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March  2009, 8(2): 533-557. doi: 10.3934/cpaa.2009.8.533

Nodal solutions to critical growth elliptic problems under Steklov boundary conditions

Elvise Berchio 1, , Filippo Gazzola 2, and  Dario Pierotti 3,

1. 

Dipartimento SEMEQ, Università del Piemonte Orientale, via E. Perrone 18, Novara, 28100, Italy
2. 

Dipartimento di Matematica Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano
3. 

Dipartimento di Matematica del Politecnico, Piazza L. da Vinci 32, Milano, 20133, Italy

Received  January 2008 Revised  July 2008 Published  December 2008

We study elliptic problems at critical growth under Steklov boundary conditions in bounded domains. For a second order problem we prove existence of nontrivial nodal solutions. These are obtained by combining a suitable linking argument with fine estimates on the concentration of Sobolev minimizers on the boundary. When the domain is the unit ball, we obtain a multiplicity result by taking advantage of the explicit form of the Steklov eigenfunctions. We also partially extend the results in the ball to the case of fourth order Steklov boundary value problems.
Keywords: nodal solutions, Steklov, Critical growth.
Mathematics Subject Classification: Primary: 35J20, 35J35; Secondary: 35J40.
Citation: Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
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