Existence of solutions to a perturbed critical biharmonic equation with Hardy potential

TL;DR

This paper proves the existence of solutions for a critical biharmonic equation with Hardy potential using variational methods, specifically the Mountain Pass Lemma, under certain conditions on the nonlinearity.

Contribution

It establishes the existence of at least one mountain pass solution for a perturbed critical biharmonic problem involving Hardy potential, expanding understanding of such elliptic equations.

Findings

Existence of at least one solution under certain conditions.

Application of fibering maps and Mountain Pass Lemma.

Solution existence depends on parameter λ and growth conditions.

Abstract

\ In this paper, the following biharmonic elliptic problem \begin{eqnarray*} \begin{cases} \Delta^2u-\lambda\frac{|u|^{q-2}u}{|x|^s}=|u|^{2^{**}-2}u+ f(x,u), &x\in\Omega,\\ u=\dfrac{\partial u}{\partial n}=0, &x\in\partial\Omega \end{cases} \end{eqnarray*} is considered. The main feature of the equation is that it involves a Hardy term and a nonlinearity with critical Sobolev exponent. By combining a careful analysis of the fibering maps of the energy functional associated with the problem with the Mountain Pass Lemma, it is shown, for some positive parameter $\lambda$ depending on $s$ and $q$, that the problem admits at least one mountain pass type solution under appropriate growth conditions on the nonlinearity $f(x,u)$.