## 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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