Constant sign and nodal solutions for nonlinear Robin equations with locally defined source term
Abstract
We consider a parametric Robin problem driven by a nonlinear, nonhomogeneous differential operator which includes as special cases the p-Laplacian and the (p,q)-Laplacian. The source term is parametric and only locally defined (that is, in a neighborhood of zero). Using suitable cut-off techniques together with variational tools and comparison principles, we show that for all big values of the parameter, the problem has at least three nontrivial smooth solutions, all with sign information (positive, negative and nodal).
Keyword : locally defined reaction, nonlinear regularity, nonlinear maximum principle, constant sign and nodal solutions, critical groups
How to Cite
Papageorgiou, N. S., Vetro, C., & Vetro, F. (2020). Constant sign and nodal solutions for nonlinear Robin equations with locally defined source term. Mathematical Modelling and Analysis, 25(3), 374-390. https://doi.org/10.3846/mma.2020.11031
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
A. Ambrosetti and P.H. Rabinowitz. Dual variational methods in critical point theory and applications. J. Functional Anal., 14(4):349–381, 1973. https://doi.org/10.1016/0022-1236(73)90051-7
V. Benci, P. D’Avenia, D. Fortunato and L. Pisani. Solitons in several space dimensions: Derrick’s problem and infinitely many solutions. Arch. Ration. Mech. Anal., 154:297–324, 2000. https://doi.org/10.1007/s002050000101
L. Cherfils and Y. Il’yasov. On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian. Commun. Pure Appl. Anal., 4(1):9–22, 2005. https://doi.org/10.3934/cpaa.2005.4.9
L. Gasiński and N.S. Papageorgiou. Positive solutions for the Robin p-Laplacian problem with competing nonlinearities. Adv. Calc. Var., 12(1):31–58, 2019. https://doi.org/10.1515/acv-2016-0039
U. Guarnotta, S.A. Marano and N.S. Papageorgiou. Multiple nodal solutions to a Robin problem with sign-changing potential and locally defined reaction. Rend. Lincei Mat. Appl., 30(2):269–294, 2019. https://doi.org/10.4171/RLM/847
S. Hu and N.S. Papageorgiou. Handbook of Multivalued Analysis. Vol. I. Theory. Mathematics and its Applications 419. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. https://doi.org/10.1007/978-1-4615-4665-8_17
Z. Li and Z.-Q. Wang. Schrödinger equations with concave and convex nonlinearities. Z. Angew. Math. Phys., 56:609–629, 2005. https://doi.org/10.1007/s00033-005-3115-6
G.M. Lieberman. The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Comm. Partial Diff. Equations, 16(2-3):311–361, 1991. https://doi.org/10.1080/03605309108820761
S.A. Marano and S. Mosconi. Some recent results on the Dirichlet problem for (p,q)-Laplace equations. Discr. Cont. Dyn. Systems, Ser. S, 11(2):279–291, 2018. https://doi.org/10.3934/dcdss.2018015
D. Mugnai and N.S. Papageorgiou. Resonant nonlinear Neumann problems with indefinite weight. Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 11(4):729–788, 2012. https://doi.org/10.2422/2036-2145.201012_003
N.S. Papageorgiou and V.D. Rădulescu. Coercive and noncoercive nonlinear Neumann problems with indefinite potential. Forum Math., 28(3):545–571, 2016. https://doi.org/10.1515/forum-2014-0094
N.S. Papageorgiou and V.D. Rădulescu. Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear. Stud., 16(4):737–764, 2016. https://doi.org/10.1515/ans-2016-0023
N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete Contin. Dyn. Syst. - A, 37(5):2589–2618, 2017. https://doi.org/10.3934/dcds.2017111
N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš. Double-phase problems with reaction of arbitrary growth. Z. Angew. Math. Phys., 69(108), 2018. https://doi.org/10.1007/s00033-018-1001-2
N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš. Nodal solutions for nonlinear nonhomogeneous Robin problems. Rend. Lincei Mat. Appl., 29(4):721–738, 2018. https://doi.org/10.4171/RLM/831
N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš. Nonlinear Analysis Theory and Methods. Springer, Switzerland, 2019. https://doi.org/10.1007/978-3-030-03430-6
N.S. Papageorgiou and C. Vetro. Superlinear (p(z),q(z))equations. Complex Var. Ellip. Equ., 64(1):8–25, 2019. https://doi.org/10.1080/17476933.2017.1409743
N.S. Papageorgiou, C. Vetro and F. Vetro. Multiple nodal solutions for semilinear Robin problems with indefinite linear part and concave terms. Topol. Methods Nonlinear Anal., 50(1):269–286, 2017. https://doi.org/10.12775/TMNA.2017.029
