doi: 10.3934/dcdss.2022114
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Sequences of high and low energy solutions for weighted (p, q)-equations

1. 

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

2. 

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland

3. 

Department of Mathematics, University of Craiova, Craiova 200585, Romania, China-Romania Research Center in Applied Mathematics

4. 

College of Science, Hunan University of Technology and Business, Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation, Changsha, Hunan 410205, China

* Corresponding author: Jian Zhang (zhangjian433130@163.com)

Received  February 2022 Early access May 2022

We consider a Dirichlet elliptic equation driven by a weighted $ (p,q) $-Laplace differential operator. The weights are in general different. When the reaction is "superlinear", using the fountain theorem, we show the existence of a sequence of distinct smooth solutions with energies diverging to $ +\infty $. When the reaction is "sublinear" (possibly resonant), we establish the existence of a sequence of nodal solutions converging to zero in $ C^1_0(\bar{\Omega}) $ (in particular, the energies converge to zero).

Citation: Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Jian Zhang. Sequences of high and low energy solutions for weighted (p, q)-equations. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022114
References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc., 196 (2008), no. 915. doi: 10.1090/memo/0915.

[2]

V. I. Bogachev and O. G. Smolyanov, Real and Functional Analysis, Moscow Lectures, 4. Springer, Cham, 2020. doi: 10.1007/978-3-030-38219-3.

[3]

S. Chen and X. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, 268 (2020), 945-976.  doi: 10.1016/j.jde.2019.08.036.

[4]

G. FragnelliD. Mugnai and N. S. Papageorgiou, The Brezis-Oswald result for quasilinear Robin problems, Adv. Nonlinear Stud., 16 (2016), 603-622.  doi: 10.1515/ans-2016-0010.

[5]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall$/$CRC, Boca Raton, FL, 2006.

[6]

L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis, Springer, Cham, 2016. doi: 10.1007/978-3-319-27817-9.

[7]

L. Gasiński and N. S. Papageorgiou, On a nonlinear parametric Robin problem with a locally defined reaction, Nonlinear Anal., 185 (2019), 374-387.  doi: 10.1016/j.na.2019.03.019.

[8]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory, Mathematics and its Applications, 419. Kluwer Academic Publishers, Dordrecht, 1997.

[9]

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.

[10] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. 
[11]

S. Leonardi and N. S. Papageorgiou, Arbitrarily small nodal solutions for parametric Robin $(p, q)$-equations plus an indefinite potential, Acta Math. Sci. Ser. B (Engl. Ed.), 42 (2022), 561-574.  doi: 10.1007/s10473-022-0210-0.

[12]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361.  doi: 10.1080/03605309108820761.

[13]

D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 729-788. 

[14]

H.-L. Pan and C.-L. Tang, Existence of infinitely many solutions for semilinear elliptic equations, Electron. J. Differential Equations, (2016), Paper No. 167, 11 pp.

[15]

N. S. Papageorgiou and V. D. Rădulescu, An infinity of nodal solutions for superlinear Robin problems with an indefinite and unbounded potential, Bull. Sci. Math., 141 (2017), 251-266.  doi: 10.1016/j.bulsci.2017.03.001.

[16]

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

[17]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys., 69 (2018), Paper No. 108, 21 pp. doi: 10.1007/s00033-018-1001-2.

[18]

N. S. PapageorgiouV. D. Rădulescu 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.

[19]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Nonlinear Analysis–Theory and Methods, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.

[20]

N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for parametric double phase Dirichlet problems, Commun. Contemp. Math., 23 (2021), Paper No. 2050006, 18 pp. doi: 10.1142/S0219199720500066.

[21]

N. S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis, De Gruyter, Berlin, 2018.

[22]

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

[23]

X. H. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 110, 25 pp. doi: 10.1007/s00526-017-1214-9.

[24]

X. H. Tang and B. T. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.  doi: 10.1016/j.jde.2016.04.032.

[25]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[26]

J. Zhang and W. Zhang, Semiclassical states for coupled nonlinear Schrödinger system with competing potentials, J. Geom. Anal., 32 (2022), Paper No. 114, 36 pp. doi: 10.1007/s12220-022-00870-x.

