A construction of singular solutions for a semilinear elliptic equation\n using asymptotic analysis

1994/10/10 by Rafe Mazzeo, Frank Pacard, Mazzeo, Rafe +1 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.dg-ga/9410004

Abstract

The aim of this paper is to prove the existence of weak solutions to the\nequation \Δ u + u^p = 0 which are positive in a domain \Ω \⊂\n Bbb R^N, vanish at the boundary, and have prescribed isolated\nsingularities. The exponent p is required to lie in the interval (N/(N-2),\n(N+2)/(N-2)). We also prove the existence of solutions to the equation \Δν + u^p = 0 which are positive in a domain \Ω \⊂ Bbb Rⁿ and\nwhich are singular along arbitrary smooth k-dimensional submanifolds in the\ninterior of these domains provided p lie in the interval ((n - k)/(n-k-2),\n(n-k+2)/(n-k-2)). A particular case is when p = (n+2)/(n-2), in which case\nsolutions correspond to solutions of the singular Yamabe problem. The method\nused is a mixture of different ingredients used by both authors in their\nseparate constructions of solutions to the singular Yamabe problem, along with\na new set of scaling techniques.\n

Citations

• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
• The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation

Cited by

• Solutions of the singular Yamabe problem near singular boundaries
• Nonhomothetic complete periodic metrics with constant scalar curvature

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• Boundary singularities for weak solutions of semilinear elliptic problems
• Stable and singular solutions of the equation Δu = 1/u
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• Isolated Boundary Singularities of Semilinear Elliptic Equations
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