## Topological Methods in Nonlinear Analysis

### Nonlinear noncoercive Neumann problems with a reaction concave near the origin

#### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 47, Number 1 (2016), 289-317.

Dates
First available in Project Euclid: 23 March 2016

https://projecteuclid.org/euclid.tmna/1458740740

Digital Object Identifier
doi:10.12775/TMNA.2016.007

Mathematical Reviews number (MathSciNet)
MR3469058

Zentralblatt MATH identifier
1373.35148

#### Citation

Candito, Pasquale; D'Aguí, Giuseppina; Papageorgiou, Nikolaos S. Nonlinear noncoercive Neumann problems with a reaction concave near the origin. Topol. Methods Nonlinear Anal. 47 (2016), no. 1, 289--317. doi:10.12775/TMNA.2016.007. https://projecteuclid.org/euclid.tmna/1458740740

#### 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. Saá, Existence et unicité de solutions positives pour certaines equations elliptiques quasilinéaires, C.R. Acad. Sci. Paris Sér. 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.
• ––––, Nonlinear elliptic equations with singular terms and combined nonlinearities, Ann. H. Poincaré 13 (2012), no. 3, 481–512.
• ––––, 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, $W^{1,p}$ versus $C^1$ 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.
• ––––, 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.
• ––––, 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.
• ––––, 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. Vázquez, 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.