Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential

Abstract

In this paper we study nonexistence of non-negative distributional supersolutions for a class of semilinear elliptic equations involving inverse-square potentials.

Semilinear elliptic equations for the fractional Laplacian with Hardy potential

“…and p ≥ N +2−2β N −2−2β . Some related results and problems are in [6], [7], [16], [18], [19], [24], [45], [22], [23], [21], [8], [9]. Our results in this paper extends the one of Brezis-Dupaigne-Tesei in [7] to the case s ∈ (0, 1).…”

“…The problem of existence and nonexistence of (0.1), for s = 1, was studied by Brezis-Dupaigne-Tesei in [7] where the authors showed that for β ∈ 0, N −2 and p ≥ N +2−2β N −2−2β . Some related results and problems are in [6], [7], [16], [18], [19], [24], [45], [22], [23], [21], [8], [9]. Our results in this paper extends the one of Brezis-Dupaigne-Tesei in [7] to the case s ∈ (0, 1).…”

Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function

“…for k ∈ (1, N − 2). According to [11], for δ < 1 the function δ −A 1/2 k (1−δ 1/2 ) 1 + 1 log δ , A k := N − k − 2 2 2 (B.2) satisfies (B.1). Hence, for any x ∈ Ω r 0 with r 0 ≤ 1 we define v := φz for z ∈ C ∞ 0 (Ω r 0 ); in particular, v ∈ C ∞ 0 (Ω r 0 ) and |∇v| 2 = φ 2 |∇z| 2 + z 2 |∇φ| 2 + 1 2 ∇(φ 2 ) · ∇(z 2 ).…”

“…We split the proof in two parts: firstly, we derive (2.3) in Ω r 0 and, in a second moment, we extend the result to the whole Ω. for k ∈ (1, N − 2). According to [11], for δ < 1 the function…”

Weighted Hardy inequality with higher dimensional singularity on the boundary

“…We would like to mention that some of the results in this paper might of interest in the study of semilinear equations with a Hardy potential singular at a submanifold of the boundary. We refer to [9,2,3], where existence and nonexistence for semilinear problems were studied via the method of super/sub-solutions.…”