# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022114
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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. Fragnelli, D. 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. 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., 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. Zhang, W. 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. Fragnelli, D. 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. 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., 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. Zhang, W. 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.
 [1] Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 [2] Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub. Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1033-1050. doi: 10.3934/dcds.2010.28.1033 [3] Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151 [4] A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure and Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253 [5] Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897 [6] Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2653-2676. doi: 10.3934/dcds.2020379 [7] Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023 [8] Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 2020, 28 (1) : 205-220. doi: 10.3934/era.2020014 [9] Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6133-6166. doi: 10.3934/dcds.2016068 [10] Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125 [11] Daomin Cao, Hang Li. High energy solutions of the Choquard equation. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3023-3032. doi: 10.3934/dcds.2018129 [12] Ruyun Ma, Yanqiong Lu. Disconjugacy and extremal solutions of nonlinear third-order equations. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1223-1236. doi: 10.3934/cpaa.2014.13.1223 [13] Mostafa Fazly. Regularity of extremal solutions of nonlocal elliptic systems. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 107-131. doi: 10.3934/dcds.2020005 [14] M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057 [15] Vladimir Lubyshev. Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem. Communications on Pure and Applied Analysis, 2009, 8 (3) : 999-1018. doi: 10.3934/cpaa.2009.8.999 [16] Geng Chen, Yannan Shen. Existence and regularity of solutions in nonlinear wave equations. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3327-3342. doi: 10.3934/dcds.2015.35.3327 [17] Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6553-6605. doi: 10.3934/dcdsb.2019155 [18] Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921 [19] Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561 [20] Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445

2021 Impact Factor: 1.865