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Communications on Pure and Applied Analysis (CPAA)
 

Uniqueness results for a Dirichlet problem with variable exponent

Pages: 1399 - 1410, Volume 9, Issue 5, September 2010      doi:10.3934/cpaa.2010.9.1399

 
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V. V. Motreanu - Universität Zürich, Institut für Mathematik, CH-8057 Zürich, Switzerland (email)

Abstract: We study the uniqueness of weak solutions for Dirichlet problems with variable exponent and non-standard growth conditions. First, we provide two uniqueness results under ellipticity type hypotheses. Next, we provide a uniqueness result when the operator driving the problem is in the form of the divergence of a monotone map. Finally, we derive a fourth uniqueness result under homogeneity type hypotheses, by means of a comparison result and approximation.

Keywords:  Variable exponent Dirichlet problem, p(x)-Laplacian equation, uniqueness of weak solution, subsolution, supersolution, approximation.
Mathematics Subject Classification:  Primary: 35J60, 35J70; Secondary: 35D30.

Received: September 2009;      Revised: September 2009;      Published: May 2010.