A nonlinear problem witha weight and a nonvanishing boundary datum

TL;DR

This paper proves the existence of minimizers for a nonlinear weighted variational problem with nonvanishing boundary data, analyzing cases based on the weight's behavior and boundary conditions.

Contribution

It introduces a novel approach to establish minimizer existence for a weighted nonlinear problem with non-zero boundary data, distinguishing cases by weight behavior.

Findings

Existence of minimizers is proven for the problem.

Different methods are used depending on the weight's behavior.

The boundary datum being non-zero is crucial for the analysis.

Abstract

We consider the problem: $$\inf_{{u}\in {H}^{1}_{g}(\Omega),\|u\|_{q}=1} \int_{\Omega}{p(x)}|\nabla{u(x)}|^{2}dx-\lambda\int_{\Omega}| u(x)|^{2}dx$$ where $\Omega$ is a bounded domain in $\R^{n}$, ${n}\geq{4}$, $ p : \bar{\Omega}\longrightarrow \R$ is a given positive weight such that $p\in H^{1}(\Omega)\cap C(\bar{\Omega})$, $0< c_1 \leq p(x) \leq c_2$, $\lambda$ is a real constant and $q=\frac{2n}{n-2}$ and $g$ a given positive boundary data. The goal of this present paper is to show that minimizers do exist. We distinguish two cases, the first is solved by a convex argument while the second is not so straightforward and will be treated using the behavior of the weight near its minimum and the fact that the boundary datum is not zero.