Infinitely many sign changing solutions of an elliptic problem involving critical Sobolev and Hardy-Sobolev exponent

TL;DR

This paper proves the existence of infinitely many sign-changing solutions for a nonlinear elliptic PDE involving critical Sobolev and Hardy-Sobolev exponents in a bounded domain with specific geometric conditions.

Contribution

It establishes the existence of infinitely many sign-changing solutions for an elliptic problem with critical exponents, considering boundary curvature and Hardy-Sobolev terms, which is a novel extension.

Findings

Proved existence of infinitely many sign-changing solutions.

Analyzed solutions in domains with negative principal curvatures at boundary points.

Addressed the problem involving critical Sobolev and Hardy-Sobolev exponents.

Abstract

We study the existence and multiplicity of sign changing solutions of the following equation $ \begin{cases} -\Delta u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^t}+a(x)u \quad\text{in}\quad \Omega, u=0 \quad\text{on}\quad\partial\Omega, \end{cases} $where $\Omega$ is a bounded domain in $R^N$, $0\in\partial\Omega$, all the principal curvatures of $\partial\Omega$ at $0$ are negative and $\mu\geq 0, \ \ a>0, \ \ N\geq 7, \ \ 0<t<2, \ \ 2^{\star}=\frac{2N}{N-2}$ and $2^{\star}(t)=\frac{2(N-t)}{N-2}$.