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

Abstract

Let Ω⊂Rn\Omega \subset {{\bf{R}}}^{n} be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let q∈]0,1[q\in ]0,1{[}, α∈L∞(Ω)\alpha \in {L}^{\infty }\left(\Omega ), with α>0\alpha \gt 0, and k∈Nk\in {\bf{N}}. Then, the problem −tan∫Ω∣∇u(x)∣2dxΔu=α(x)uqinΩu>0inΩu=0on∂Ω(k−1)π<∫Ω∣∇u(x)∣2dx<(k−1)π+π2\left\{\begin{array}{ll}-\tan \left(\mathop{\displaystyle \int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x\right)\Delta u=\alpha \left(x){u}^{q}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \\ u\gt 0\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \\ u=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega \\ \left(k-1)\pi \lt \mathop{\displaystyle \int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x\lt \left(k-1)\pi +\frac{\pi }{2}\hspace{1.0em}\end{array}\right. has a unique weak solution u˜\tilde{u}, which is the unique global minimum in H01(Ω){H}_{0}^{1}\left(\Omega ) of the functional u→12tan∫Ω∣∇u˜(x)∣2dx∫Ω∣∇u(x)∣2dx−1q+1∫Ωα(x)∣u+(x)∣q+1dx,u\to \frac{1}{2}\tan \left(\mathop{\int }\limits_{\Omega }| \nabla \tilde{u}\left(x){| }^{2}{\rm{d}}x\right)\mathop{\int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x-\frac{1}{q+1}\mathop{\int }\limits_{\Omega }\alpha \left(x)| {u}^{+}\left(x){| }^{q+1}{\rm{d}}x, where u+=max{0,u}{u}^{+}=\max \left\{0,u\right\}.

Keywords

• discontinuous kirchhoff function
• existence
• uniqueness
• localization
• minimization property
• 35j15
• 35j25
• 35j61
• 49j35