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Multiple solutions for a Neumann problem in an exterior domain. (English) Zbl 0803.35049
From the introduction: We prove the existence of multiple solutions for the Neumann boundary value problem $$- \Delta u + u = Q(x) \vert u \vert\sp{p - 1} u \quad \text{in} \quad \bbfR\sp N \backslash \Omega, \qquad {\partial u \over \partial n} = 0 \quad \text{on} \quad \partial \Omega \tag 1 $$ where $\Omega$ is a smooth bounded domain of $\bbfR\sp N$, $N \ge 3$, $1<p<(N + 2)/(N - 2)$, $Q(x) \in C (\bbfR\sp N)$, $Q(x)>0$ in $\bbfR\sp N \backslash \Omega$. We always assume that $Q(x) \to \overline Q>0$ as $\vert x \vert \to \infty$. In the present paper, for general domains $\Omega$, with the help of the concentration-compactness argument, we first obtain the “ground state solution” and then combine it with some ideas of {\it G. Cerami}, {\it S. Solimini} and {\it M. Struwe} [J. Funct. Anal. 69, 289-306 (1986; Zbl 0614.35035)] to prove the existence of another solution which changes sign.

35J65Nonlinear boundary value problems for linear elliptic equations
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