Nontrivial solutions for a class of Hamiltonian strongly degenerate elliptic system

References

• Franchi B, Lanconelli E. Une métrique associée à une classe d'opérateurs elliptiques dégénérés (French) [A metric associated with a class of degenerate elliptic operators] Conference on linear partial and pseudodifferential operators (Torino, 1982). Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue, 105–114 (1984).
   Google Scholar
• Kogoj AE, Lancenolli E. On semilinear Δλ-Laplace equation. Nonlinear Anal. 2012;75:4637–4649.
   Web of Science ®Google Scholar
• Anh CT, My BK. Existence of solutions to Δλ-Laplace equations without the Ambrosetti-Rabinowitz condition. Complex Var Elliptic Equ. 2016;61:137–150.
   Web of Science ®Google Scholar
• Anh CT, My BK. Liouville type theorems for elliptic inequalities involving the Δλ-Laplace operator. Complex Var Elliptic Equ. 2016;61:1002–1013.
   Web of Science ®Google Scholar
• Anh CT, My BK. Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system. Adv Nonlinear Anal. 2019;8:661–678.
   Web of Science ®Google Scholar
• Anh CT, Lee J, My BK. On a class of Hamiltonian strongly degenerate elliptic systems with concave and convex nonlinearities. Complex Var Elliptic Equ. 2020;65:648–671.
   Web of Science ®Google Scholar
• Hamdani MK. Multiple solutions for Grushin operator without odd nonlinearity. Asia-Eur J Math. 2020;13:2050131. DOI:10.1142/S1793557120501314
   Google Scholar
• Luyen DT. Multiple solutions for semilinear Δγ differential equations in RN with sign-changing potential. Commun Math Anal. 2019;22:61–75.
   Google Scholar
• Luyen DT, Tri NM. Existence of infinitely many solutions for semilinear degenerate Schrödinger equations. J Math Anal Appl. 2018;461:1271–1286.
   Web of Science ®Google Scholar
• Rahal B, Hamdani MK. Infinitely many solutions for Δα-Laplace equations with sign-changing potential. J Fixed Point Theory Appl. 2018;20:349. DOI:10.1007/s11784-018-0617-3
   Google Scholar
• Tang X. Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity. J Math Anal Appl. 2013;401:407–415.
   Web of Science ®Google Scholar
• Kogoj AE, Lanconelli E. Linear and semilinear problems involving Δλ-Laplacians. Proceedings of the International Conference “Two nonlinear days in Urbino 2017”, 167–178, Electron. J. Differ. Equ. Conf., 25, Texas State Univ.-San Marcos, Dept. Math., San Marcos, TX, 2018.
   Google Scholar
• Bartsch T, De Figueiredo DG. Infinitely many solutions of nonlinear elliptic systems. In: Progress in nonlinear differential equations and their applications. Vol. 35. Basel, Switzerland: Birkhäuser; 1999. p. 51–67.
   Google Scholar
• Benci V, Rabinowitz PH. Critical point theorems for indefinite functionals. Invent Math. 1979;52:241–273.
   Web of Science ®Google Scholar
• Clément PH, de Fegueiredo D, Mitedieri E. Positive solutions of semilinear elliptic systems. Comm Partial Differ Equ. 1992;17:923–940.
   Web of Science ®Google Scholar
• De Figueiredo DG, Ding YH. Strongly indefinite functions and multiple solutions of elliptic systems. Trans Am Math Soc. 2003;355:2973–2989.
   Web of Science ®Google Scholar
• De Figueiredo DG, Felmer PL. On superquadratic elliptic systems. Trans Am Math Soc. 1994;343:99–116.
   Web of Science ®Google Scholar
• De Figueiredo DG, Yang J. Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal. 1998;33:211–234.
   Web of Science ®Google Scholar
• Hulshof J, Van De Vorst RCAM. Differential systems with strongly variational structure. J Funct Anal. 1993;114:32–58.
   Web of Science ®Google Scholar
• Kryszewski W, Szulkin A. An infinite dimensional Morse theory with applications. Trans Am Math Soc. 1997;349:3181–3234.
