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

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 \(\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}_{a,b}(u)=\left( \int \limits _{\Omega }\frac{|u(y)|^{p}}{|x-y|^{\mu }}dy\right) |u(x)|^{p-2}u(x)+\lambda |u(x)|^{q-2}u(x), &{}\quad x\in \Omega ,\\ ~~~u(x)\ge 0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~&{}\quad x\in \Omega ,\\ ~u(x)=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~&{}\quad x\in \mathbb {R}^{N}\setminus \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathcal{L}}_{a,b}(u)=\left(\underset{\mathrm{\Omega }}{\int },\frac{{|u(y)|}^p}{{|x-y|}^{\mu }},d,y\right){|u(x)|}^{p-2}u(x)+\lambda {|u(x)|}^{q-2}u(x),\hfill & x\in \mathrm{\Omega },\hfill \\ u(x)\ge 0,\hfill & x\in \mathrm{\Omega },\hfill \\ u(x)=0,\hfill & x\in {\mathbb{R}}^N\backslash \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\mathcal {L}_{a,b}(u)=-\left( a+b \Vert \nabla u\Vert ^{2(\gamma -1)}_{L^{2}(\Omega )}\right) \Delta u(x)+(-\Delta )^s u(x)\) ${\mathcal{L}}_{a,b}(u)=-\left(a,+,{b\Vert \mathrm{\nabla }u\Vert }_{L^2(\mathrm{\Omega })}^{2(\gamma -1)}\right)\mathrm{\Delta }u(x)+{(-\mathrm{\Delta })}^{s}u(x)$ , \(\gamma \in \left( 1,\frac{N+4s+2}{N-2}\right) \) $\gamma \in \left(1,,,\frac{N+4s+2}{N-2}\right)$ , \(a>0\) $a>0$ , \(b>0\) $b>0$ are constants, \((-\Delta )^{s}\) ${(-\mathrm{\Delta })}^s$ is the restricted fractional Laplacian, \(0<s<1\) $0<s<1$ , \(1<q<2<2p\) $1<q<2<2p$ , \(0<\mu <N\) $0<\mu <N$ . The main contribution of this paper is giving a new supercritical range of \(2p-1\) $2p-1$ and \(\gamma \) $\gamma$ .