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

Abstract

<abstract><p>In this paper, by the Stampacchia method, we consider the boundedness of positive solutions to the following mixed local and nonlocal quasilinear elliptic operator</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p> <p>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.</p></abstract>