Ground state solutions for a class of semilinear elliptic systems with sum of periodic and vanishing potentials

Guofeng Che, Haibo Chen


In this paper, we consider the following semilinear elliptic systems: $$ \begin{cases} -\Delta u+V(x)u=F_{u}(x, u, v)-\Gamma(x)|u|^{q-2}u & \mbox{in }\mathbb{R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v)-\Gamma(x)|v|^{q-2}v & \mbox{in }\mathbb{R}^{N},\\ \end{cases} $$% where $q\in[2,2^{*})$, $V=V_{\rom{per}}+V_{\rom{loc}}\in L^{\infty}(\mathbb{R}^{N})$ is the sum of a periodic potential $V_{\rom{per}}$ and a localized potential $V_{\rom{loc}}$ 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.


Semilinear elliptic systems; ground state; periodic potential; localized potential; variational methods

Full Text:



A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.

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.

T. Bartsch and J. Mederski, Ground and bound state solutions of semilinear timeharmonic Maxwell equations in a bounded domain, Arch. Rational Mech. Anal. 215 (2015), 283–306.

B. Bieganowski and J. Mederski, Nonlinear Schrödinger equations with sum of periodic and vanishig potentials and sign-changning nonlinearities, preprint, arXiv:1602.05078.

D. Cao and Z. Tang, Solutions with prescribed number of nodes to superlinear elliptic systems, Nonlinear Anal. 55 (2003), 702–722.

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.

V. Coti-Zelati and P. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on RN , Comm. Pure Appl. Math. 45 (1992), 1217–1269.

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.

S. Duan and X. Wu, The existence of solutions for a class of semilinear elliptic systems, Nonlinear Anal. 73 (2010), 2842–2854.

G. Figueiredo and H.R. Quoirin, Ground states of elliptic problems involving nonhomogeneous operators, Indiana Univ. Math. J. 65 (2016), 779–795.

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.

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on RN , Indiana Univ. Math. J. 54 (2005), 443–464.

G. Li and X. Tang, Nehari-type state solutions for Schrödinger equations including critical exponent, Appl. Math. Lett. 37 (2014), 101–106.

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.

F.F. Liao, X.H. Tang, J. Zhang and D.D. Qin, Super-quadratic conditions for periodic elliptic system on RN , Electron. J. Differential Equations 127 (2015), 1–11.

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.

L. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations 229 (2006), 743–767.

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.

J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Comm. Partial Differential Equations 41 (2016), 1426–1440.

A. Pankov, On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc. 136 (2008), 2565–2570.

Z. Qu and C. Tang, Existence and multiplicity results for some elliptic systems at resonance, Nonlinear Anal. 71 (2009), 2660–2666.

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Analysis of Operators, Vol. IV, Academic Press, New York, 1978.

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.

E. Silva, Existence and multiplicity of solutions for semilinear elliptic systems, NoDEA Nonlinear Differential Equations Appl. 1 (1994), 339–363.

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009), 3802–3822.

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.

L.R. Xia, J. Zhang and F.K. Zhao, Ground state solutions for superlinear elliptic systems on RN , J. Math. Anal. Appl. 401 (2013), 518–525.

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.

J. Zhang and Z. Zhang, Existence results for some nonlinear elliptic systems, Nonlinear Anal. 71 (2009), 2840–2846.

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.


  • There are currently no refbacks.

Partnerzy platformy czasopism