A system with weights and with critical Sobolev exponent

TL;DR

This paper studies a weighted minimization problem involving critical Sobolev exponents, establishing the existence of solutions under certain conditions on weights, dimension, and parameters.

Contribution

It introduces a new weighted minimization framework with critical Sobolev exponents and proves existence results for solutions under specific conditions.

Findings

Existence of solutions under certain weight and dimension conditions

Conditions on the parameter λ ensuring solvability

Extension of Sobolev minimization problems with weights

Abstract

In this paper, we investigate the minimization problem : $$ \inf_{ \displaystyle{\begin{array}{lll} u \in H_0^1(\Omega), v \in H_0^1(\Omega),\\ \quad \| u \|_{L^{q}} =1, \quad \| v \|_{L^{q}} = 1 \end{array}}} \left[ \frac{1}{2} \int_{\Omega} a(x) \vert \nabla u(x) \vert^2dx + \displaystyle{ \frac{1}{2} \int_{\Omega} b(x) \vert \nabla v (x)\vert^2dx } - \lambda \displaystyle{\int_{\Omega} u(x)v (x)dx} \right] $$ where $q=\frac{2N}{N-2}$, $ N \geq 4$, $a$ and $b$ are two continuous positive weight functions. We show the existence of solutions of the previous minimizing problem under some conditions on $a$, $b$, the dimension of the space and the parameter $\lambda$.