Multiplicity of solutions for a biharmonic equation with subcritical or critical growth

Abstract

We consider the fourth-order problem $$ \left\{ \begin{array}{l} \epsilon^4\Delta^2u + V(x)u = f(u) + \gamma|u|^{2_{**}-2}u \mbox{in $\mathbb{R}^N$}\\ u\in H^2(\mathbb{R}^N), \end{array} \right. $$ where $\epsilon > 0$, $N\geq 5$, $V$ is a positive continuous potential, $f$ is a function with subcritical growth and $\gamma \in \{0,1\}$. We relate the number of solutions with the topology of the set where $V$ attain its minimum values. We consider the subcritical case $\gamma=0$ and the critical case $\gamma=1$. In the proofs we apply Ljusternik-Schnirelmann theory.