Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains

TL;DR

This paper studies the existence and nonexistence of positive (super-)solutions to the singular semilinear elliptic equation $-\\nabla\\cdot(|x|^A\\nabla u)-B|x|^{A-2}u=C|x|^{A-\\sigma}u^p$ on cone-like domains in $\\mathbb{R}^N$ ($N\\ge 2$) for all real parameters $A,B,\\sigma,p$ with $C>0$. By the simple transformation $w(x)=|x|^{A/2}u(x)$ they reduce the problem to a uniformly elliptic equation with a critical potential and shifted power, and then use a blend of barrier constructions, Phragm\\'en-Lindel\\'of arguments, Kelvin transform, and a sharp Hardy inequality to analyze existence and nonexistence. A main achievement is a complete characterization of the no-solution region via a critical line $\\Lambda_*(p,A,B,\\Omega)$ depending on the cross-section eigenvalue, together with an improved Hardy inequality on cone-like domains with optimal constants. They also establish precise asymptotics at infinity for positive solutions in exterior cones, derive sharp two-sided growth rates governed by angular eigenvalues, and separate subcritical, critical, and supercritical regimes for both $p\\ge 1$ and $p<1$, proving nonexistence below threshold lines and existence above via barrier methods.

Abstract

We study the existence and nonexistence of positive (super-)solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\R^N$ ($N\ge 2$), for the full range of parameters $A,B,\sigma,p\in\R$ and $C>0$. We provide a complete characterization of the set of $(p,\sigma)\in\R^2$ such that the equation has no positive (super-)solutions, depending on the values of $A,B$ and the principle Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the Laplace operator with critical potentials, Phragmen--Lindel\"of type comparison arguments and an improved version of Hardy's inequality in cone--like domains.

Figures (1)

• Figure 1: The nonexistence set ${\mathcal{N}}$ of equation \ref{['e:MAIN']} for typical values of $\gamma^-$ and $\gamma^+$.

Theorems & Definitions (76)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Theorem 2.2
• Theorem 3.1
• proof