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Multiple sign-changing solutions for a semilinear Neumann problem and the topology of the configuration space of the domain boundary. (English) Zbl 1200.35076

Calc. Var. Partial Differ. Equ. 38, No. 3-4, 317-356 (2010); erratum ibid. 40, No. 1-2, 293-294 (2011).
In this paper, the main aim of the author is to establish a lower estimate for the number of sign-changing solutions to the following Neumann problem
\[ -d^2 \Delta u + u =f(u)\quad \text{in }\Omega,\qquad\frac{\partial u}{\partial \nu}=0 \quad \text{in }\partial \Omega, \tag{P} \]
where \(d > 0\) is small enough, \(\Omega \subset\mathbb R^N\) with \( (N \geq 2)\) is a bounded or unbounded domain whose boundary is nonempty, compact and smooth, and the nonlinearity \(f \in C(\mathbb R,\mathbb R)\) is Sobolev subcritical. Under some further conditions, the author proved that there exists \(d_0>0\) such that for each \(d\in (0, d_0)\), problem (P) has at least \(\text{cat}(C(\partial\Omega)\times [0,1]^2, C(\partial\Omega)\times [0,1]^2)\) sign-changing solutions, and each of them has precisely two nodal domains. Furthermore, if \(\Omega\) is bounded, then there is at least one other sign-changing solution which has at most four nodal domains.

MSC:

35J20 Variational methods for second-order elliptic equations
35J61 Semilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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