# American Institute of Mathematical Sciences

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

## Perturbations of nonlinear eigenvalue problems

 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.

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. [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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##### 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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