Nonlinear elliptic boundary value problems at resonance with nonlinear Wentzell boundary conditions

TL;DR

This paper investigates the solvability, uniqueness, and boundedness of solutions for nonlinear elliptic boundary value problems with nonlinear Wentzell boundary conditions involving the q-Laplace-Beltrami operator, providing explicit estimates.

Contribution

It establishes necessary and sufficient conditions for solvability and analyzes existence, uniqueness, and boundedness of solutions in this class of nonlinear problems.

Findings

Derived solvability conditions for nonlinear elliptic problems with Wentzell boundary conditions.

Proved existence and uniqueness of weak solutions.

Provided explicit bounds and continuous dependence estimates.

Abstract

In the first part of the article, we give necessary and sufficient conditions for the solvability of a class of nonlinear elliptic boundary value problems with nonlinear boundary conditions involving the q-Laplace-Beltrami operator. In the second part, we give some additional results on existence, uniqueness and continuous dependence estimates of the weak solutions for these classes of nonlinear problems. Finally, we show that these solutions are also bounded and provide explicit estimates.