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Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator

Authors: D. Motreanu and N. S. Papageorgiou
Journal: Proc. Amer. Math. Soc. 139 (2011), 3527-3535
MSC (2010): Primary 35J40; Secondary 35J70
Published electronically: February 18, 2011
MathSciNet review: 2813384
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Abstract: We consider a nonlinear Neumann problem driven by a nonhomogeneous quasilinear degenerate elliptic differential operator $ \operatorname{div} a(x,\nabla u)$, a special case of which is the $ p$-Laplacian. The reaction term is a Carathéodory function $ f(x,s)$ which exhibits subcritical growth in $ s$. Using variational methods based on the mountain pass and second deformation theorems, together with truncation and minimization techniques, we show that the problem has three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). A crucial tool in our analysis is a result of independent interest which we prove here and which relates $ W^{1,p}$ and $ C^1$ local minimizers of a $ C^1$-functional constructed with the general differential operator $ \operatorname{div} a(x,\nabla u)$.

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Additional Information

D. Motreanu
Affiliation: Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France

N. S. Papageorgiou
Affiliation: Department of Mathematics, National Technical University, Athens 15780, Greece

Keywords: Nonlinear Neumann problem, $p$-Laplacian, local minimizers, mountain pass theorem, second deformation theorem, nonlinear regularity theory
Received by editor(s): March 17, 2010
Received by editor(s) in revised form: July 23, 2010, and August 24, 2010
Published electronically: February 18, 2011
Communicated by: Walter Craig
Article copyright: © Copyright 2011 American Mathematical Society