P. Pucci and J. Serrin. The Maximum Principle. Birkhäuser Verlag, Basel, 2007. https://doi.org/10.1007/978-3-7643-8145-5
Z.-Q. Wang. Nonlinear boundary value problems with concave nonlinearities near the origin. NoDEA Nonlinear Differential Equations Appl., 8(1):15–33, 2001. https://doi.org/10.1007/PL00001436
V.V. Zhikov. Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR Izv., 29(1):33–66, 1987. https://doi.org/10.1070/IM1987v029n01ABEH000958
V.V. Zhikov. On variational problems and nonlinear elliptic equations with nonstandard growth conditions. J. Math. Sci. (N.Y.), 173(5):463–570, 2011. https://doi.org/10.1007/s10958-011-0260-7
V. Benci, P. D’Avenia, D. Fortunato and L. Pisani. Solitons in several space dimensions: Derrick’s problem and infinitely many solutions. Arch. Ration. Mech. Anal., 154:297–324, 2000. https://doi.org/10.1007/s002050000101
L. Cherfils and Y. Il’yasov. On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian. Commun. Pure Appl. Anal., 4(1):9–22, 2005. https://doi.org/10.3934/cpaa.2005.4.9
L. Gasiński and N.S. Papageorgiou. Positive solutions for the Robin p-Laplacian problem with competing nonlinearities. Adv. Calc. Var., 12(1):31–58, 2019. https://doi.org/10.1515/acv-2016-0039
U. Guarnotta, S.A. Marano and N.S. Papageorgiou. Multiple nodal solutions to a Robin problem with sign-changing potential and locally defined reaction. Rend. Lincei Mat. Appl., 30(2):269–294, 2019. https://doi.org/10.4171/RLM/847
S. Hu and N.S. Papageorgiou. Handbook of Multivalued Analysis. Vol. I. Theory. Mathematics and its Applications 419. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. https://doi.org/10.1007/978-1-4615-4665-8_17
Z. Li and Z.-Q. Wang. Schrödinger equations with concave and convex nonlinearities. Z. Angew. Math. Phys., 56:609–629, 2005. https://doi.org/10.1007/s00033-005-3115-6
G.M. Lieberman. The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Comm. Partial Diff. Equations, 16(2-3):311–361, 1991. https://doi.org/10.1080/03605309108820761
S.A. Marano and S. Mosconi. Some recent results on the Dirichlet problem for (p,q)-Laplace equations. Discr. Cont. Dyn. Systems, Ser. S, 11(2):279–291, 2018. https://doi.org/10.3934/dcdss.2018015
D. Mugnai and N.S. Papageorgiou. Resonant nonlinear Neumann problems with indefinite weight. Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 11(4):729–788, 2012. https://doi.org/10.2422/2036-2145.201012_003
N.S. Papageorgiou and V.D. Rădulescu. Coercive and noncoercive nonlinear Neumann problems with indefinite potential. Forum Math., 28(3):545–571, 2016. https://doi.org/10.1515/forum-2014-0094
N.S. Papageorgiou and V.D. Rădulescu. Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear. Stud., 16(4):737–764, 2016. https://doi.org/10.1515/ans-2016-0023
N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete Contin. Dyn. Syst. - A, 37(5):2589–2618, 2017. https://doi.org/10.3934/dcds.2017111
N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš. Double-phase problems with reaction of arbitrary growth. Z. Angew. Math. Phys., 69(108), 2018. https://doi.org/10.1007/s00033-018-1001-2
N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš. Nodal solutions for nonlinear nonhomogeneous Robin problems. Rend. Lincei Mat. Appl., 29(4):721–738, 2018. https://doi.org/10.4171/RLM/831
N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš. Nonlinear Analysis Theory and Methods. Springer, Switzerland, 2019. https://doi.org/10.1007/978-3-030-03430-6
N.S. Papageorgiou and C. Vetro. Superlinear (p(z),q(z))equations. Complex Var. Ellip. Equ., 64(1):8–25, 2019. https://doi.org/10.1080/17476933.2017.1409743
N.S. Papageorgiou, C. Vetro and F. Vetro. Multiple nodal solutions for semilinear Robin problems with indefinite linear part and concave terms. Topol. Methods Nonlinear Anal., 50(1):269–286, 2017. https://doi.org/10.12775/TMNA.2017.029
P. Pucci and J. Serrin. The Maximum Principle. Birkhäuser Verlag, Basel, 2007. https://doi.org/10.1007/978-3-7643-8145-5
Z.-Q. Wang. Nonlinear boundary value problems with concave nonlinearities near the origin. NoDEA Nonlinear Differential Equations Appl., 8(1):15–33, 2001. https://doi.org/10.1007/PL00001436
V.V. Zhikov. Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR Izv., 29(1):33–66, 1987. https://doi.org/10.1070/IM1987v029n01ABEH000958
V.V. Zhikov. On variational problems and nonlinear elliptic equations with nonstandard growth conditions. J. Math. Sci. (N.Y.), 173(5):463–570, 2011. https://doi.org/10.1007/s10958-011-0260-7