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Nodal Solutions for Nonlinear Non-Homogeneous Robin Problems with an Indefinite Potential

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  13 June 2018

Leszek Gasiński  and
Nikolaos S. Papageorgiou
Leszek Gasiński*
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland (Leszek.Gasinski@ii.uj.edu.pl)
Nikolaos S. Papageorgiou
Affiliation:
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece (npapg@math.ntua.gr)
*
*Corresponding author.
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Abstract

We consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

Keywords

Robin boundary conditionnonlinear non-homogeneous differential operatornonlinear regularity theorynodal solutions

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 943 - 959
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Aizicovici, S., Papageorgiou, N. S. and Staicu, V., Nodal solutions for (p, 2)-equations, Trans. Amer. Math. Soc. 367 (2015), 7343–7372.Google Scholar
2.Diazand, J. I. Saa, J. E., Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 521–524.Google Scholar
3.Gasiński, L. and Papageorgiou, N. S., Nonlinear analysis (Chapman and Hall/CRC, Boca Raton, FL, 2006).Google Scholar
4.Gasiński, L. and Papageorgiou, N. S., Existence and multiplicity of solutions for Neumann p-Laplacian-type equations, Adv. Nonlinear Stud. 8 (2008), 843–870.Google Scholar
5.Gasiński, L. and Papageorgiou, N. S., Multiplicity of positive solutions for eigenvalue problems of (p, 2)-equations, Bound. Value Probl. 152 (2012), 1–17.Google Scholar
6.Gasiński, L. and Papageorgiou, N. S., Nonhomogeneous nonlinear Dirichlet problems with a p-superlinear reaction, Abstr. Appl. Anal. (2012), 1–28.Google Scholar
7.Gasiński, L. and Papageorgiou, N. S., Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set-Valued Var. Anal. 20 (2012), 417–443.Google Scholar
8.Gasiński, L. and Papageorgiou, N. S., Nonlinear periodic equations driven by a nonhomogeneous differential operator, J. Nonlinear Convex Anal. 14 (2013), 583–600.Google Scholar
9.Gasiński, L. and Papageorgiou, N. S., A pair of positive solutions for (p, q)-equations with combined nonlinearities, Commun. Pure Appl. Anal. 13 (2014), 203–215.Google Scholar
10.Gasiński, L. and Papageorgiou, N. S., Dirichlet (p, q)-equations at resonance, Dis. Contin. Dyn. Syst. 34 (2014), 2037–2060.Google Scholar
11.Gasiński, L. and Papageorgiou, N. S., Nodal and multiple solutions for nonlinear elliptic equations involving a reaction with zeros, Dyn. Partial Differ. Equ. 12 (2015), 13–42.Google Scholar
12.Gasiński, L. and Papageorgiou, N. S., Nonlinear elliptic equations with a jumping reaction, J. Math. Anal. Appl. 443 (2016), 1033–1070.Google Scholar
13.Gasiński, L. and Papageorgiou, N. S., Positive solutions for the generalized nonlinear logistic equations, Canad. Math. Bull. 59 (2016), 73–86.Google Scholar
14.Gasiński, L., O'Regan, D. and Papageorgiou, N. S., Positive solutions for nonlinear nonhomogeneous Robin problems, Z. Anal. Anwend. 34 (2015), 435–458.Google Scholar
15.He, T., Chen, C., Huang, Y. and Hou, C., Infinitely many sign-changing solutions for p-Laplacian Neumann problems with indefinite weight, Appl. Math. Lett. 39 (2015), 73–79.Google Scholar
16.Heinz, H. P., Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems, J. Differ. Equ. 66 (1987), 263–300.Google Scholar
17.Kajikiya, R., A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005), 352–370.Google Scholar
18.Lieberman, G. M., The natural generalizations of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Commun. Partial Differ. Equ. 16 (1991), 311–361.Google Scholar
19.Mugnai, D. and Papageorgiou, N. S., Resonant nonlinear Neumann problems with indefinite weight, Ann. Sci. Norm. Super. Pisa Cl. Sci. 11(5) (2012), 729–788.Google Scholar
20.Mugnai, D. and Papageorgiou, N. S., Wang's multiplicity result for superlinear (p, q)-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc. 366 (2014), 4919–4937.Google Scholar
21.Papageorgiou, N. S. and Rădulescu, V. D., Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim. 69 (2014), 393–430.Google Scholar
22.Papageorgiou, N. S. and Rădulescu, V. D., Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differ. Equ. 256(7) (2014), 2449–2479.Google Scholar
23.Papageorgiou, N. S. and Rădulescu, V. D., Resonant (p, 2)-equations with asymmetric reaction, Anal. Appl. (Singap.) 13 (2015), 481–506.Google Scholar
24.Papageorgiou, N. S. and Rădulescu, V. D., Nonlinear parametric Robin problems with combined nonlinearities, Adv. Nonlinear Stud. 15 (2015), 715–748.Google Scholar
25.Papageorgiou, N. S. and Rădulescu, V. D., Coercive and noncoercive nonlinear Neumann problems with an indefinite potential, Forum Math. 28 (2016), 545–571.Google Scholar
26.Papageorgiou, N. S. and Rădulescu, V. D., Infinitely many nodal solutions for nonlinear nonhomogeneous Robin problems, Adv. Nonlinear Stud. 16 (2016), 287–299.Google Scholar
27.Papageorgiou, N. S. and Rădulescu, V. D., Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud. 16 (2016), 737–764.Google Scholar
28.Papageorgiou, N. S. and Rădulescu, V. D., Multiplicity theorems for nonlinear nonhomogeneous Robin problems, Rev. Mat. Iberoam. 31 (2017), 251–289.Google Scholar
29.Pucci, P. and Serrin, J., The maximum principle (Birkhäuser Verlag, Basel, 2007).Google Scholar
30.Sun, M., Zhang, M. and Su, J., Critical groups at zero and multiple solutions for a quasilinear elliptic equation, J. Math. Anal. Appl. 428 (2015), 696–712.Google Scholar
31.Wang, Z.-Q., Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear Differ. Equ. Appl. 8 (2001), 15–33.Google Scholar
32.Yang, D. and Bai, C., Nonlinear elliptic problem of 2-q-Laplacian type with asymmetric nonlinearities, Electron. J. Differ. Equ. 170 (2014), 1–13.Google Scholar