Elliptic problem involving finite many critical exponents in $\mathbb{R}^{N}$

TL;DR

This paper proves the existence of nontrivial solutions for a nonlinear elliptic PDE involving multiple critical exponents and Hardy potential in ^N, using advanced variational methods and refined inequalities.

Contribution

It introduces a new approach to handle multiple critical exponents in elliptic equations with Hardy potential, extending previous results to more complex nonlinearities.

Findings

Existence of nontrivial solutions established.

Generalization of previous results by Yang and Wu.

Application of Coulomb--Sobolev space and endpoint refined Sobolev inequality.

Abstract

In this paper, we consider the following problem $$ -\Delta u -\zeta \frac{u}{|x|^{2}} = \sum_{i=1}^{k} \left( \int_{\mathbb{R}^{N}} \frac{|u|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}} \mathrm{d}y \right) |u|^{2^{*}_{\alpha_{i}}-2}u + |u|^{2^{*}-2}u , \mathrm{~in~} \mathbb{R}^{N}, $$ where $N\geqslant3$, $\zeta\in(0,\frac{(N-2)^{2}}{4})$, $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent, and $2^{*}_{\alpha_{i}}=\frac{2N-\alpha_{i}}{N-2}$ ($i=1,\ldots,k$) are the critical Hardy--Littlewood--Sobolev upper exponents. The parameters $\alpha_{i}$ ($i=1,\ldots,k$) satisfy some suitable assumptions. By using Coulomb--Sobolev space, endpoint refined Sobolev inequality and variational methods, we establish the existence of nontrivial solutions. Our result generalizes the result obtained by Yang and Wu [Adv. Nonlinear Stud. (2017)].