Semilinear elliptic Schrödinger equations with singular potentials and absorption terms

Abstract

Let Ω ⊂ R N (N ≥ 3) be a C 2 bounded domain and Σ ⊂ Ω be a compact, C 2 submanifold without boundary, of dimensionin Ω \ Σ, where dΣ(x) = dist(x, Σ) and µ is a parameter. We investigate the boundary value problem (P) −Lµu + g(u) = τ in Ω \ Σ with condition u = ν on ∂Ω ∪ Σ, where g : R → 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 −2 Σ , 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).

Poisson problems involving fractional Hardy operators and measures

“…[7,9,11,28]. The topic on elliptic equations has been diversified in different directions, including [18] concerning spectral properties of Hardy potentials with multipolar inverse-square potentials, [10] for semilinear equations with Hardy potentials singular on the boundary, and [15,[25][26][27] for equations involving more general potentials blowing up on a submanifold.…”

Semilinear elliptic equations involving fractional Hardy operators

“…Moreover, when g(u) = |u| p−1 u with p > 1, they gave necessary and sufficient conditions for the existence of a weak solution to (1.2). For semilinear elliptic equations with more general potentials, we refer to [22].…”

Poisson Problems involving fractional Hardy operators and measures

“…Singular solutions to semilinear elliptic equations with Hardy potentials have been studied in many papers; see, e.g., [28,12,10,8]. The topic on elliptic equations has been diversified in different directions, including [18] concerning spectral properties of Hardy potentials with multipolar inverse-square potentials, [11] for semilinear equations with Hardy potentials singular on the boundary, and [15,25,26,27] for equations involving more general potentials blowing up on a submanifold.…”