Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential

TL;DR

This paper investigates the existence and nonexistence of stable and finite Morse index solutions for a nonlinear elliptic equation with Hardy potential, identifying a critical exponent that determines solution behavior.

Contribution

It introduces a new critical exponent $p_c(l,)$ that governs the existence of stable solutions in equations with Hardy potential, providing a comprehensive Liouville-type classification.

Findings

No nontrivial stable solutions for $1<p<p_c(l,)$.

Existence of positive radial stable solutions for $p>p_c(l,)$.

Dependence of solution behavior on the Hardy potential parameter $$.

Abstract

We are concerned with Liouville-type results of stable solutions and finite Morse index solutions for the following nonlinear elliptic equation with Hardy potential: \begin{displaymath} \Delta u+\dfrac{\mu}{|x|^2}u+|x|^l |u|^{p-1}u=0 \qquad \textrm{in}\ \ \Omega, \end{displaymath} where $\Omega=\RN$, $\RN\setminus\{0\}$ for $N\geq3$, $p>1$, $l>-2$ and $\mu<(N-2)^2/4$. Our results depend crucially on a new critical exponent $p=p_c(l,\mu)$ and the parameter $\mu$ in Hardy term. We prove that there exist no nontrivial stable solution and finite Morse index solution for $1<p<p_c(l,\mu)$. We also observe a range of the exponent $p$ larger than $p_c(l,\mu)$ satisfying that our equation admits a positive radial stable solution.