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

DOI:
https://doi.org/10.1090/S0002-9939-2011-10884-0

Published electronically:
February 18, 2011

MathSciNet review:
2813384

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a nonlinear Neumann problem driven by a nonhomogeneous quasilinear degenerate elliptic differential operator , a special case of which is the -Laplacian. The reaction term is a Carathéodory function which exhibits subcritical growth in . 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 and local minimizers of a -functional constructed with the general differential operator .

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Additional Information

**D. Motreanu**

Affiliation:
Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France

Email:
motreanu@univ-perp.fr

**N. S. Papageorgiou**

Affiliation:
Department of Mathematics, National Technical University, Athens 15780, Greece

Email:
npapg@math.ntua.gr

DOI:
https://doi.org/10.1090/S0002-9939-2011-10884-0

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