Existence results for superlinear elliptic equations with nonlinear boundary value conditions

TL;DR

This paper establishes the existence of solutions for superlinear elliptic equations with nonlinear boundary conditions across various criticality cases, expanding understanding of such boundary value problems in bounded domains.

Contribution

It provides new existence results for superlinear elliptic equations with nonlinear boundary conditions under four different criticality scenarios.

Findings

Existence results for subcritical q and r

Existence results when r is critical and q is subcritical

Existence results when r is subcritical and q is critical

Abstract

In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition $$ \left\{ \begin{array}{ll} -\Delta u+u=|u|^{r-2}u &\text{in} \; \Omega,\\ \\ \frac{\partial u}{\partial \nu}=|u|^{q-2}u &\text{on}\;\partial\Omega, \end{array} \right. $$ where $\Omega\subset \mathbb R^N, N\geq 3$ is a bounded domain with smooth boundary. We will prove the existence results for the above equation under four different cases: (i) Both $q$ and $r$ are subcritical; (ii) $r$ is critical and $q$ is subcritical; (iii) $r$ is subcritical and $q$ is critical; (iv) Both $q$ and $r$ are critical.