Existence results for singular p-biharmonic problem with Hardy potential and critical Hardy-Sobolev exponent

Abstract

In this article, we consider the singular $p-$biharmonic problem involving Hardy potential and citical Hardy-Sobolev exponent. We study the existence of ground state solutions and least energy sign-changing solutions of the following problem \begin{equation*} Δ_{p}^{2} u -λ_{1} \frac{|u|^{p-2}u}{|x|^{2p}}= \frac{|u|^{p_{*}(α)-2}}{|x|^α}u+λ_{2}\Big(|x|^{-β}*|u|^{q}\Big)|u|^{q-2}u \quad\mbox{ in }\R^{N}, \end{equation*} where $p>2$, $2<q<p_{\ast }(\hat{I}\pm )$, $\hat{I}{»}_1>0$, $\hat{I}{»}_2\in \mathbb{R}$, $\hat{I}\pm ,{\hat{I}}^2\in (0,N)$, $p_{\ast }(\hat{I}\pm )=\frac{p(N-\hat{I}\pm )}{N-2p}$ and $N\ge 5$. Firstly, we study existence of ground state solutions by using the minimization method on the associated Nehari manifold. Then, we investigate the least energy sign-changing solutions by considering the Nehari nodal set.

19 pages