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.


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


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