Very weak solutions with boundary singularities for semilinear elliptic Dirichlet problems in domains with conical corners

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:352

Article

Horak, J.²  McKenna, P. J.³  Reichel, W.¹ 

[1] Univ Karlsruhe, Inst Anal, D-76128 Karlsruhe, Germany

[2] Univ Cologne, Math Inst, D-50931 Cologne, Germany

[3] Univ Connecticut, Dept Math, Storrs, CT 06269 USA

关键词: Very weak solutions;    Critical exponents;    Conical corners;   

DOI  :  10.1016/j.jmaa.2008.06.008

来源: Elsevier

【 摘 要 】

Let Omega subset of R-n be a bounded Lipschitz domain with a cone-like corner at 0 is an element of partial derivative Omega. We prove existence of at least two positive unbounded very weak solutions of the problem -Delta u = u(p) in Omega, u = 0 on partial derivative Omega, which have a singularity at 0, for any p slightly bigger that the generalized Brezis-Turner exponent p*. On an example of a planar polygonal domain the actual size of the p-interval on which the existence result holds is computed. The solutions are found variationally as perturbations of explicitly constructed singular solutions in cones. This approach also makes it possible to find numerical approximations of the two very weak solutions on Omega following a gradient flow of a suitable functional and using the mountain pass algorithm. Two-dimensional examples are presented. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

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