Combined effects in mixed local-nonlocal stationary problems

TL;DR

This paper investigates a mixed local-nonlocal elliptic problem, analyzing existence, regularity, and boundary behavior of solutions, especially considering singular nonlinearities and the interplay of data summability and nonlinear exponents.

Contribution

It introduces a comprehensive analysis of elliptic problems with combined local and nonlocal operators, including regularity and boundary behavior under singular nonlinearities.

Findings

Existence and non-existence conditions identified.

Sobolev regularity results established.

Boundary behavior of solutions characterized.

Abstract

In this work, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlinearity. We investigate existence or non-existence properties, power and exponential type Sobolev regularity results, and the boundary behavior of the weak solution, in the light of the interplay between the summability of the datum and the power exponent in singular nonlinearities.