System with weights and with critical Sobolev exponent

Asma Benhamida, Rejeb Hadiji

Published 2021-10-27, updated 2022-03-02Version 2

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