Existence of solutions of quasilinear elliptic equations on manifolds with conic points

by Thi Thu Huong Nguyen
Doctoral thesis
Date of Examination:2013-12-13
Date of issue:2014-05-15
Advisor:Prof. Dr. Ingo Witt
Referee:Prof. Dr. Ingo Witt
Referee:Prof. Dr. Dorothea Bahns
Persistent Address: http://dx.doi.org/10.53846/goediss-4507

 

 

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Abstract

English

Existence and regularity of solutions of quasilinear elliptic equations in nonsmooth domains have been interesting topics in the development of partial differential equations. The existence of finite-energy solutions of higher-order equations, also those with degenerations and singularities, can be shown by theories of monotone operators and topological methods. There are few results about singular solutions of second-order equations involving the p-Laplacian with the Dirac distribution on the right-hand side. So far the existence of singular solutions of higher-order equations with a prescribed asymptotic behavior has not been investigated.   The aims of my dissertation are to look for finite-energy and singular solutions of quasilinear elliptic equations on manifolds with conic points. We single out realizations of the p-Laplacian in particular, (p>= 2), and a cone-degenerate operator in general, which are shown to belong to the class (S)_+. Assuming further coercivity conditions and employing mapping degree theory for generalized monotone mappings, we obtain existence for the prototypical example of the p-Laplacian and for general higher-order equations.    
Keywords: Singular solutions; Quasilinear elliptic equations; Manifolds with conic points; Topological methods; Conic p-Laplacian
 
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