Brezis Nirenberg type results for local non-local problems under mixed boundary conditions

Abstract

<p>In this paper, we are concerned with an elliptic problem with mixed Dirichlet and Neumann boundary conditions that involve a mixed operator (i.e., the combination of classical Laplace operator and fractional Laplace operator) and critical nonlinearity. Also, we focus on identifying the optimal constant in the mixed Sobolev inequality, which we show is never achieved. Furthermore, by using variational methods, we provide an existence and nonexistence theory for both linear and superlinear perturbation cases.</p>