Rabinowitz, Paul H. On a class of nonlinear Schrödinger equations. (English) Zbl 0763.35087 Z. Angew. Math. Phys. 43, No. 2, 270-291 (1992). 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 \(f(x,u)\). Reviewer: Guo Boling (Beijing) Cited in 14 ReviewsCited in 994 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction Keywords:mountain pass; existence of standing wave solutions; comparison arguments × Cite Format Result Cite Review PDF Full Text: DOI References: [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). · 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). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.