Nonlinear, nonhomogeneous parametric Neumann problems

Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu



We consider a parametric nonlinear Neumann problem driven by a nonlinear nonhomogeneous differential operator, with a Carathéodory reaction $f$ which is $p$-superlinear in the second variable, but not necessarily satisfying the usual in such cases Ambrosetti-Rabinowitz condition. We prove a bifurcation type result describing the dependence of positive solutions on the parameter $\lambda> 0$, show the existence of a smallest positive solution $\overline{u}_{\lambda}$ and investigate properties of the map $\lambda\mapsto\overline{u}_{\lambda}$. Finally, we show the existence of nodal solutions.


Positive solutions; nonlinear nonhomogeneous differential operator; nonlinear regularity; nonlinear maximum principle; bifurcation type result; nodal solutions

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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 (915), 2008.

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neumann problems, Ann. Mat. Pura Appl. 188 (2009), 679–719.

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Multiple solutions for super-linear p-Laplacian Neumann problems, Osaka J. Math. 49 (2012), 699–740.

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nodal solutions for (p, 2)-equations, Trans. Amer. Math. Soc. (in print), DOI:

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nodal solutions for nonlinear nonhomogeneous Neumann equations, Topol. Merthods Nonlinear Anal. 43 (2014), 421–438.

L. Cherfils and Y. Ilyasov, On the stationary solutions of generalized reaction-diffusion equations with p and q Laplacian, Commun. Pure Appl. Anal. 4 (2005), 9–22.

S. Cingolani and M. Degiovanni, Nontrivial solutions for p-Laplacian equations with right hand side having p-linear growth, Comm. Partial Differential Equations 30 (2005), 1191–1203.

D. G. Costa and C.A. Magalhães, Existence results for perturbations of the p-Laplacian, Nonlinear Anal. 24 (1995), 409–418.

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. NonLinéaire 15 (1998), 493–516.

M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nonzero solutions with constant sign for a p-Laplacian equation, Discrete Contin. Dyn. Syst. 24 (2009), 405–440.

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman &Hall/ CRC Press, Boca Raton, 2006.

L. Gasinski and N. S. Papageorgiou, Bifurcation-type results for nonlinear parametric elliptic equations, Proc. Royal Soc. Edinburgh Sect. A 142 (2012), 515–623.

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer Academic Publishers, Dordrecht, The Netherland, 1997.

S. Hu and N. S. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Commun. Pure Appl. Anal. 9 (2010), 1801–1827.

G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), 311–361.

G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti–Rabinawitz condition, Nonlinear Anal. 72 (2010), 4602–4613.

S. Marano and N. S. Papageorgiou, On the Neumann problem with p-Laplacian and noncoercitive resonant nonlinearity, Pacific J. Math. 253 (2011), 103–123.

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. 10 (5) (2011), 729–755.

D. Motreanu, V. Motreanu and N.S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.

D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems, J. Differential Equations 232 (2007), 1–35.

D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator, Proc. Amer. Math. Soc. 72 (2010), 4602–4613.

D. Mugnai and N. S. Papageorgiou, Wang’s multiplicity result for superlinear (p,q) equations without the Ambrosetti–Rabinowitz condition, Trans. Amer. Math. Soc. 366 (2014), 4919–4937.

N. S. Papageorgiou and V.D. Radulescu, Qualitative phenomena for some classes of quaselinear elliptic equations with multiple resonance, Appl. Math. Optim. 69 (2014), 393–430.

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 (2008), 1150–1163.

P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.

M. Sun, Multiplicity of solutions for a class of quasilinear elliptic equations at resonance, J. Math. Anal. Appl. 386 (2012), 661–668.

P. Winkert, L∞ -estimates for nonlinear elliptic Neumann boundary value problems, Nonlinear Differential Equations Appl. (NoDEA) 17 (2010), 289–302.


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