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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 \[ - \Delta u + u = Q(x) | u |^{p - 1} u \quad \text{in} \quad \mathbb{R}^ N \backslash \Omega, \qquad {\partial u \over \partial n} = 0 \quad \text{on} \quad \partial \Omega \tag{1} \] where \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^ N\), \(N \geq 3\), \(1<p<(N + 2)/(N - 2)\), \(Q(x) \in C (\mathbb{R}^ N)\), \(Q(x)>0\) in \(\mathbb{R}^ N \backslash \Omega\). We always assume that \(Q(x) \to \overline Q>0\) as \(| x | \to \infty\). 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 G. Cerami, S. Solimini and M. Struwe [J. Funct. Anal. 69, 289-306 (1986; Zbl 0614.35035)] to prove the existence of another solution which changes sign.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations

Citations:

Zbl 0614.35035
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References:

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