Two constant sign solutions for a nonhomogeneous Neumann boundary value problem

Liliana Klimczak

Abstract


We consider a nonlinear Neumann problem with a nonhomogeneous elliptic differential operator. With some natural conditions for its structure and some general assumptions on the growth of the reaction term we prove that the problem has two nontrivial solutions of constant sign. In the proof we use variational methods with truncation and minimization techniques.

Keywords


local minimizers, truncations, constant sign solutions

Mathematics Subject Classification (2010)


35J20; 35J60

References


Casas, E. and Fernández, L.A. "A Green’s formula for quasilinear elliptic operators." J. Math. Anal. Appl. 142.1 (1989): 62-73.

Damascelli, L. "Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results." Ann. Inst. H. Poincaré. Analyse non linéaire 15.4 (1998): 493–516.

Gasinski, L. and Papageorgiou, N.S. "Nonlinear Analysis." Series in Mathematical Analysis and Applications 9. Boca Raton, FL: Chapman and Hall/CRC Press 2006.

Leoni, G. "A First Course in Sobolev Spaces." Graduate Studies in Mathematics 105. Providence, RI: American Mathematical Society, 2009.

Lieberman, G.M. "The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations." Commun. Partial Differential Equations 16.1-2 (1991): 311-361.

Montenegro, M. "Strong maximum principles for supersolutions of quasilinear elliptic equations." Nonlinear Anal. 37.4 (1999): 431-448.

Motreanu, D. and Papageorgiou, N.S. "Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operators." Proc. Amer. Math. Soc. 139.10 (2011): 3527-3535.

Pucci, P. and Serrin, J. "The Maximum Principle." Progress in Nonlinear Differential Equations and their Applications 73. Basel: Birkhäuser Verlag, 2007.

Zeidler, E. "Nonlinear Functional Analysis and Its Applications II/B." Nonlinear Monotone operators. New York: Springer-Verlag, 1990.


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