Boundary singularities of solutions of semilinear elliptic equations in the half-space with a Hardy potential

TL;DR

This paper analyzes boundary singularities of solutions to a semilinear elliptic equation with Hardy potential in the half-space, classifying boundary behaviors and constructing solutions with prescribed asymptotics.

Contribution

It provides a detailed classification of boundary behaviors and constructs solutions with specific asymptotics for the nonlinear problem with Hardy potential.

Findings

Classified admissible boundary behaviors of solutions.

Constructed solutions with prescribed boundary asymptotics.

Analyzed linear and nonlinear separable solutions across parameter ranges.

Abstract

We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-\Delta u -\frac{\mu}{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<\mu<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1\to 0$, or is determined by the solutions of the linear problem $-\Delta h -\frac{\mu}{x_1^2}h=0$. In the first part we study in full detail the separable solutions of the linear equations for the whole range of $\mu$. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like $O(x_1^{-2/(p-1)})$ near the origin and which, away from the origin have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like $O(x_1^{-2/(p-1)})$ at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.