$L^{\infty}$-estimates for elliptic equations involving critical exponents and convolution

Greta Marino, Dumitru Motreanu

Published 2019-08-04Version 1

In this paper we study the boundedness of weak solutions to a nonlinear boundary value problem with an operator in divergence form, in which the functions are allowed to exhibit a critical growth even on the boundary. The main novel relies in the fact that the nonlinearity exhibits also the presence of the convolution of the solution with an integrable function. Through a modified version of Moser iteration up to the boundary initiated in some works of the first author, we prove that any weak solution to our problem is bounded. Combining this with a result due to the second author, the existence of a bounded solution to the corresponding problem with homogeneous Dirichlet boundary condition is obtained.