MATHEMATICA BOHEMICA, Vol. 124, No. 2–3, pp. 245-254 (1999)

Local solvability and regularity results for a class of semilinear elliptic problems
in nonsmooth domains

M. Bochniak, A.-M. Sändig

M. Bochniak, A.-M. Sändig, Mathematisches Institut A/6, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany, e-mails: bochniak,

Abstract: We consider a class of semilinear elliptic problems in two- and three-dimensional domains with conical points. We introduce Sobolev spaces with detached asymptotics generated by the asymptotical behaviour of solutions of corresponding linearized problems near conical boundary points. We show that the corresponding nonlinear operator acting between these spaces is Fréchet differentiable. Applying the local invertibility theorem we prove that the solution of the semilinear problem has the same asymptotic behaviour near the conical points as the solution of the linearized problem if the norms of the given right hand sides are small enough. Estimates for the difference between the solution of the semilinear and of the linearized problem are derived.

Keywords: semilinear elliptic problems, spaces with detached asymptotics, asymptotic behaviour near conical points

Classification (MSC2000): 35A07, 35B65, 35C20, 35J60

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