Existence of solutions to elliptic equation with mixed local and nonlocal operators

Abstract

In this paper, making use of a new non-smooth variational approach established by Moameni[13,14,15,16], we establish the existence of solutions to the following mixed local and nonlocal elliptic problem $ \begin{equation*} \begin{cases} -\Delta u+(-\Delta)^s u = \mu g(x,u)+b(x), &x\in\Omega,\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; u\geq0,\; \; \; \; \; &x\in\Omega,\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; u = 0,\; \; \; \; \; &x\in\mathbb{R}^{N}\setminus\Omega, \end{cases} \end{equation*} $ where $ \Omega \subset \mathbb{R}^{N} $ is a bounded smooth domain, $ (-\Delta)^{s} $ is the restricted fractional Laplacian, $ \mu > 0 $, $ 0 < s < 1 $, $ N > 2s $, $ g $ satisfies some growth condition and $ b(x)\in L^m(\Omega) $ for $ m\geq 2 $. The interesting feature of our work is that we show that the nonlocal operator has an important influence in the existence of solutions to the above equation since $ g $ has new growth condition.

Keywords

• non-smooth variational approach
• mixed local and nonlocal operators