Differential Inclusions, Control and Optimization 24 (2004) 31-40
doi: 10.7151/dmdico.1050

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ON NEUMANN BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATIONS

Dimitrios A. Kandilakis

Department of Sciences
Technical University of Crete
Chania, Crete 73100, Greece

e-mail: dkan@science.tuc.gr

Abstract

We provide two existence results for the nonlinear Neumann problem
ě
ď
ď
í
ď
ď
î
-div(a(x)Ń u(x)) = f(x,u)
in W
u
n
= 0
on W,
where W is a smooth bounded domain in IRN, a is a weight function and f a nonlinear perturbation. Our approach is variational in character.

Keywords and phrases: variational methods, Palais-Smale condition, saddle point theorem, mountain pass theorem.

2000 Mathematics Subject Classification: 35J20, 35J60.

References

[1] D. Arcoya and L. Orsina, Landesman-Laser conditions and quasilinear elliptic equations, Nonlin. Anal. TMA 28 (1997), 1623-1632.
[2] J. Bouchala and P. Drabek, Strong resonance for some quasilinear elliptic equations, J. Math. Anal. Appl. 245 (2000), 7-19.
[3] P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, NoDEA 7 (2000), 187-199.
[4] F. Cîrstea, D. Motreanu and V. Radulescu, Weak solutions of quasilinear problems with nonlinear boundary condition, Nonlin. Anal. 43 (2001), 623-636.
[5] P. Drabek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singulaties, W. De Gruyter 1997.
[6] W. Li and H. Zhen, The applications of sums of ranges of accretive operators to nonlinear equations involving the p-Laplacian operator, Nonlin. Anal. TMA 24 (2) (1995), 185-193.
[7] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Amer. Math. Soc. Prividence, 1976.

Received 7 March 2004