## Topological Methods in Nonlinear Analysis

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

#### Abstract

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 $q\in[2,2^{*})$, $V=V_{{\rm per}}+V_{{\rm loc}}\in L^{\infty}(\mathbb{R}^{N})$ is the sum of a periodic potential $V_{{\rm per}}$ and a localized potential $V_{{\rm loc}}$ 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 51, Number 1 (2018), 215-242.

Dates
First available in Project Euclid: 27 November 2017

https://projecteuclid.org/euclid.tmna/1511751649

Digital Object Identifier
doi:10.12775/TMNA.2017.046

Mathematical Reviews number (MathSciNet)
MR3784743

Zentralblatt MATH identifier
06887979

#### Citation

Che, Guofeng F.; Chen, Haibo B. Ground state solutions for a class of semilinear elliptic systems with sum of periodic and vanishing potentials. Topol. Methods Nonlinear Anal. 51 (2018), no. 1, 215--242. doi:10.12775/TMNA.2017.046. https://projecteuclid.org/euclid.tmna/1511751649

#### References

• 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 time-harmonic 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 \romPDE on $\mathbb{R}^{N}$, 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 $\mathbb{R}^{N}$, 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 $\mathbb{R}^{N}$, 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 $\mathbb{R}^{N}$, 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.