Existence and multiplicity for perturbations of an equation involving Hardy inequality and critical Sobolev exponent in the whole R^N

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a perturbed elliptic equation involving Hardy potential and critical Sobolev exponent in the entire space, emphasizing the influence of perturbation functions.

Contribution

It establishes conditions under which solutions exist and are multiple, considering the shape and size of perturbation functions in a Hardy-Sobolev critical problem.

Findings

Solutions exist under specific size and shape conditions of perturbations.

Multiplicity of solutions depends on local behavior of the perturbation near maximum points.

Results extend understanding of Hardy-Sobolev equations with perturbations.

Abstract

In order to obtain solutions to problem $$ {{array}{c} -\Delta u=\dfrac{A+h(x)} {|x|^2}u+k(x)u^{2^*-1}, x\in {\mathbb R}^N, u>0 \hbox{in}{\mathbb R}^N, {and}u\in {\mathcal D}^{1,2}({\mathbb R}^N), {array}. $$ $h$ and $k$ must be chosen taking into account not only the size of some norm but the shape. Moreover, if $h(x)\equiv 0$, to reach multiplicity of solution, some hypotheses about the local behaviour of $k$ close to the points of maximum are needed.