Multiple solutions for semilinear elliptic equations in unbounded cylinder domains. (English) Zbl 1152.35371

Hsu, Tsing-San

Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 4, 719-731 (2004).

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

Keywords:

Dirichlet problem; semilinear elliptic equations; positive solution; sign changes