Infinite multiplicity of positive solutions of an inhomogeneous supercritical elliptic equation on $\mathbb{R}^N$

TL;DR

This paper proves the existence of infinitely many positive solutions for a class of inhomogeneous supercritical elliptic equations on br^N, revealing a rich solution structure under specific conditions on the coefficients and exponents.

Contribution

It establishes a link between the existence of singular solutions and infinitely many bounded solutions for supercritical elliptic equations with inhomogeneous terms.

Findings

Existence of a unique positive radial singular solution under certain conditions.

Infinitely many positive bounded solutions exist between critical exponents.

Solutions are not uniformly bounded, indicating complex solution behavior.

Abstract

We are concerned with positive radial solutions of the inhomogeneous elliptic equation $\Delta u+K(|x|)u^p+\mu f(|x|)=0$ on $\mathbb{R}^N$, where $N\ge 3$, $\mu>0$ and $K$ and $f$ are nonnegative nontrivial functions. If $K(r)\sim r^{\alpha}$, $\alpha>-2$, near $r=0$, $K(r)\sim r^{\beta}$, $\beta>-2$, near $r=\infty$ and certain assumptions on $f$ are imposed, then the problem has a unique positive radial singular solution for a certain range of $\mu$. We show that existence of a positive radial singular solution is equivalent to existence of infinitely many positive bounded solutions which are not uniformly bounded, if $p$ is between the critical Sobolev exponent $p_S(\alpha)$ and Joseph-Lundgren exponent $p_{JL}(\alpha)$. Using these theorems, we establish existence of infinitely many positive bounded solutions which are not uniformly bounded, for $p_S(\alpha)<p<p_{JL}(\alpha)$ if $K(r)=r^{-\alpha}$, $\alpha>-2$.