Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions

Abstract

Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain. In this paper, we prove a result of which the following is a by-product: Let $q\in ]0,1[$, $\alpha\in L^{\infty}(\Omega)$, with $\alpha>0$, and $k\in {\bf N}$. Then, the problem $$\cases {-\tan\left(\int_{\Omega}|\nabla u(x)|^2dx\right)\Delta u= \alpha(x)u^q & in $\Omega$\cr & \cr u>0 & in $\Omega$\cr & \cr u=0 & on $\partial \Omega$ \cr & \cr (k-1)\pi<\int_{\Omega}|\nabla u(x)|^2dx<(k-1)\pi+{{\pi}\over {2}} \cr}$$ has a unique weak solution $\tilde u$ which is the unique global minimum in $H^1_0(\Omega)$ of the functional $$u\to {{1}\over {2}}\tan\left (\int_{\Omega}|\nabla\tilde u(x)|^2dx\right)\int_{\Omega}|\nabla u(x)|^2dx-{{1}\over {q+1}}\int_{\Omega}\alpha(x)|u^+(x)|^{q+1}dx\ ,$$ where $u^+=\max\{0,u\}$.