Multiple positive solutions to elliptic boundary blow-up problems

TL;DR

This paper establishes the existence of multiple positive radial solutions for a class of elliptic boundary blow-up problems with sign-changing weights, using shooting methods and analyzing the influence of the weight's nodal behavior.

Contribution

It introduces a novel shooting-type approach to prove multiple solutions for elliptic problems with sign-changing weights and boundary blow-up conditions.

Findings

Multiple positive solutions are proven to exist.

The number of solutions depends on the nodal behavior of the weight function.

The technique extends to find homoclinic solutions in e space.

Abstract

We prove the existence of multiple positive radial solutions to the sign-indefinite elliptic boundary blow-up problem \[ \left\{\begin{array}{ll} \Delta u + \bigl(a^+(\vert x \vert) - \mu a^-(\vert x \vert)\bigr) g(u) = 0, & \; \vert x \vert < 1, \\ u(x) \to \infty, & \; \vert x \vert \to 1, \end{array} \right. \] where $g$ is a function superlinear at zero and at infinity, $a^+$ and $a^-$ are the positive/negative part, respectively, of a sign-changing function $a$ and $\mu > 0$ is a large parameter. In particular, we show how the number of solutions is affected by the nodal behavior of the weight function $a$. The proof is based on a careful shooting-type argument for the equivalent singular ODE problem. As a further application of this technique, the existence of multiple positive radial homoclinic solutions to $$ \Delta u + \bigl(a^+(\vert x \vert) - \mu a^-(\vert x \vert)\bigr) g(u) = 0, \qquad x \in \mathbb{R}^N, $$ is also considered.