Ground state solutions for a class of semilinear elliptic systems with sum of periodic and vanishing potentials. (English) Zbl 1400.35135
Summary: In this paper, we consider the following semilinear elliptic systems:
\[
\begin{cases} -\Delta u+V(x)u=F_{u}(x, u, v)-\Gamma(x)|u|^{q-2}u \quad \mathrm{in}\mathbb{R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v)-\Gamma(x)|v|^{q-2}v \quad \mathrm{in}\mathbb{R}^{N},\\ \end{cases}
\]
where \(q\in[2,2^{*})\), \(V=V_{{\mathrm per}}+V_{{\mathrm loc}}\in L^{\infty}(\mathbb{R}^{N})\) is the sum of a periodic potential \(V_{{\mathrm {per}}}\) and a localized potential \(V_{\mathrm{loc}}\) and \(\Gamma\in L^{\infty}(\mathbb{R}^{N})\) is periodic and \(\Gamma(x)\geq0\) for almost every \(x\in\mathbb{R}^{N}\). Under some appropriate assumptions on \(F\), we investigate the existence and nonexistence of ground state solutions for the above system. Recent results from the literature are improved and extended.
MSC:
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35J47 | Second-order elliptic systems |
References:
| [1] | A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7 |
| [2] | T. Bartsch and Y.H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr. 279 (2006), 1-22. · Zbl 1117.58007 · doi:10.1002/mana.200410420 |
| [3] | T. Bartsch and J. Mederski, Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain, Arch. Rational Mech. Anal. 215 (2015), 283-306. · Zbl 1315.35214 · doi:10.1007/s00205-014-0778-1 |
| [4] | B. Bieganowski and J. Mederski, Nonlinear Schrödinger equations with sum of periodic and vanishig potentials and sign-changning nonlinearities, preprint, arXiv:1602.05078. · Zbl 1375.35475 |
| [5] | D. Cao and Z. Tang, Solutions with prescribed number of nodes to superlinear elliptic systems, Nonlinear Anal. 55 (2003), 702-722. · Zbl 1073.35078 · doi:10.1016/j.na.2003.07.014 |
| [6] | G.F. Che and H.B. Chen, Multiplicity of small negative-energy solutions for a class of semilinear elliptic systems, Bound. Value Probl. 2016 (2016), 1-12. · Zbl 1343.35095 |
| [7] | V. Coti-Zelati and P. Rabinowitz, Homoclinic type solutions for a semilinear elliptic \romPDE on \(\mathbb{R}^{N}\), Comm. Pure Appl. Math. 45 (1992), 1217-1269. · Zbl 0785.35029 |
| [8] | P. D’Avenia and J. Mederski, Positive ground states for a system of Schrödinger equations with critically growing nonlinearities, Calc. Var. Partial Differential Equations 53 (2015), 879-900. · Zbl 1317.35067 |
| [9] | S. Duan and X. Wu, The existence of solutions for a class of semilinear elliptic systems, Nonlinear Anal. 73 (2010), 2842-2854. · Zbl 1198.35080 · doi:10.1016/j.na.2010.06.031 |
| [10] | G. Figueiredo and H.R. Quoirin, Ground states of elliptic problems involving nonhomogeneous operators, Indiana Univ. Math. J. 65 (2016), 779-795. · Zbl 1355.35060 · doi:10.1512/iumj.2016.65.5828 |
| [11] | Q. Guo and J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inverse square potentials, J. Differential Equations 260 (2016), 4180-4202. · Zbl 1335.35232 · doi:10.1016/j.jde.2015.11.006 |
| [12] | L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on \(\mathbb{R}^{N}\), Indiana Univ. Math. J. 54 (2005), 443-464. · Zbl 1143.35321 |
| [13] | G. Li and X. Tang, Nehari-type state solutions for Schrödinger equations including critical exponent, Appl. Math. Lett. 37 (2014), 101-106. · Zbl 1320.35167 |
| [14] | F.F. Liao, X.H. Tang, J. Zhang and D.D. Qin, Semi-classical solutions of perturbed elliptic system with general superlinear nonlinearity, Bound. Value Probl. 2014 (2014), 1-13. · Zbl 1304.35270 |
| [15] | F.F. Liao, X.H. Tang, J. Zhang and D.D. Qin, Super-quadratic conditions for periodic elliptic system on \(\mathbb{R}^{N}\), Electron. J. Differential Equations 127 (2015), 1-11. · Zbl 1322.35022 |
| [16] | P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Parts I and II, Ann. Inst. H. Poincaré Anal. Non Linéare 1 (1984), 109-145; 223-283. · Zbl 0704.49004 · doi:10.1016/S0294-1449(16)30422-X |
| [17] | L. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations 229 (2006), 743-767. · Zbl 1104.35053 · doi:10.1016/j.jde.2006.07.002 |
| [18] | J. Mederski, Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum, Topol. Methods Nonlinear Anal. 46 (2015), 755-771. · Zbl 1362.35289 |
| [19] | J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Comm. Partial Differential Equations 41 (2016), 1426-1440. · Zbl 1353.35267 · doi:10.1080/03605302.2016.1209520 |
| [20] | A. Pankov, On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc. 136 (2008), 2565-2570. · Zbl 1143.35093 |
| [21] | Z. Qu and C. Tang, Existence and multiplicity results for some elliptic systems at resonance, Nonlinear Anal. 71 (2009), 2660-2666. · Zbl 1172.35382 · doi:10.1016/j.na.2009.01.106 |
| [22] | M. Reed and B. Simon, Methods of Modern Mathematical Physics, Analysis of Operators, Vol. IV, Academic Press, New York, 1978. · Zbl 0401.47001 |
| [23] | H.X. Shi and H.B. Chen, Ground state solutions for resonant cooperative elliptic systems with general superlinear terms, Mediterr. J. Math. 13 (2016), 2897-2909. · Zbl 1355.35065 · doi:10.1007/s00009-015-0663-7 |
| [24] | E. Silva, Existence and multiplicity of solutions for semilinear elliptic systems, NoDEA Nonlinear Differential Equations Appl. 1 (1994), 339-363. · Zbl 0814.35036 · doi:10.1007/BF01194985 |
| [25] | A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009), 3802-3822. · Zbl 1178.35352 · doi:10.1016/j.jfa.2009.09.013 |
| [26] | M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. · Zbl 0856.49001 |
| [27] | L.R. Xia, J. Zhang and F.K. Zhao, Ground state solutions for superlinear elliptic systems on \(\mathbb{R}^{N}\), J. Math. Anal. Appl. 401 (2013), 518-525. · Zbl 1266.35051 |
| [28] | J. Zhang, W.P. Qin and F.K. Zhao, Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system, J. Math. Anal. Appl. 399 (2013), 433-441. · Zbl 1345.35031 · doi:10.1016/j.jmaa.2012.10.030 |
| [29] | J. Zhang and Z. Zhang, Existence results for some nonlinear elliptic systems, Nonlinear Anal. 71 (2009), 2840-2846. · Zbl 1167.35360 · doi:10.1016/j.na.2009.01.158 |
| [30] | F.K. Zhao, L.G. Zhao and Y.H. Ding, Infinitely many solutions for asymptotically linear periodic Hamiltonian system, ESAIM Control Optim. Calc. Var. 16 (2010), 77-91. · Zbl 1189.35091 · doi:10.1051/cocv:2008064 |
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