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

TL;DR

This work analyzes radial positive solutions of the inhomogeneous elliptic equation $\Delta u + K(r)u^p + \mu f(r) = 0$ on $\mathbb{R}^N$, $N\ge3$, with power-type behavior of the coefficients and forcing. Using an Emden–Fowler transform and a detailed study of a singular solution, the authors connect the existence of a singular solution to the existence of infinitely many positive bounded solutions that are not uniformly bounded in the supercritical regime $p_S(\alpha) < p < p_{JL}(\alpha)$. In particular, for $K(r)=r^{\alpha}$ with $\alpha>-2$, they prove the existence of infinitely many such bounded solutions when $p_S(\alpha) < p < p_{JL}(\alpha)$, by showing that large regular solutions converge to a singular profile and that the intersection number with the singular solution tends to infinity as the initial data grows. They further classify the possible asymptotic behaviors of singular solutions in terms of fast-decay and slow-decay regimes, and establish finiteness/open-ness properties for the sets of forcing parameters yielding various singular behaviors, providing a comprehensive multiplicity picture in the inhomogeneous, supercritical setting.

Abstract

We are concerned with positive radial solutions of the inhomogeneous elliptic equation $Δu+K(|x|)u^p+μf(|x|)=0$ on $\mathbb{R}^N$, where $N\ge 3$, $μ>0$ and $K$ and $f$ are nonnegative nontrivial functions. If $K(r)\sim r^α$, $α>-2$, near $r=0$, $K(r)\sim r^β$, $β>-2$, 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 $μ$. 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 $p_S(α)$ and Joseph-Lundgren exponent $p_{JL}(α)$. Using these theorems, we establish existence of infinitely many positive bounded solutions which are not uniformly bounded, for $p_S(α)<p<p_{JL}(α)$ if $K(r)=r^{-α}$, $α>-2$.

Theorems & Definitions (51)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Example 1.6
• Corollary 1.7
• proof
• Corollary 1.8
• Theorem 1.9