## 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

### Keywords:

nonlinear elliptic euqations; strip-like domains

### Citations:

Zbl 0533.35029; Zbl 0544.35042
Full Text:

### References:

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