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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 \(b(x) \geq b^\infty+C\,\exp(-\delta| z|)\) for \(| z| \geq R_0\) uniformly for \(y\in\overline\omega\) where \(x = (y, z)\in\mathbb R^N\) with \(y \in\mathbb R^m\), \(z\in\mathbb R^n\), \(N = m+n\geq 3\), \(m\geq 2\), \(n\geq 1\), \(1 < p < (N+2)/(N-2)\), \(\omega \subseteq\mathbb R^m\) a bounded \(C^{1,1}\) domain and \(\Omega = \omega \times \mathbb R^n\), 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.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
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