Abstract
In this paper, we consider the following semilinear elliptic systems:
where \(V:\mathbb {R}^{N}\rightarrow \mathbb {R},~F_{u}(x,u,v)\) and \(F_{v}(x,u,v)\) are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of \(-\triangle +V\). Under appropriate assumptions on \(F_{u}(x, u, v)\) and \(F_{v}(x, u, v)\), we prove the above system has a ground-state solution by using the Nehari-type technique in a strongly indefinite setting. Furthermore, the existence of infinitely many geometrically distinct solutions is obtained via variational methods. Recent results from the literature are improved and extended.
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References
Silva, E.: Existence and multiplicity of solutions for semilinear elliptic systems. NoDEA 1, 339–363 (1994)
Duan, S., Wu, X.: The existence of solutions for a class of semilinear elliptic systems. Nonlinear Anal. 73, 2842–2854 (2010)
Qu, Z., Tang, C.: Existence and multiplicity results for some elliptic systems at resonance. Nonlinear Anal. 71, 2660–2666 (2009)
Shi, H.X., Chen, H.B.: Ground state solutions for resonant cooperative elliptic systems with general superlinear terms. Mediterr. J. Math. 13, 2897–2909 (2016)
Zhang, J., Zhang, Z.: Existence results for some nonlinear elliptic systems. Nonlinear Anal. 71, 2840–2846 (2009)
Li, G., Tang, X.: Nehari-type state solutions for Schrödinger equations including critical exponent. Appl. Math. Lett. 37, 101–106 (2014)
Che, G.F., Chen, H.B.: Multiplicity of small negative-energy solutions for a class of semilinear elliptic systems. Bound. Value. Probl. 2016, 1–12 (2016)
Liao, F.F., Tang, X.H., Qin, D.D.: Super-quadratic conditions for periodic elliptic system on \(\mathbb{R}^{N}\). Electron. J. Differ. Equ. 127, 1–11 (2015)
Maia, L., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229, 743–767 (2006)
Zhang, J., Tang, X.H., Zhang, W.: Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms. Nonlinear Anal. 95, 1–10 (2014)
Zhang, J., Qin, W.P., Zhao, F.K.: Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system. J. Math. Anal. Appl. 399, 433–441 (2013)
Chen, G.W., Ma, S.W.: Nonexistence and multiplicity of solutions for nonlinear elliptic systems in \(\mathbb{R}^{N}\). Nonlinear Anal. 36, 233–248 (2017)
Zhang, W., Zhang, J., Zhao, F.K.: Multiple solutions for asymptotically quadratic and superquadratic elliptic system of Hamiltonian type. Appl. Math. Comput. 263, 36–46 (2015)
Shi, H.X., Chen, H.B.: Ground state solutions for asymptotically periodic coupled Kirchhoff-type systems with critical growth. Math. Methods Appl. Sci. 39, 2193–2201 (2016)
Che, G.F., Chen, H.B.: Existence and multiplicity of systems of Kirchhoff-type equations with general potentials. Math. Methods Appl. Sci. 40, 775–785 (2017)
Tang, X.H.: Ground state solutions of Nehari–Pankov type for a superlinear Hamiltonian elliptic system on \(\mathbb{R}^{N}\). Canad. Math. Bull. 58(3), 651–663 (2015)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Analysis of Operators, vol. IV. Academic Press, New York (1978)
Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257(12), 3802–3822 (2009)
Pankov, A.: On decay of solutions to nonlinear Schrödinger equations. Proc. Am. Math. Soc. 136, 2565–2570 (2008)
Mederski, J.: Ground states of a system of nonlinear Schrödinger equations with periodic potentials. Commun. Partial Diff. Equ. 41(9), 1426–1440 (2016)
Guo, Q., Mederski, J.: Ground states of nonlinear Schrödinger equations with sum of periodic and inverse square potentials. J. Differ. Equ. 260, 4180–4202 (2016)
Mederski, J.: Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum. Topol. Methods Nonlinear Anal. 46(2), 755–771 (2015)
Bieganowski, B., Mederski, J.: Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities. arXiv preprint arXiv:1602.05078
Bartsch, T., Ding, Y.: On a nonlinear Schrödinger equation with periodic potential. Math. Ann. 313(1), 15–37 (1999)
Pankov, A.: Periodic nonlinear Schröinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005)
Badiale, M., Pisani, L., Rolando, S.: Sum of weighted Lebesgue spaces and nonlinear elliptic equations. Nodea Nonlinear Differ. Equ. Appl. 18, 369–405 (2011)
Bartsch, T., Mederski, J.: Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain. Arch. Ration. Mech. Anal. 215(1), 283–306 (2015)
Willem, M.: Minimax Theorems. Birkhäuser Verlag, Basel (1996)
Struwe, M.: Variational Methods. Springer, Berlin (2008)
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Communicated by Norhashidah Hj. Mohd. Ali.
This work is partially supported by National Natural Science Foundation of China 11671403, by the Fundamental Research Funds for the Central Universities of Central South University 2017zzts058, and by the Mathematics and Interdisciplinary Sciences Project of CSU.
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Che, G., Chen, H. & Yang, L. Existence and Multiplicity of Solutions for Semilinear Elliptic Systems with Periodic Potential. Bull. Malays. Math. Sci. Soc. 42, 1329–1348 (2019). https://doi.org/10.1007/s40840-017-0551-3
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DOI: https://doi.org/10.1007/s40840-017-0551-3
Keywords
- Semilinear elliptic systems
- Strongly indefinite functional
- Ground state
- Nehari–Pankov manifold
- Variational methods