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May  2019, 18(3): 1403-1431. doi: 10.3934/cpaa.2019068

Perturbations of nonlinear eigenvalue problems

Nikolaos S. Papageorgiou 1,2, , Vicenţiu D. Rădulescu 2,3,4,, and  Dušan D. Repovš 2,5,

1. 

National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
2. 

Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
3. 

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
4. 

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
5. 

Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia

* Corresponding author

Received  July 2018 Revised  July 2018 Published  November 2018

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive solutions changes as the real parameter $λ$ varies. We also show that there exists a minimal positive solution $\overline{u}_λ$ and determine the monotonicity and continuity properties of the map $λ\mapsto\overline{u}_λ$. Special attention is given to the particular case of the $p$-Laplacian.

Keywords: Nonhomogeneous differential operator, sublinear and superlinear perturbation, nonlinear regularity, nonlinear maximum principle, comparison principle, minimal positive solution.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60.
Citation: Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Perturbations of nonlinear eigenvalue problems. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1403-1431. doi: 10.3934/cpaa.2019068
References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Memoirs Amer. Math. Soc., 196 (2008), nr. 915, ⅵ+70 pp.

[2]

W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Lapacian and applications, Nonlinear Anal., 32 (1998), 819-830.  doi: 10.1016/S0362-546X(97)00530-0.

[3]

G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 977-1009.  doi: 10.1007/s00030-012-0193-y.

[4]

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

[5]

F. Colasuonno, P. Pucci and C. Varga, Multiple solutions for an eigenvalue problem involving $p$-Laplacian type operators, Nonlinear Anal., 75 (2012), 4496-4512.  doi: 10.1016/j.na.2011.09.048.

[6]

J. I. Diaz and J. E. Saa, Existence et unicité des solutions positives pour certaines équations elliptiques quasilinéaires, C.R. Acad. Sci. Paris, 305 (1987), 521-524. 

[7]

G. Fragnelli, D. Mugnai and N. S. Papageorgiou, Brezis-Oswald result for quasilinear Robin problems, Adv. Nonlin. Studies, 16 (2016), 403-422.  doi: 10.1515/ans-2016-0010.

[8]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.

[9]

L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian type equations, Adv. Nonlin. Studies, 8 (2008), 843-870.  doi: 10.1515/ans-2008-0411.

[10]

L. Gasinski and N. S. Papageorgiou, Exercises in Analysis, Part 2: Nonlinear Analysis, Problem Books in Mathematics, Springer, Cham, 2016.

[11]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Mathematics and its Applications, 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.

[12]

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-Rabinowitz condition, Nonlinear Anal., 72 (2010), 4602-4613.  doi: 10.1016/j.na.2010.02.037.

[13]

G. Lieberman, On the natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Diff. Equations, 16 (1991), 311-361.  doi: 10.1080/03605309108820761.

[14]

S. A. Marano and S. Mosconi, Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 279-291.  doi: 10.3934/dcdss.2018015.

[15]

S. A. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Comm. Pure Appl. Anal., 12 (2013), 815-829.  doi: 10.3934/cpaa.2013.12.815.

[16]

D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 729-788. 

[17]

N. S. Papageorgiou and V. D. Rădulescu, Multiple solutions precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.  doi: 10.1016/j.jde.2014.01.010.

[18]

N. S. Papageorgiou and V. D. Rădulescu, Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math., 28 (2016), 545-571.  doi: 10.1515/forum-2014-0094.

[19]

N. S. Papageorgiou and V. D. Rădulescu, Multiplicity theorems for nonlinear nonhomogeneous Robin problems, Revista Mat. Iberoam., 33 (2017), 251-289.  doi: 10.4171/RMI/936.

[20]

N. S. Papageorgiou and V. D. Rădulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlin. Studies, 16 (2016), 737-764. 

[21]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discr. Cont. Dynam. Systems, Ser. A, 37 (2017), 2589-2618.  doi: 10.3934/dcds.2017111.

