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

TL;DR

This paper establishes the existence of various solutions for a class of elliptic equations involving both local and nonlocal operators, singularities, and measure data, using fixed point theorems and inequalities.

Contribution

It introduces the concept of duality solutions for mixed local-nonlocal elliptic problems and proves their equivalence with weak solutions, extending previous studies.

Findings

Existence of weak, veryweak, and duality solutions for the problem.

Proved a veryweak maximum principle and a Kato-type inequality for the mixed operator.

Extended prior work on local-nonlocal elliptic equations to include singularities and measure 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.