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Multiple solutions for semilinear elliptic equations in unbounded cylinder domains. (English) Zbl 1152.35371
Summary: In this paper, we show that if $b(x) \geq b^\infty > 0$ in $\Omega^-$ and there exist positive constants $C, \delta, R_0$ such that $b(x) \geq b^\infty+C\,\exp(-\delta\vert z\vert)$ for $\vert z\vert \geq R_0$ uniformly for $y\in\overline\omega$ where $x = (y, z)\in\Bbb R^N$ with $y \in\Bbb R^m$, $z\in\Bbb R^n$, $N = m+n\geq 3$, $m\geq 2$, $n\geq 1$, $1 < p < (N+2)/(N-2)$, $\omega \subseteq\Bbb R^m$ a bounded $C^{1,1}$ domain and $\Omega = \omega \times \Bbb R^n$, then the Dirichlet problem $-\Delta u+u = b(x)|u|^{p-1}u$ in $\Omega$ has a solution that changes sign in $\Omega$, in addition to a positive solution.

35J60Nonlinear elliptic equations
35J20Second order elliptic equations, variational methods
35J25Second order elliptic equations, boundary value problems
47J30Variational methods (nonlinear operator equations)
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