[27]

J. ZhangW. Zhang and X. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565-4583.  doi: 10.3934/dcds.2017195.

[28]

J.-H. Zhao and P.-H. Zhao, Existence of infinitely many weak solutions for the $p$-Laplacian with nonlinear boundary conditions, Nonlinear Anal., 69 (2008), 1343-1355.  doi: 10.1016/j.na.2007.06.036.

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, Mem. Amer. Math. Soc., 196 (2008), no. 915. doi: 10.1090/memo/0915.

[2]

V. I. Bogachev and O. G. Smolyanov, Real and Functional Analysis, Moscow Lectures, 4. Springer, Cham, 2020. doi: 10.1007/978-3-030-38219-3.

[3]

S. Chen and X. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, 268 (2020), 945-976.  doi: 10.1016/j.jde.2019.08.036.

[4]

G. FragnelliD. Mugnai and N. S. Papageorgiou, The Brezis-Oswald result for quasilinear Robin problems, Adv. Nonlinear Stud., 16 (2016), 603-622.  doi: 10.1515/ans-2016-0010.

[5]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall$/$CRC, Boca Raton, FL, 2006.

[6]

L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis, Springer, Cham, 2016. doi: 10.1007/978-3-319-27817-9.

[7]

L. Gasiński and N. S. Papageorgiou, On a nonlinear parametric Robin problem with a locally defined reaction, Nonlinear Anal., 185 (2019), 374-387.  doi: 10.1016/j.na.2019.03.019.

[8]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory, Mathematics and its Applications, 419. Kluwer Academic Publishers, Dordrecht, 1997.

[9]

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.

[10] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. 
[11]

S. Leonardi and N. S. Papageorgiou, Arbitrarily small nodal solutions for parametric Robin $(p, q)$-equations plus an indefinite potential, Acta Math. Sci. Ser. B (Engl. Ed.), 42 (2022), 561-574.  doi: 10.1007/s10473-022-0210-0.

[12]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361.  doi: 10.1080/03605309108820761.

[13]

D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 729-788. 

[14]

H.-L. Pan and C.-L. Tang, Existence of infinitely many solutions for semilinear elliptic equations, Electron. J. Differential Equations, (2016), Paper No. 167, 11 pp.

[15]

N. S. Papageorgiou and V. D. Rădulescu, An infinity of nodal solutions for superlinear Robin problems with an indefinite and unbounded potential, Bull. Sci. Math., 141 (2017), 251-266.  doi: 10.1016/j.bulsci.2017.03.001.

[16]

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

[17]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys., 69 (2018), Paper No. 108, 21 pp. doi: 10.1007/s00033-018-1001-2.

[18]

N. S. PapageorgiouV. D. Rădulescu 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.

[19]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Nonlinear Analysis–Theory and Methods, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.

[20]

N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for parametric double phase Dirichlet problems, Commun. Contemp. Math., 23 (2021), Paper No. 2050006, 18 pp. doi: 10.1142/S0219199720500066.

[21]

N. S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis, De Gruyter, Berlin, 2018.

[22]

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

[23]

X. H. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 110, 25 pp. doi: 10.1007/s00526-017-1214-9.

[24]

X. H. Tang and B. T. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.  doi: 10.1016/j.jde.2016.04.032.

[25]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[26]

J. Zhang and W. Zhang, Semiclassical states for coupled nonlinear Schrödinger system with competing potentials, J. Geom. Anal., 32 (2022), Paper No. 114, 36 pp. doi: 10.1007/s12220-022-00870-x.

[27]

J. ZhangW. Zhang and X. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565-4583.  doi: 10.3934/dcds.2017195.

[28]

J.-H. Zhao and P.-H. Zhao, Existence of infinitely many weak solutions for the $p$-Laplacian with nonlinear boundary conditions, Nonlinear Anal., 69 (2008), 1343-1355.  doi: 10.1016/j.na.2007.06.036.

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