Kirchhoff type mixed local and nonlocal elliptic problems with concave–convex and Choquard nonlinearities

Abstract

In this paper, making use of non-smooth variational principle, we establish the existence of solution to the following Kirchhoff type mixed local and nonlocal elliptic problem with concave–convex and Choquard nonlinearities $$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}_{a,b}(u)=\left( \int \limits _{\Omega }\frac{|u(y)|^{p}}{|x-y|^{\mu }}dy\right) |u(x)|^{p-2}u(x)+\lambda |u(x)|^{q-2}u(x), &{}\quad x\in \Omega ,\\ ~~~u(x)\ge 0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~&{}\quad x\in \Omega ,\\ ~u(x)=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~&{}\quad x\in \mathbb {R}^{N}\setminus \Omega , \end{array} \right. \end{aligned}$$ where \(\mathcal {L}_{a,b}(u)=-\left( a+b \Vert \nabla u\Vert ^{2(\gamma -1)}_{L^{2}(\Omega )}\right) \Delta u(x)+(-\Delta )^s u(x)\) , \(\gamma \in \left( 1,\frac{N+4s+2}{N-2}\right) \) , \(a>0\) , \(b>0\) are constants, \((-\Delta )^{s}\) is the restricted fractional Laplacian, \(0<s<1\) , \(1<q<2<2p\) , \(0<\mu <N\) . The main contribution of this paper is giving a new supercritical range of \(2p-1\) and \(\gamma \) .