January 2018, 17(1): 231-241. doi: 10.3934/cpaa.2018014

Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term

1. 

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

2. 

Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

3. 

Department of Mathematics, University of Craiova, 200585 Craiova, Romania

4. 

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

* Corresponding author:Vicenţiu D. Rǎdulescu

Received  April 2017 Revised  June 2017 Published  September 2017

Fund Project: This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025. V.D. Rǎdulescu acknowledges the support through a grant of Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-Ⅲ-P4-ID-PCE-2016-0130, within PNCDI Ⅲ

We consider a nonlinear Robin problem driven by the p-Laplacian plus an indefinite potential. The reaction term is of arbitrary growth and only conditions near zero are imposed. Using critical point theory together with suitable truncation and perturbation techniques and comparison principles, we show that the problem admits a sequence of distinct smooth nodal solutions converging to zero in $C^1(\overline{Ω})$.

Citation: Nikolaos S. Papageorgiou, Vicenţiu D. Rǎdulescu, Dušan D. Repovš. Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term. Communications on Pure & Applied Analysis, 2018, 17 (1) : 231-241. doi: 10.3934/cpaa.2018014
References:
[1]

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

[2]

M. Filippakis and N. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the $p$-Laplacian, J. Differential Equations, 245 (2008), 1883-1922. doi: 10.1016/j.jde.2008.07.004.

[3]

G. FragnelliD. Mugnai and N. S. Papageorgiou, Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential, Discrete Contin. Dyn. Syst., 36 (2016), 6133-6166. doi: 10.3934/dcds.2016068.

[4]

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

[5]

R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370. doi: 10.1016/j.jfa.2005.04.005.

[6]

Z. Li and Z. Q. Wang, On Clark's theorem and its applications to partially sublinear problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1015-1037. doi: 10.1016/j.anihpc.2014.05.002.

[7]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.

[8]

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

[9]

N.S. Papageorgiou and V. D. Rǎdulescu, Infinitely many nodal solutions for nonlinear, nonhomogeneous Robin problems, Adv. Nonlinear Stud., 16 (2016), 287-299. doi: 10.1515/ans-2015-5040.

[10]

N. S. Papageorgiou and V. D. Rǎdulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud., 16 (2016), 737-764. doi: 10.1515/ans-2016-0023.

[11]

N. S. PapageorgiouV. D. Rǎdulescu and D. D. Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dyn. Syst., 37 (2017), 2589-2618. doi: 10.3934/dcds.2017111.

[12]

N. S. PapageorgiouV. D. Rǎdulescu and D. D. Repovš, Robin problems with indefinite linear part and competition phenomena, Commun. Pure Appl. Anal., 16 (2017), 1293-1314. doi: 10.3934/cpaa.2017063.

[13]

Z-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 15-33. doi: 10.1007/PL00001436.

show all references

References:
[1]

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

[2]

M. Filippakis and N. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the $p$-Laplacian, J. Differential Equations, 245 (2008), 1883-1922. doi: 10.1016/j.jde.2008.07.004.

[3]

G. FragnelliD. Mugnai and N. S. Papageorgiou, Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential, Discrete Contin. Dyn. Syst., 36 (2016), 6133-6166. doi: 10.3934/dcds.2016068.

[4]

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

[5]

R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370. doi: 10.1016/j.jfa.2005.04.005.

[6]

Z. Li and Z. Q. Wang, On Clark's theorem and its applications to partially sublinear problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1015-1037. doi: 10.1016/j.anihpc.2014.05.002.

[7]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.

[8]

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

[9]

N.S. Papageorgiou and V. D. Rǎdulescu, Infinitely many nodal solutions for nonlinear, nonhomogeneous Robin problems, Adv. Nonlinear Stud., 16 (2016), 287-299. doi: 10.1515/ans-2015-5040.

[10]

N. S. Papageorgiou and V. D. Rǎdulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud., 16 (2016), 737-764. doi: 10.1515/ans-2016-0023.

[11]

N. S. PapageorgiouV. D. Rǎdulescu and D. D. Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dyn. Syst., 37 (2017), 2589-2618. doi: 10.3934/dcds.2017111.

[12]

N. S. PapageorgiouV. D. Rǎdulescu and D. D. Repovš, Robin problems with indefinite linear part and competition phenomena, Commun. Pure Appl. Anal., 16 (2017), 1293-1314. doi: 10.3934/cpaa.2017063.

[13]

Z-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 15-33. doi: 10.1007/PL00001436.

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