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

Let $\mathrm{\Omega }\subset {\mathbf{R}}^n$ be a smooth bounded domain. In this paper, we prove a result of which the following is a by-product: Let $q\in ]0,1[$, $\alpha \in L^{\mathrm{\infty }}(\mathrm{\Omega })$, with $\alpha >0$, and $k\in \mathbf{N}$. Then, the problem \cases {-\tan\left(\int_Ω|\nabla u(x)|^2dx\right)Δu= α(x)u^q & in $Ω$\cr & \cr u>0 & in $Ω$\cr & \cr u=0 & on $\partial Ω$ \cr & \cr (k-1)π<\int_Ω|\nabla u(x)|^2dx<(k-1)π+{π\over {2}} \cr} has a unique weak solution $\tilde{u}$ which is the unique global minimum in $H_0^1(\mathrm{\Omega })$ of the functional $$u\to \frac{1}{2}\tan ({\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }\tilde{u}(x){\mathrm{\mid }}^{2}dx){\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u(x){\mathrm{\mid }}^{2}dx-\frac{1}{q+1}{\int }_{\mathrm{\Omega }}\alpha (x)\mathrm{\mid }{u}^{+}(x){\mathrm{\mid }}^{q+1}dx,$$ where $u^{+}=\max \{0,u\}$.