Boundary singularities of solutions of semilinear elliptic equations in the half-space with a Hardy potential

Let Ω ⊂ R N be a bounded domain and δ(x) be the distance of a point x ∈ Ω to the boundary. We study the positive solutions of the problem ∆u + µ δ(x) 2 u = u p in Ω, where p > 0, p = 1 and µ ∈ R, µ = 0 is smaller then the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper we first give the complete picture of the radial solutions in balls. In particular we establish for p > 1 the existence of a unique large solution behaving like δ − 2 p−1 at the boundary. In general domains we extend results of [4] and show that there exists a unique singular solutions u such that u/δ β − → c on the boundary for an arbitrary positive function c ∈ C 2+γ (∂Ω) (γ ∈ (0, 1)), c ≥ 0. Here β − is the smaller root of β(β − 1) + µ = 0.

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“…It was shown in [3] that for p > 1 and μ < −μ * problem (1.1) has no nontrivial solutions. Related results are found in [2,5,6,8].…”

Section: Introductionsupporting

confidence: 72%

Positive solutions of semilinear elliptic problems with a Hardy potential

Bandle

Pozio

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Further results on (1.3) can be found in [6,7] and the references therein. When Ω is a half space in (1.3), results of similar nature have been obtained in [1,9] recently.…”

Section: Introductionsupporting

confidence: 65%

Positive solutions of elliptic equations with a strong singular potential

Lei

2018

Bull. London Math. Soc.

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In this paper, we study positive solutions of the elliptic equationis often called a Hardy potential, and the equation in this case has been extensively investigated. Here we consider the case α > 2, which gives a stronger singularity than the Hardy potential near ∂Ω. We show that when λ < 0, the equation has no positive solution, while when λ > 0, the equation has a unique positive solution, and it satisfies

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“…They also provide a framework for the study of positive solutions of (1.1) that may blow up at some parts of the boundary. The existence of such solutions in the subcritical case has been studied (by different methods) in [5]. Corresponding boundary value problems -including a study of solutions with strong isolated singularities -will be presented in a forthcoming paper [19].…”

Section: Stability If {ν N } Is a Sequence Of Measures Inmentioning

confidence: 99%

Moderate solutions of semilinear elliptic equations with Hardy potential

Marcus

Nguyen

2017

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Let Ω be a bounded smooth domain in \mathbb{R}^{N} . We study positive solutions of equation (E) −L_{\mu }u + u^{q} = 0 in Ω where L_{\mu } = \mathrm{\Delta } + \frac{\mu }{\delta ^{2}} , 0 < \mu , q > 1 and \delta (x) = \mathrm{dist}\:(x,\partial \mathrm{\Omega }) . A positive solution of (E) is moderate if it is dominated by an L_{\mu } -harmonic function. If \mu < C_{H}(\mathrm{\Omega }) (the Hardy constant for Ω ) every positive L_{\mu } -harmonic function can be represented in terms of a finite measure on ∂Ω via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary trace by which we obtain a complete classification of the positive moderate solutions of (E) in the subcritical case, 1 < q < q_{\mu ,c} . (The critical value depends only on N and μ .) For q \geq q_{\mu ,c} there exists no moderate solution with an isolated singularity on the boundary. The normalized boundary trace and associated boundary value problems are also discussed in detail for the linear operator L_{\mu } . These results form the basis for the study of the nonlinear problem.

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Boundary singularities of solutions of semilinear elliptic equations in the half-space with a Hardy potential

Abstract: We study a nonlinear equation in the half-space {x1 > 0} with a Hardy potential, specifically −∆u − µ x 2 ) at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.

Cited by 10 publications

References 14 publications

Positive solutions of semilinear elliptic problems with a Hardy potential

Positive solutions of semilinear elliptic problems with a Hardy potential

Positive solutions of elliptic equations with a strong singular potential

Moderate solutions of semilinear elliptic equations with Hardy potential

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