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Nonlinear elliptic equations in strip-like domains. (English) Zbl 1336.35169

Summary: We study the existence of a positive solution of a nonlinear elliptic equation \[ -\Delta u=f(u)\quad\text{in }\mathbb R^k\times D,\quad u=0\quad\text{on }\mathbb R^k\times\partial D, \] where \(k\geq 2\) and \(D\) is a bounded domain domain in \(\mathbb R^\ell\), \(\ell\geq 1\). We give almost necessary and sufficient condition of \(f(\xi)\) for the existence of a positive solution, which is inspired by the works of H. Berestycki and P.-L. Lions [Arch. Ration. Mech. Anal. 82, 313–345 (1983; Zbl 0533.35029)] and H. Berestycki, T. Gallouët and O. Kavian [C. R. Acad. Sci., Paris, Sér. I 297, 307–310 (1983; Zbl 0544.35042)].

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J20 Variational methods for second-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B09 Positive solutions to PDEs
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References:

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