   Web of Science ®Google Scholar
• Li G, Szulkin A. An asymptotically periodic Schrödinger equation with indefinite linear part. Commun Contemp Math. 2002;4(4):763–776.
   Web of Science ®Google Scholar
• Van der Vorst RCAM. Variational identities and applications to differential systems. Arch Rat. Mech Anal. 1992;116:375–398.
   Web of Science ®Google Scholar
• Alves CO, de Moraes Filho DC, Souto MA. On systems of elliptic equations involving subcritical or critical Sobolev exponents. Nonlinear Anal TMA. 2000;42:771–787.
   Web of Science ®Google Scholar
• Alves CO, de Moraes Filho DC. Multiple solutions for an elliptic system on bounded and unbounded domains. Nonlinear Anal. 2004;56:555–568.
   Web of Science ®Google Scholar
• Sirakov B. On the existence of solutions of Hamiltonian elliptic systems in RN. Adv Differential Equations. 2000;5:1445–1464.
   Google Scholar
• Zhao FK, Ding YH. On Hamiltonian elliptic system with periodic or non-periodic potentials. J Differential Equations. 2010;249:2964–2985.
   Web of Science ®Google Scholar
• Zhao FK, Zhao LG, Ding YH. Infinitely many solutions for asymptotically linear periodic Hamiltonian system. ESAIM Control Optim Calc Var. 2010;16:77–91.
   Web of Science ®Google Scholar
• Ávila AI, Yang J. On the existence and shape of least energy solutions for some elliptic systems. J Differential Equations. 2003;191:348–376.
   Web of Science ®Google Scholar
• Bonheure D, dos Santos E, Tavares H. Hamiltonian elliptic systems: a guide to variational frameworks. Port Math. 2014;71:301–395.
   Web of Science ®Google Scholar
• Albuquerque FSB, do Ó J, Medeiros ES. On a class of Hamiltonian elliptic systems involving unbounded or decaying potentials in dimension two. Math Nachr. 2016;289(13):1568–1584.
   Web of Science ®Google Scholar
• Albuquerque FSB, Alves CO, Medeiros ES. Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in R2. J Math Anal Appl. 2014;409:1021–1031.
   Web of Science ®Google Scholar
• Toon E, Ubilla P. Hamiltonian systems of Schrödinger equations with vanishing potentials. Commun Contem Math. 2020;254:2050074 (20 pages). DOI:10.1142/S0219199720500741
   Google Scholar
• Alves CO, Souto MAS. Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity. J Differential Equations. 2013;254:1977–1991.
   Web of Science ®Google Scholar
• Chen J, Huang X, Cheng B, et al. Existence and multiplicity of nontrivial solutions for nonlinear Schrödinger equations with unbounded potentials. Filomat. 2018;32:2465–2481.
   Web of Science ®Google Scholar
• de Souza M. Existence and multiplicity of solutions for a class of fractional elliptic systems. Collect Math. 2020;71:103–122.
   Web of Science ®Google Scholar
• Toon E, Ubilla P. Existence of positive solutions of Schrödinger equations with vanishing potentials. Discrete Contin Dyn Syst. 2020;40:5831–5843.
   Web of Science ®Google Scholar
• Lam N, Lu G. Superlinear elliptic equations with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition. J Geom Anal. 2014;24:118–143.
   Web of Science ®Google Scholar
• Li G, Ye H. Existence of positive solutions to semilinear elliptic system in RN with zero mass. Acta Math Scientia. 2013;33(4):913–928.
   Web of Science ®Google Scholar
• Liu Z, Wang ZQ. On the Ambrosetti-Rabinowitz superlinear condition. Adv Nonlinear Stud. 2004;4(4):563–574.
   Web of Science ®Google Scholar
• Anh CT. Global attractor for a semilinear strongly degenerate parabolic equation on RN. NoDEA Nonlinear Differential Equations Appl. 2014;21:663–678.
   Web of Science ®Google Scholar
• Kryszewski W, Szulkin A. Generalized linking theorem with an application to semilinear Schrödinger equation. Adv Differential Equations. 1998;3:441–471.
   Google Scholar