Topological Methods in Nonlinear Analysis

Nodal solutions for nonlinear nonhomogeneous Neumann equations

Sergiu Aizicovici, Nikolaos S. Papageorgiou, and Vasile Staicu

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We consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator with a Caratheodory reaction which is $(p-1)$-sublinear near $\pm\infty$. Using variational tools we show that the problem has at least three nontrivial smooth solutions (one positive, one negative and a third nodal). Our formulation unifies problems driven by the $p$-Laplacian, the $(p,q)$ Laplacian and the $p$-generalized mean curvature operator.

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Topol. Methods Nonlinear Anal., Volume 43, Number 2 (2014), 421-438.

First available in Project Euclid: 11 April 2016

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Aizicovici, Sergiu; Papageorgiou, Nikolaos S.; Staicu, Vasile. Nodal solutions for nonlinear nonhomogeneous Neumann equations. Topol. Methods Nonlinear Anal. 43 (2014), no. 2, 421--438.

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  • S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints , Mem. Amer. Math. Soc., 196 (2008) \ref\key 2 ––––, Existence of multiple solutions with precise sign information for superlinear Neumann problems , Ann. Mat. Pura Appl., 188 , 679–719 (2009) \ref\key 3
  • V. Benci, D. Fortunato and L. Pisani, Solitons like solutions of a Lorentz invariant equation in dimension $3$ , Rev. Math. Phys., 10 , 315–344 (1998) \ref\key 4
  • R. Benguria, H. Brezis and E.H. Lieb, The Thomas–Fermi–von Weizsacker theory of atoms and molecules , Comm. Math. Phys., 79 , 167–180 (1981) \ref\key 5
  • Z. Chen and Y. Shen, Infinitely many solutions of Dirichlet problem for $p$-mean curvature operator , Appl. Math. J. Chinese Univ. Ser. B, 18 , 161–172 (2003) \ref\key 6
  • J.I. Diaz and J.E. Saa, Existence et unicité de solutions positives pour certaines equations elliptiques quasilineaires , C.R. Math. Acad. Sci. Paris, 305 , 521–524 (1987) \ref\key 7
  • N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, New York (1958) \ref\key 8
  • L. Gasinski and N.S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC Press, Boca Raton (2006) \ref\key 9 ––––, Anisotropic nonlinear Neumann problems , Calc. Var. Partial Differential Equations, 42 (2011), 323–354 \ref\key 10
  • S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, New York (1994) \ref\key 11
  • Q. Jiu and J. Su, Existence and multiplicity results for perturbations of the $p$-Laplacian , J. Math. Anal. Appl., 281 , 587–601 (2003) \ref\key 12
  • G. Li and H. Zhou, Multiple solutions to $p$-Laplacian problems with asymptotic nonlinearity as $u^{p-1}$ at infinity , J. London Math. Soc., 65 , 123–138 (2002) \ref\key 13
  • G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations , Comm. Partial Differential Equations, 16 , 311–361 (1991) \ref\key 14
  • J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems , J. Math. Anal. Appl., 258 , 209–222 (2001) \ref\key 15
  • S. Liu, Multiple solutions for coercive $p$-Laplacian equations , J. Math. Anal. Appl., 316 , 229–236 (2006) \ref\key 16
  • S.A. Marano and N.S. Papageorgiou, Constant-sign and nodal solutions of coercive $(p,q)$-Laplacian problems , Nonlinear Anal., 77 , 118–129 (2013) \ref\key 17
  • D. Motreanu, V. Motreanu and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems , Ann. Scuola Norm. Sup. Pisa, Cl. Sci., X , 729–755 (2011) \ref\key 18
  • D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator , Proc. Amer. Math. Soc., 139 , 3527–3535 (2011) \ref\key 19
  • E. Papageorgiou and N.S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian , J. Funct. Anal., 244 , 63–77 (2007) \ref\key 20
  • N.S. Papageorgiou, E.M. Rocha and V. Staicu, A multiplicity theorem for hemivariational inequalities with a $p$-Laplacian-like differential operator , Nonlinear Anal., 69 , 1150–1163 (2008) \ref\key 21
  • P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel (2007)