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Solutions of semilinear elliptic equations with one isolated singularity. (English) Zbl 1064.35510
The paper is devoted to the study of the existence of singular positive solutions of semilinear elliptic equations of the type \(\Delta u + \lambda f(u) = 0.\) The author is concerned with the supercritical case, namely when the nonlinearity \(f(u)\) is stronger than \(u\to u^{\frac {N+2}{N-2}}\). The author proves the existence of a positive weak solution which is defined in the unit ball of \(\mathbb R^N\), has zero boundary data and has a nonremovable prescribed singularity at some point close to the origin. The tools used are a. o. the implicit function theorem, the weighted Hölder spaces, as well as the eigenfunction expansion which transforms the problem to the ordinary differential equation.

35J60 Nonlinear elliptic equations
35A20 Analyticity in context of PDEs