Existence of positive ground solutions for biharmonic equations via Pohožaev-Nehari manifold

Abstract

We investigate the following nonlinear biharmonic equations with pure power nonlinearities: \begin{equation*} \begin{cases} \triangle^2u-\triangle u+V(x)u= u^{p-1}u & \text{in } \mathbb{R}^N,\\ u>0 &\text{for } u\in H^2(\mathbb{R}^N), \end{cases} \end{equation*} where $2 < p< 2^*={2N}/({N-4})$. Under some suitable assumptions on $V(x)$, we obtain the existence of ground state solutions. The proof relies on the Pohožaev-Nehari manifold, the monotonic trick and the global compactness lemma, which is possibly different to other papers on this problem. Some recent results are extended.