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Nonautonomous and non periodic Schrödinger equation with indefinite linear part. (English) Zbl 1366.35028

Summary: The existence of solution of the nonlinear Schrödinger equation \[ -\Delta u + V(x) u = f(x,u), \] is established in \(\mathbb {R}^N\), where \(V\) changes sign and \(f\) is an asymptotically linear function at infinity, with \(V\) and \(f\) non periodic in \(x\). Spectral theory, a classical linking theorem and interaction between translated solutions of the problem at infinity are employed.

MSC:

35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI

References:

[1] Ackermann, N., Clapp, M., Pacella, F.: Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains. Commun. Partial Differ. Equ. 38(5), 751-779 (2013) · Zbl 1273.35132 · doi:10.1080/03605302.2013.771657
[2] Azzollini, A., Pomponio, A.: On the Schrödinger equation in \[\mathbb{R}^N\] RN under the effect of a general nonlinear term. Indiana Univ. Math. J. 58, 1361-1378 (2009) · Zbl 1170.35038 · doi:10.1512/iumj.2009.58.3576
[3] Berestycki, H., Lions, P.L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313-345 (1983) · Zbl 0533.35029
[4] Cerami, G.: Un criterio di ezistenza per i punti critici su varietà illimitate. Rend. Accad. Sci. Lett. Inst. Lombardo 112, 332-336 (1978) · Zbl 0436.58006
[5] Costa, D.G., Tehrani, H.: Existence and multiplicity results for a class of Schrödinger equations with indefinite nonlinearities. Adv. Differ. Equ. 8, 1319-1340 (2003) · Zbl 1158.35348
[6] Egorov, Y., Kondratiev, V.: On spectral theorey of elliptic operators. Birkhäuser, Basel (1996) · Zbl 0855.35001 · doi:10.1007/978-3-0348-9029-8
[7] Jeanjean, L.: On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on \[\mathbb{R}^N\] RN. Proc. R. Soc. Edinburgh Sect. A 129, 787-809 (1999) · Zbl 0935.35044 · doi:10.1017/S0308210500013147
[8] Jeanjean, L., Tanaka, K.: A positive solution for an asymptotically linear elliptic problem on \[\mathbb{R}^N\] RN autonomous at infinity. ESAIM Control Optim. Calc. Var. 7, 597-614 (2002) · Zbl 1225.35088 · doi:10.1051/cocv:2002068
[9] Kryszewski, W., Szulkin, A.: Generalized linking theorem with an application to semilinear Schrödinger equations. Adv. Differ. Equ. 3, 441-472 (1998) · Zbl 0947.35061
[10] Li, G., Szulkin, A.: An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math. 4, 763-776 (2002) · Zbl 1056.35065 · doi:10.1142/S0219199702000853
[11] Li, G., Wang, C.: The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition. Annales Academiae Scientiarum Fennicae 36, 461-480 (2011) · Zbl 1234.35095 · doi:10.5186/aasfm.2011.3627
[12] Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case. Parts I and II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 109-145 and 223-283 (1984) · Zbl 0541.49009
[13] Maia, L.A, Oliveira Junior, J.C., Ruviaro, R.: A non periodic and asymptotically linear indefinite variational problem in \[{\mathbb{R}^N}\] RN, online Indiana University Mathematics Journal (2015) · Zbl 1366.35028
[14] Pankov, A.A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259-287 (2005) · Zbl 1225.35222 · doi:10.1007/s00032-005-0047-8
[15] Rabinowitz, P.H.: Some critical point theorems and applications to semilinear elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5, 215-223 (1978) · Zbl 0375.35026
[16] Stuart, C.A.: An introduction to elliptic equation in \[{\mathbb{R}^N}\] RN, Trieste Notes (1998) · Zbl 0533.35029
[17] Stuart, C.A., Zhou, H.S.: Applying the mountain pass theorem to an asymptotically linear elliptic equation on \[\mathbb{R}^N\] RN. Commun. Partial Differ. Equ. 24, 1731-1758 (1999) · Zbl 0935.35043 · doi:10.1080/03605309908821481
[18] Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Func. Anal. 257, 3802-3822 (2009) · Zbl 1178.35352 · doi:10.1016/j.jfa.2009.09.013
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