Abstract
This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation:
The functionf is assumed to be “superlinear”. A special case is the power nonlinearityf(x, z)=∥z∥s−1 z where 1<s<(n+2)(n−2)−1. Making different assumptions onb(x), mainly at infinity, various sufficient conditions for the existence of nontrivial solutionsu ∈W 1,2(ℝn) are established.
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Dedicated to Klaus Kirchgässner on the occasion of his 60th birthday
This research was sponsored in part by the U.S. Army Research Office under Contract No. DAAL03-87-K-0043, the National Science Foundation under Grant No. MCS-8110556, and the Office of Naval Research under Grant No. N00014-88-K-0134. Any reproduction for the purposes of the U.S. Government is permitted.
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Rabinowitz, P.H. On a class of nonlinear Schrödinger equations. Z. angew. Math. Phys. 43, 270–291 (1992). https://doi.org/10.1007/BF00946631
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DOI: https://doi.org/10.1007/BF00946631