Positive solutions of semilinear elliptic problems with a Hardy\n potential

2018/03/22 by Catherine Bandle, Bandle, Catherine, Maria Assunta Pozio +1
Mathematics · #34B16 #35B09 #35B51 #35J75 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1803.08397

openalex publication_date 2018/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \Ω \⊂ \ℝ^N be a bounded domain and \δ(x) be the\ndistance of a point x\∈ \Ω to the boundary. We study the positive\nsolutions of the problem \Δ u +\(\μ)/(\δ(x)²)u=u^p in \Ω,\nwhere p>0, ,p\≠ 1 and \μ \∈ \ℝ, ,\μ\≠ 0 is smaller then the\nHardy constant. The interplay between the singular potential and the\nnonlinearity leads to interesting structures of the solution sets. In this\npaper we first give the complete picture of the radial solutions in balls. In\nparticular we establish for p>1 the existence of a unique large solution\nbehaving like \δ^- frac2p-1 at the boundary. In general domains we\nextend results of arXiv:arch-ive/1407.0288 and show that there exists a unique\nsingular solutions u such that u/\δ^\β_-\→ c on the boundary for\nan arbitrary positive function c \∈ C^2+\γ(\∂\Ω) , (\γ\n\∈ (0,1)), c \≥ 0. Here \β_- is the smaller root of\n\β(\β-1)+\μ=0.\n

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