Uniform boundedness results of solutions to mixed local and nonlocal elliptic operator

Abstract

In this paper, by the Stampacchia method, we consider the boundedness of positive solutions to the following mixed local and nonlocal quasilinear elliptic operator $ \begin{align*} \left\{\begin{array}{rl} -\Delta_{p}u+(-\Delta)_{p}^su = f(x)u^{\gamma},&x\in\Omega,\\ u = 0,\; \; \; \; \; \; \; \; &x\in \mathbb{R}^{N}\setminus\Omega, \end{array} \right. \end{align*} $ where $ s\in(0, 1) $, $ 1 < p < N $, $ f\in L^{m}(\Omega) $ with $ m > \frac{Np}{p(s+p-1)-\gamma(N-sp)} $, $ 0\leqslant\gamma < p_s^*-1 $, $ p_s^{*} = \frac{Np}{N-sp} $ is the critical Sobolev exponent.

Keywords

• mixed local and nonlocal operator
• bioundedness
• stampacchia method