathbb {R}}^{3}$$ , and $$a,b>0$$ . Inspired by an anti-example, we find a new superlinear growth condition which unifies the known Ambrosetti–Rabinowitz type conditions and get the existence and multiplicity of nontrivial solutions. We also prove the existence of positive solution, negative solution and sign-changing solution to the problem without any symmetry. Our argument depends on the newly established condition and the method of critical point theory in the setting of invariant sets of the descending flows. We improve and extend some recent results in the literature.