Abstract
We study the existence and the number of decaying solutions for the semilinear Schrödinger equations \({-\varepsilon^{2}\Delta u + V(x)u = g(x,u)}\), \({\varepsilon > 0}\) small, and \({-\Delta u + \lambda V(x)u = g(x,u)}\), \({\lambda > 0}\) large. The potential V may change sign and g is either asymptotically linear or superlinear (but subcritical) in u as \({|u| \to \infty}\) .
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References
Ackermann N. (2004). On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248: 423–443
Ambrosetti A., Malchiodi A., Secchi S. (2001). Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Rat. Mech. Anal. 159: 253–271
Bartolo P., Benci V., Fortunato D. (1983). Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlin. Anal. 7: 981–1012
Bartsch T., Pankov A., Wang Z.Q. (2001). Nonlinear Schrödinger equations with steep potential well. Comm. Contemp. Math. 3: 549–569
Bartsch T., Wang Z.Q. (1995). Existence and multiplicity results for some superlinear elliptic problems on \(\mathbf{R}^N\) Comm. PDE 20: 1725–1741
Bartsch T., Wang Z.Q. (2000). Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys. 51: 366–384
Byeon, J., Jeanjean, L.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity (preprint)
Byeon J., Wang Z.Q. (2003). Standing waves with a critical frequency for nonlinear Schrödinger equations, II. Calc. Var. PDE 18: 207–219
Chabrowski J., Szulkin A. (2005). On the Schrödinger equation involving a critical Sobolev exponent and magnetic field. Top. Meth. Nonl. Anal. 25: 3–21
del Pino M., Felmer P. (1996). Local mountain passes for semilinear problems in unbounded domains. Calc. Var. PDE 4: 121–137
Ding Y.H., Lee C. (2006). Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. Diff. Eq. 222: 137–163
Ding, Y.H., Szulkin, A.: Existence and number of solutions for a class of semilinear Schrödinger equations. In: Cazenave, T., et al. (eds.) Contributions to Nonlinear Analysis. A tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday, pp. 221–231. Birkhäuser, Basel (2006)
Felmer P., Torres J.J. (2002). Semi-classical limit for the one dimensional nonlinear Schrödinger equation. Comm. Contemp. Math. 4: 481–512
Floer A., Weinstein A. (1986). Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Func. Anal. 69: 397–408
Jeanjean L., Tanaka K. (2004). Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. PDE 21: 287–318
Lions P.L. (1984). The concentration–compactness principle in the calculus of variations. The locally compact case. Part I. Ann. IHP, Analyse Non Linéaire 1: 109–145
Liu Z.L., van Heerden F.A., Wang Z.Q. (2005). Nodal type bound states of Schrödinger equations via invariant set and minimax methods. J. Diff. Eq. 214: 358–390
Oh Y.G. (1988). Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V) a . Comm. PDE 13: 1499–1519
Oh Y.G. (1990). On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys. 131: 223–253
Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS 65, American Mathematical Society, Providence, R.I. (1986)
Rabinowitz P.H. (1992). On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43: 270–291
Willem M. (1995). Analyse Harmonique Réelle. Hermann, Paris
Willem M. (1996). Minimax Theorems. Birkhäuser, Boston
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Ding, Y., Szulkin, A. Bound states for semilinear Schrödinger equations with sign-changing potential. Calc. Var. 29, 397–419 (2007). https://doi.org/10.1007/s00526-006-0071-8
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DOI: https://doi.org/10.1007/s00526-006-0071-8