## 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
Full Text:

### References:

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