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Semilinear elliptic eigenvalue problems on an infinite strip with an application to stratified fluids. (English) Zbl 0568.35076
Consider the semilinear elliptic eigenvalue problem \(-\nabla (a(y)\nabla u)=\lambda b(y)u+F(y,u,\lambda)\) in \((x,y)\in R\times (0,1),\) \(u(x,0)=u(x,1)=0\) for \(x\in R\), and u(x,y)\(\to 0\) as (x,y)\(\to \infty\) in the strip under certain assumptions on a,b, and F which ensure the existence of an unbounded connected set of non-trivial classical solutions (\(\lambda\),u). The author states and proves a global version of K. Kirchgässner’s local results [J. Differ. Equations 45, 113-127 (1982; Zbl 0507.35033)] concerning the branching of solutions when \(\lambda <\mu\), where \(\mu\) is the simple first eigenvalue of the associated x-independent eigenvalue problem. The condition \(F(y,u,\lambda)/| u|^{1+\sigma}\to A(y,\lambda)\) as \(u\to 0\) on F, when \(\sigma =1\), arises in the problem of solitary waves in stratified fluids. The introduction to this paper is very informative, cites motivation, technique, and physical relevancy, and is well referenced.
Reviewer: P.W.Schaefer

MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
76B25 Solitary waves for incompressible inviscid fluids
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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