Universität zu Köln

Schnieders, Inka and Sweers, Guido ORCID: 0000-0003-0180-5890 (2021). Classical solutions up to the boundary to some higher order semilinear Dirichlet problems. Nonlinear Anal.-Theory Methods Appl., 207. OXFORD: PERGAMON-ELSEVIER SCIENCE LTD. ISSN 1873-5215

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Abstract

We consider the semilinear Dirichlet problem (-Delta)(m)u + g(., u) = f in bounded domains Omega subset of R-n under homogeneous Dirichlet boundary conditions (partial derivative/partial derivative v)(i)u = 0 for i = 0, ..., m - 1 on the smooth boundary partial derivative Omega. We assume that g fulfils a sign condition u g(x, u) >= 0 for all (x, u) is an element of Omega x R, which gives the coercivity of the corresponding variational formulation. For the second order problem, i.e. m = 1, with this sign condition on g the maximum principle will then imply that such a solution is classical no matter the growth of g. For higher order problems the existence of a classical solution is not obvious, unless g has a growth rate below the critical value for a Sobolev imbedding. Grunau and Sweers (1997), proved that there is a solution that is classical in the interior, allowing a one-sided larger growth rate, when compensated by a lower growth rate on the other side. We improve this result by showing the existence of a solution, that is classical up to the boundary, i.e. u is an element of C-2m,C-gamma((Omega) over bar) boolean AND C-0(m-1) ((Omega) over bar). For this we use recent estimates for the Green function of the polyharmonic Dirichlet problem and an approximation procedure. (C) 2021 Elsevier Ltd. All rights reserved.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Schnieders, Inka UNSPECIFIED UNSPECIFIED UNSPECIFIED Sweers, Guido UNSPECIFIED orcid.org/0000-0003-0180-5890 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-565738
DOI:	10.1016/j.na.2021.112265
Journal or Publication Title:	Nonlinear Anal.-Theory Methods Appl.
Volume:	207
Date:	2021
Publisher:	PERGAMON-ELSEVIER SCIENCE LTD
Place of Publication:	OXFORD
ISSN:	1873-5215
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language Mathematics, Applied; Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/56573