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

TL;DR

This paper proves the existence of ground state and least energy sign-changing solutions for a singular p-biharmonic equation involving Hardy potential and critical Hardy-Sobolev exponent in high-dimensional space.

Contribution

It introduces new existence results for solutions to a complex nonlinear PDE with Hardy potential and critical exponent, using variational methods on Nehari manifolds.

Findings

Existence of ground state solutions established.

Existence of least energy sign-changing solutions demonstrated.

Solutions found using minimization on Nehari manifolds.

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*} \Delta_{p}^{2} u -\lambda_{1} \frac{|u|^{p-2}u}{|x|^{2p}}= \frac{|u|^{p_{*}(\alpha)-2}}{|x|^{\alpha}}u+\lambda_{2}\Big(|x|^{-\beta}*|u|^{q}\Big)|u|^{q-2}u \quad\mbox{ in }\R^{N}, \end{equation*} where $p>2$, $2<q< p_{*}(\alpha)$, $\lambda_{1}>0$, $\lambda_{2} \in \R$, $\alpha, \beta \in (0,N)$, $p_{*}(\alpha)=\frac{p(N-\alpha)}{N-2p}$ and $N\geq 5$. 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.