MULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH

Abstract

We prove the existence of infinitely many solutions $u\in W_{0}^{1,2}(\unicode[STIX]{x1D6FA})$ for the Kirchhoff equation $$\begin{eqnarray}\displaystyle -\biggl(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{2}\,dx\biggr)\unicode[STIX]{x1D6E5}u=a(x)|u|^{q-1}u+\unicode[STIX]{x1D707}f(x,u)\quad \text{in }\unicode[STIX]{x1D6FA}, & & \displaystyle \nonumber\end{eqnarray}$$ where $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}$ is a bounded smooth domain, $a(x)$ is a (possibly) sign-changing potential, $0<q<1$, $\unicode[STIX]{x1D6FC}>0$, $\unicode[STIX]{x1D6FD}\geq 0$, $\unicode[STIX]{x1D707}>0$ and the function $f$ has arbitrary growth at infinity. In the proof, we apply variational methods together with a truncation argument.