Nonlinear noncoercive Neumann problems with a reaction concave near the origin

Pasquale Candito, Giuseppina D'Aguì, Nikolaos S. Papageorgiou

Abstract


We consider a nonlinear Neumann problem driven by the $p$-Laplacian with a concave parametric reaction term and an asymptotically linear perturbation. We prove a multiplicity theorem producing five nontrivial solutions all with sign information when the parameter is small. For the semilinear case $(p=2)$ we produce six solutions, but we are unable to determine the sign of the sixth solution. Our approach uses critical point theory, truncation and comparison techniques, and Morse theory.

Keywords


Concave term;, constant sign solutions; nodal solutions; nonlinear regularity; critical groups

Full Text:

PREVIEW FULL TEXT

References


S. Aizicovici, N. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neumann problems, Ann. Mat. Pura Appl. (4) 188 (2009), no. 4, 679-719.

T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001), no. 1, 117-152.

T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 28 (1997), no. 3, 419-441.

F.O. de Paiva and E. Massa, Multiple solutions for some elliptic equations with a nonlinearity concave at the origin, Nonlinear Anal. 66 (2007), no. 12, 2940-2946.

J.I. Daz and J.E. Saa, Existence et unicite de solutions positives pour certaines equations elliptiques quasilineaires, C.R. Acad. Sci. Paris Ser. I Math. 305 (1987), no. 12, 521-524.

N. Dunford and J. Schwartz, Linear Operators, Wiles-Interscience, New York (1958).

L. Gasinski and N.S. Papageorgiou, Nonlinear Analysis, Ser. Math. Anal. Appl. 9, Chapman and Hall/CRC Press, Boca Raton, 2006.

L. Gasinski and N.S. Papageorgiou, Nonlinear elliptic equations with singular terms and combined nonlinearities, Ann. H. Poincare 13 (2012), no. 3, 481-512.

L. Gasinski and N.S. Papageorgiou, A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities, J. Global Optim. 56 (2013), no. 4, 1347-1360.

Z. Guo and Z. Zhang, W1;p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl. 286 (2003), no. 1, 32-50.

S. Hu and N. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Commun. Pure Appl. Anal. 10 (2011), no. 4, 1055-1078.

S. Hu and N. Papageorgiou, Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities, Commun. Pure Appl. Anal. 11 (2012), no. 5, 2005-2021.

Q. Jiu and J. Su, Existence and multiplicity results for Dirichlet problems with p-Laplacian, J. Math. Anal. Appl. 281 (2003), no. 2, 587-601.

S. Kyritsi and N.S. Papageorgiou, Pairs of positive solutions for p-Laplacian equations with combined nonlinearities, Commun. Pure Appl. Anal. 8 (2009), no. 3, 1031-1051.

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219.

S. Li, S. Wu and H.Z. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations 185 (2002), no. 1, 200-224.

S.A. Marano and N.S. Papageorgiou, Positive solutions to a Dirichlet problem with p-Laplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal. 12 (2013), no. 2, 815-829.

D. Motreanu and N.S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3527-3535.

D. Motreanu, V.V. Motreanu and N.S. Papageorgiou, On p-Laplace equations with concave terms and asymmetric perturbations, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 1, 171-192.

D. Motreanu, V.V. Motreanu and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), no. 3, 729-755.

D. Motreanu, V.V. Motreanu and N.S. Papageorgiou, On resonant Neumann problems, Math. Ann. 354 (2012), no. 3, 1117-1145.

D. Mugnai and N.S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Sup. Pisa Cl. Scu. Vol. XI, 4 (2012), 729-788.

R. Palais, Homotopy theory of indefinite dimensional manifolds, Topology 5 (1966), 1-16.

N.S. Papageorgiou and G. Smyrlis, Positive solutions for nonlinear Neumann problems with concave and convex terms, Positivity 16 (2012), no. 2, 271-296.

K. Perera, Multiplicity results for some elliptic problems with concave nonlinearities, J. Differential Equations 140 (1997), 133-141.

J.L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202.

S.P. Wu and H. Yang, A class of resonant elliptic problems with sublinear nonlinearity at origin and at infinity, Nonlinear Anal. 45 (2001), 925-935.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism