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Spectral density estimates with partial symmetries and an application to Bahri-Lions-type results. (English) Zbl 1372.35078

The boundary value problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega\), \(u=u_0\) on \(\partial \Omega\), where \(\Omega\) is a domain in \(\mathbb R^N\) (\(N\geq 3\)) with a sufficiently smooth boundary \(\partial \Omega\), is discussed. The authors consider conditions on \(p\), \(f\) and \(u_0\) when the problem has infinitely many solutions extending the known results for \(f=0\) and \(u_0=0\). The authors determine the maximal possible \(p\) employing improved Sobolev embeddings for spaces of invariant functions. Spectral density estimates for Schrödinger operators are used.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J61 Semilinear elliptic equations
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