Critical Neumann problems with asymmetric nonlinearity

Francisco Odair de Paiva, Wallisom Rosa

DOI: http://dx.doi.org/10.12775/TMNA.2019.120

Abstract


We prove an existence result for a semilinear elliptic equation with superlinear and asymmetric nonlinearity. The asymmetry that we consider is of the type: linear at $-\infty$ and superlinear at $+\infty$. To obtain these results we apply a Linking Theorem.

Keywords


Neumann problem; critical nonlinearity; asymmetric nonlinearity; varialtional methods

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References


Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critically nonlinearity, A tribute in honour of G. Prodi, Scuola Norm. Sup. Pisa, (1991), 9–25.

D. Arcoya and S. Villegas, Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at −∞ and superlinear at +∞, Math. Z. 219 (1995), no. 4, 499–513.

H. Brezis and L. Niremberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev expoents. Comm. Pure Appl. Math. 36 (1983), 437–477.

M. Calanchi and B. Ruf, Elliptic equations with one-side critical growth, Electron. J. Differential Equations 2002 (2002), no. 89, 1–21.

A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 6, 463–470.

G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 5, 341–350.

J. Chabrowski and B. Ruf, On the critical Neumann problem with perturbations of lower order, Colloq. Math. 108 (2007), no. 2, 225–246.

J. Chabrowski and J. Yang, Multiple solutions of a nonlinear elliptic equation involving Neumann conditions and a critical Sobolev exponent, Rend. Sem. Mat. Univ. Padova 110 (2003), 1–24.

D.G. De Figueiredo and J. Yang, Critical superlinear Ambrosetti–Prodi problems, Topol. Methods Nonlinear Anal. 14 (1999), 59–80.

L. Gasinski and N. S. Papageorgiou , Multiple solutions for nonlinear Neumann problems with asymmetric reaction via Morse theory, Adv. Nonlinear Stud. 11 (2011), 781–808.

L. Gasinski and N.S. Papageorgiou, Multiple solutions for a class of nonlinear Neumann eigenvalue problems, Commun. Pure Appl. Anal. 13 (2014), 1491–1512.

L. Gasinski and N.S. Papageorgiou, Resonant equations with the Neumann p-Laplacian plus an indefinite potential, J. Math. Anal. Appl. 422 (2015), 1146–1179.

F. Gazzola and B. Ruf, Lower-order perturbations of critical growth non-linearities in semilinear elliptic equations. Adv. Differential Equations 2 (1997), no. 4, 555–572.

D. Gilbargn and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer–Verlag, New York, 1983.

N.S. Papageorgiou and G. Smyrlis, A multiplicity theorems for Neumann problems with asymmetric nonlinearity, Ann. Mat. Pura Appl. (4) 189 (2010), no. 2, 253–272.

P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math. 65, Amer. Math Soc., Providence, R.I., 1986.

E.A. Silva, Linking theorems and applications to semilinear elliptic problems at resoance. Nonlinear Anal. 16 (1991), 455–477.

X.J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differentail Equations 93 (1991), 283–310.

M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, New York, Birkhäuser, 1996.


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