Nonlocal critical Kirchhoff problems in high dimension
Giovanni Anello · Electronic Journal of Differential Equations · 2025
We study the nonlocal critical Kirchhoff problem $$\displaylines{ -\Big(a+b\int_\Omega | abla u|^2dx\Big)\Delta u =|u|^{2^*-2}u +\lambda f(x,u), \quad \text{in } \Omega,\cr u=0, \quad \text{on } \partial\Omega, }$$ where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), \(N>4\), \(a,b>0\), \(\lambda\in \mathbb{R}\), \(2^*:=\frac{2N}{N-2}\) is the critical exponent for the Sobolev embedding, and \(f:\Omega\times \mathbb{R}\to \mathbb{R}\) is a Caratheodory function with subcritical growth. We establish the existence of global minimizers for the energy functional associated to this problem. In particular, we improve a recent result proved by Faraci and Silva [3] under more strict conditions on the nonlinearity \(f\) and under additional conditions on \(a\) and \(b\). For more information see https://ejde.math.txstate.edu/Volumes/2025/46/abstr.html