## 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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### References:

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