Semilinear elliptic Schrödinger equations with singular potentials and absorption terms,Journal of the London Mathematical Society

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Semilinear elliptic Schrödinger equations with singular potentials and absorption terms
Journal of the London Mathematical Society ( IF 1.2 ) Pub Date : 2023-12-19 , DOI: 10.1112/jlms.12844
Konstantinos T. Gkikas _{1,

2} , Phuoc‐Tai Nguyen ₃

Affiliation

Let

Ω \subset R^{N} $\Omega \subset \mathbb {R}^N$

(

N ⩾ 3 $N \geqslant 3$

) be a

C^{2} $C^2$

bounded domain and

Σ \subset Ω $\Sigma \subset \Omega$

be a compact,

C^{2} $C^2$

submanifold without boundary, of dimension

k $k$

with

0 ⩽ k < N - 2 $0\leqslant k < N-2$

. Put

L_{μ} = Δ + μ d_{Σ}^{- 2} $L_\mu = \Delta + \mu d_\Sigma ^{-2}$

Ω ∖ Σ $\Omega \setminus \Sigma$

, where

d_{Σ} (x) = dist (x, Σ) $d_\Sigma (x) = \mathrm{dist}(x,\Sigma)$

and

μ $\mu$

is a parameter. We investigate the boundary value problem (P)

- L_{μ} u + g (u) = τ $-L_\mu u + g(u) = \tau$

Ω ∖ Σ $\Omega \setminus \Sigma$

with condition

u = ν $u=\nu$

\partial Ω \cup Σ $\partial \Omega \cup \Sigma$

, where

g : R \to R $g: \mathbb {R} \rightarrow \mathbb {R}$

is a nondecreasing, continuous function, and

τ $\tau$

and

ν $\nu$

are positive measures. The complex interplay between the competing effects of the inverse-square potential

d_{Σ}^{- 2} $d_\Sigma ^{-2}$

, the absorption term

g (u) $g(u)$

and the measure data

τ, ν $\tau,\nu$

discloses different scenarios in which problem (P) is solvable. We provide sharp conditions on the growth of

g $g$

for the existence of solutions. When

g $g$

is a power function, namely

g (u) = {| u |}^{p - 1} u $g(u)=|u|^{p-1}u$

with

p > 1 $p>1$

, we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e.

p $p$

is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e.

p $p$

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).

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