Hamiltonian elliptic systems with nonlinearities of arbitrary growth
DOI:
https://doi.org/10.12775/TMNA.2016.018Keywords
Nonlinear Schrӧdinger equations, standing waves, variational methods, elliptic systems, nonlinearities of arbitrary growthAbstract
We study the existence of standing wave solutions for the following class of elliptic Hamiltonian-type systems: \[ \begin{cases} -\hs^2\Delta u+ V(x)u = g(v) & \mbox{in } \mathbb{R}^N, \\ -\hs^2\Delta v+ V(x)v = f(u) & \mbox{in } \mathbb{R}^N, \end{cases} \] with $N\geq2$, where $\hbar$ is a positive parameter and the nonlinearities $f,g$ are superlinear and can have arbitrary growth at infinity. This system is in variational form and the associated energy functional is strongly indefinite. Moreover, in view of unboundedness of the domain $\mathbb{R}^N$ and the arbitrary growth of nonlinearities we have lack of compactness. We use a dual variational approach in combination with a mountain-pass type procedure to prove the existence of positive solution for $\hbar$ sufficiently small.References
C.O. Alves, Existence of positive solutions for an equation involving supercritical exponent in RN , Nonlinear Anal. 42 (2000), 573–581.
C.O. Alves, J.M. do Ó and M.A.S. Souto, Local mountain-pass for a class of elliptic problems in RN involving critical growth, Nonlinear Anal. 46 (2001), 495–510.
A. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems, J. Differential Equations 191 (2003), 348–376.
D. Bonheure, E.M. dos Santos and M. Ramos, Ground state and non-ground state solutions of some strongly coupled elliptic systems, Trans. Amer. Math. Soc. 364 (2012), 447–491.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations – I and II, Arch. Ration. Mech. Anal. 82 (1983), 313–376.
D. Bonheure and M. Ramos, Multiple critical points of perturbed symmetric strongly indefinite functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 675–688.
J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal. 185 (2007), 185–200.
J. Byeon and Z. Wang, Standing waves with a critical frequency for nonlinear Schrdinger equations, Arch. Ration. Mech. Anal. 165 (2002), 295–316.
Ph. Clément, D. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations 17 (1992), 923–940.
D.G. de Figueiredo, Nonlinear elliptic systems. An. Acad. Brasil. Ciênc. 72 (2000), 453–469.
D.G. de Figueiredo, Semilinear elliptic systems: existence, multiplicity, symmetry of solutions, Handbook of differential equations, Vol. 5: Stationary Partial Differential Equations (M. Chipot, ed.), Elsevier, 2008, 1–48.
D.G. de Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal. 33 (1998), 211–234.
M. del Pino and P.L. Felmer, Local mountain-pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137.
J. Hulshof, R.C.A.M. Van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal. 114 (1993), 32–58.
T.C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations 229 (2006), 538–569.
O.H. Miyagaki, On a class of semilinear elliptic problems in RN with critical growth, Nonlinear Anal. 29 (1997), 773–781.
A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in RN , J. Differential Equations 229 (2006), 570–587.
J.D. Murray, Mathematical biology I, Spatial models and biomedical applications, third edition. Interdisciplinary Applied Mathematics, Vol. 18, Springer–Verlag, New York, 2003. xxvi+811 pp.
P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew Math. Phys. 43 (1992), 272–291.
M. Ramos and H. Tavares, Solutions with multiple spike patterns for an elliptic system, Calc. Var. Partial Differential Equations 31 (2008), 1–25.
M. Ramos and J. Yang, Spike-layered solutions for an elliptic system with Neumann boundary conditions, Trans. Amer. Math. Soc. 357 (2005), 3265–3284.
E. Schrödinger, Collected Papers on Wave Mechanics, Third Edition, AMS Chelsea Publishing, 2003.
B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in RN , Adv. Differential Equations 5 (2000), 1445–1464.
B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in RN , Ann. Mat. Pura Appl. 181 (2002), 73–83.
B. Sirakov and S.H.M. Soares, Soliton solutions to systems of coupled Schrödinger equations of Hamiltonian type, Trans. Amer. Math. Soc. 362 (2010), 5729–5744.
W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.
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