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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 $b\left(x\right)\ge {b}^{\infty }>0$ in ${{\Omega }}^{-}$ and there exist positive constants $C,\delta ,{R}_{0}$ such that $b\left(x\right)\ge {b}^{\infty }+C\phantom{\rule{0.166667em}{0ex}}exp\left(-\delta |z|\right)$ for $|z|\ge {R}_{0}$ uniformly for $y\in \overline{\omega }$ where $x=\left(y,z\right)\in {ℝ}^{N}$ with $y\in {ℝ}^{m}$, $z\in {ℝ}^{n}$, $N=m+n\ge 3$, $m\ge 2$, $n\ge 1$, $1, $\omega \subseteq {ℝ}^{m}$ a bounded ${C}^{1,1}$ domain and ${\Omega }=\omega ×{ℝ}^{n}$, then the Dirichlet problem $-{\Delta }u+u={b\left(x\right)|u|}^{p-1}u$ in ${\Omega }$ has a solution that changes sign in ${\Omega }$, in addition to a positive solution.
##### MSC:
 35J60 Nonlinear elliptic equations 35J20 Second order elliptic equations, variational methods 35J25 Second order elliptic equations, boundary value problems 47J30 Variational methods (nonlinear operator equations)