Semilinear elliptic Schrödinger equations with singular potentials and absorption terms

Abstract

Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $Σ\subset Ω$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_μ= Δ+ μd_Σ^{-2}$ in $Ω\setminus Σ$, where $d_Σ(x) = \mathrm{dist}(x,Σ)$ and $μ$ is a parameter. We investigate the boundary value problem (P) $-L_μu + g(u) = τ$ in $Ω\setminus Σ$ with condition $u=ν$ on $\partial Ω\cup Σ$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function, and $τ$ and $ν$ are positive measures. The complex interplay between the competing effects of the inverse-square potential $d_Σ^{-2}$, the absorption term $g(u)$ and the measure data $τ,ν$ discloses different scenarios in which problem (P) is solvable. We provide sharp conditions on the growth of $g$ for the existence of solutions. When $g$ is a power function, namely $g(u)=|u|^{p-1}u$ with $p>1$, we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e. $p$ is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e. $p$ is greater than a critical exponent). Finally, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities for the solvability of (P).

Theorems & Definitions (57)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
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• Theorem 1.8
• Definition 1.9
• Theorem 1.10