[22]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Positive solutions for nonlinear nonhomogeneous parametric Robin problems, Forum Math., 30 (2018), 553-580.  doi: 10.1515/forum-2017-0124.

[23]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term, Communications on Pure and Applied Analysis, 17 (2018), 231-241.  doi: 10.3934/cpaa.2018014.

[24]

K. Perera, P. Pucci and C. Varga, An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 441-451.  doi: 10.1007/s00030-013-0255-9.

[25]

P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73, Birkhäuser Verlag, Basel, 2007.

show all references

References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Memoirs Amer. Math. Soc., 196 (2008), nr. 915, ⅵ+70 pp.

[2]

W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Lapacian and applications, Nonlinear Anal., 32 (1998), 819-830.  doi: 10.1016/S0362-546X(97)00530-0.

[3]

G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 977-1009.  doi: 10.1007/s00030-012-0193-y.

[4]

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

[5]

F. Colasuonno, P. Pucci and C. Varga, Multiple solutions for an eigenvalue problem involving $p$-Laplacian type operators, Nonlinear Anal., 75 (2012), 4496-4512.  doi: 10.1016/j.na.2011.09.048.

[6]

J. I. Diaz and J. E. Saa, Existence et unicité des solutions positives pour certaines équations elliptiques quasilinéaires, C.R. Acad. Sci. Paris, 305 (1987), 521-524. 

[7]

G. Fragnelli, D. Mugnai and N. S. Papageorgiou, Brezis-Oswald result for quasilinear Robin problems, Adv. Nonlin. Studies, 16 (2016), 403-422.  doi: 10.1515/ans-2016-0010.

[8]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.

[9]

L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian type equations, Adv. Nonlin. Studies, 8 (2008), 843-870.  doi: 10.1515/ans-2008-0411.

[10]

L. Gasinski and N. S. Papageorgiou, Exercises in Analysis, Part 2: Nonlinear Analysis, Problem Books in Mathematics, Springer, Cham, 2016.

[11]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Mathematics and its Applications, 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.

[12]

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-Rabinowitz condition, Nonlinear Anal., 72 (2010), 4602-4613.  doi: 10.1016/j.na.2010.02.037.

[13]

G. Lieberman, On the natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Diff. Equations, 16 (1991), 311-361.  doi: 10.1080/03605309108820761.

[14]

S. A. Marano and S. Mosconi, Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 279-291.  doi: 10.3934/dcdss.2018015.

[15]

S. A. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Comm. Pure Appl. Anal., 12 (2013), 815-829.  doi: 10.3934/cpaa.2013.12.815.

[16]

D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 729-788. 

[17]

N. S. Papageorgiou and V. D. Rădulescu, Multiple solutions precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.  doi: 10.1016/j.jde.2014.01.010.

[18]

N. S. Papageorgiou and V. D. Rădulescu, Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math., 28 (2016), 545-571.  doi: 10.1515/forum-2014-0094.

[19]

N. S. Papageorgiou and V. D. Rădulescu, Multiplicity theorems for nonlinear nonhomogeneous Robin problems, Revista Mat. Iberoam., 33 (2017), 251-289.  doi: 10.4171/RMI/936.

[20]

N. S. Papageorgiou and V. D. Rădulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlin. Studies, 16 (2016), 737-764. 

[21]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discr. Cont. Dynam. Systems, Ser. A, 37 (2017), 2589-2618.  doi: 10.3934/dcds.2017111.

[22]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Positive solutions for nonlinear nonhomogeneous parametric Robin problems, Forum Math., 30 (2018), 553-580.  doi: 10.1515/forum-2017-0124.

[23]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term, Communications on Pure and Applied Analysis, 17 (2018), 231-241.  doi: 10.3934/cpaa.2018014.

[24]

K. Perera, P. Pucci and C. Varga, An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 441-451.  doi: 10.1007/s00030-013-0255-9.

[25]

P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73, Birkhäuser Verlag, Basel, 2007.

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