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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

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