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On a class of nonlinear Schrödinger equations. (English) Zbl 0763.35087
This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations \[ i\hslash\psi_ t=-{\hslash^ 2\over 2m}\Delta\psi+V(x)\psi-\gamma|\psi|^{p-1}\psi, \tag{1} \] where \(x\in \mathbb R^ n\), \(1<p<(n+2)/(n-2)\). Making a standing wave ansatz the problem reduces to that of studying the semilinear elliptic equation. In fact, a more general semilinear elliptic PDE \[ -\Delta V+b(x)v=f(x,v),\quad x\in \mathbb R \tag{2} \] is considered. Using “mountain pass” and comparison arguments, the author gets existence of nontrivial solutions \(u\in W^{1,2}(\mathbb R^ n)\) for (2), under some additional assumptions on \(b(x)\) and \(f(x,u)\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] A. Floer and A. Weinstein,Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal.,69, 397-408 (1986). · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0
[2] Y.-G. Oh,Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class (V) a . Comm. Partial Diff. Eq.,13, 1499-1519 (1988). · Zbl 0702.35228 · doi:10.1080/03605308808820585
[3] Y.-G. Oh,Corrections to ?Existence of semi-classical bound state of nonlinear Schrödinger equations with potentials of the class (V)a?. Comm. Partial Diff. Eq.,14, 833-834 (1989). · Zbl 0714.35078
[4] Y.-G. Oh,Stability of semiclassical bound states of nonlinear Schrödinger equations with potential. Comm. Math. Phys.,121, 11-33 (1989). · Zbl 0693.35132 · doi:10.1007/BF01218621
[5] Y.-G. Oh,On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys.,131, 223-253 (1990). · Zbl 0753.35097 · doi:10.1007/BF02161413
[6] T. Kato,Remarks on holomorphic families of Schrödinger and Dirac operators, Differential Equations (eds. I. Knowles and R. Lewis), pp. 341-352. North Holland, Amsterdam 1984. · Zbl 0565.47011
[7] A. Ambrosetti and P. H. Rabinowitz,Dual variational methods in critical point theory and applications. J. Funct. Anal.,14, 349-381 (1973). · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[8] P.-L., Lions,The concentration-compactness principle in the calculus of variations. The locally compact case, Part 2. Analyse Nonlinéaire,1, 223-283 (1984). · Zbl 0704.49004
[9] Y. Li,Remarks on a semilinear elliptic equation on ? n . J. Diff. Eq.,74, 34-49 (1988). · Zbl 0662.35038 · doi:10.1016/0022-0396(88)90017-4
[10] W.-Y. Ding and W.-M. Ni,On the existence of positive entire solutions of a semi-linear elliptic equation. Arch. Rat. Mech. Anal.,91, 283-308 (1986). · Zbl 0616.35029 · doi:10.1007/BF00282336
[11] V. Coti Zelati, and P. H. Rabinowitz,Homoclinic type solutions for a semilinear elliptic PDE on ? n , to appear, Comm. Pure Appl. Math. · Zbl 0785.35029
[12] P. H. Rabinowitz and K. Tanaka,Some results on connecting orbits for a class of Hamiltonian systems. Math. Z.,206, 473-499 (1991). · Zbl 0716.58013 · doi:10.1007/BF02571356
[13] J. Mawhin, and M. Willem,Critical Point Theory and Hamiltonian Systems. Springer Verlag, Berlin 1989. · Zbl 0676.58017
[14] D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order. Springer Verlag, Berlin 1977. · Zbl 0361.35003
[15] A. Friedman,Partial Differential Equations. Holt, Rinehart, and Winston, New York 1969. · Zbl 0224.35002
[16] C. V. Coffman,A minimum-maximum principle for a class of integral equations. J. Analyse Math.,22, 391-419 (1969). · Zbl 0179.15601 · doi:10.1007/BF02786802
[17] J. A. Hempel,Superlinear variational boundary value problems and nonuniqueness, thesis, University of New England, Australia 1970.
[18] Z. Nehari,On a class of nonlinear second order differential equations. Trans. A.M.S.,95, 101-123 (1960). · Zbl 0097.29501 · doi:10.1090/S0002-9947-1960-0111898-8
[19] W.-M. Ni,Recent progress in semilinear elliptic equations. RIMS Kokyuroku,679. (ed., T. Suzuki), pp. 1-9, (1989).
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