Existence results for mixed local and nonlocal elliptic equations involving singularity and nonregular data

Abstract

In this paper, we prove the existence of weak, veryweak and duality solutions to a class of elliptic problems involving singularity and measure data which is given by: $-\Delta u+(-\Delta)^s u = \frac{f(x)}{u^\gamma} +\mu$ in $\Omega$ with the zero Dirichlet boundary data $u=0$ in $\mathbb R^N \setminus \Omega$. The existence of weak solutions is obtained by approximating a sequence of problems for $0<\gamma\leq1$ and $\gamma>1$. We employ Schauder's fixed point theorem and embeddings of Marcinkiewicz spaces. The novelty of our work is that we prove the existence of a duality solution and its equivalence with weak solutions to the problem $\mathcal{L}u=\mu$. Moreover, we prove a veryweak maximum principle and a Kato-type inequality for the mixed local-nonlocal operator $\mathcal{L}=-\Delta +(-\Delta)^s$, which are crucial tools to guarantee the existence of veryweak solutions to the problem. Using a Kato-type inequality, maximum principle together with sub-super solution method, we prove the existence of veryweak solution for $0<\gamma<1$. Our work extends the studies due to Oliva and Petitta [ESAIM Control Optim. Calc. Var., 22(1):289--308, 2016.] and Petitta [Adv. Nonlinear Stud., 16(1):115--124, 2016.] for the mixed local-nonlocal operator.