On a class of intermediate local-nonlocal elliptic problems

Abstract

This paper is concerned with the existence of solutions for a class of intermediate local-nonlocal boundary value problems of the following type: \begin{equation} -{\rm div} \bigg[a\bigg(\mathop{\,\rlap{-}\!\!\int}\nolimits_{\Omega (x,r)}u(y)\,dy\bigg)\nabla u\bigg] = f(x,u,\nabla u ) \quad \mbox{in } \Omega, \ u\in H_{0}^{1}(\Omega ), \tag{${\rm IP}$} \end{equation} where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$, $a\colon\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, $f\colon \Omega \times \mathbb{R} \times \mathbb{R}^{N}$ is a given function, $r>0$ is a fixed number, $\Omega (x,r)=\Omega \cap B(x,r)$, where $B(x,r)=\{ y\in \mathbb{R}^{N}: |y-x|<r\}$. Here $|\,\cdot\, |$ is the Euclidian norm, $$ \mathop{\,\rlap{-}\!\!\int}\nolimits_{\Omega (x,r)}u(y)\,dy=\frac{1}{{\rm meas}\hspace{.06em}(\Omega (x,r))}\int_{\Omega (x,r)}u(y)\,dy $$ and ${\rm meas}\hspace{.06em}(X)$ denotes the Lebesgue measure of a measurable set $X\subset \mathbb{R}^{N}$.