Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem - Δ u = | u | p - 2 u + f in Ω , u = u 0 on ∂ Ω , where Ω is a bounded domain in R N , N ≥ 3 . This can be seen as a perturbation of the problem with f = 0 and u 0 = 0 , which is odd in u . If Ω is invariant with respec...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2017-02, Vol.56 (1), p.1-19, Article 6
Main Authors: Ackermann, Nils, Cano, Alfredo, Hernández-Martínez, Eric
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Control
Invariants
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Subgroups
Systems Theory
Theoretical
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Summary:We are concerned with the existence of infinitely many solutions for the problem - Δ u = | u | p - 2 u + f in Ω , u = u 0 on ∂ Ω , where Ω is a bounded domain in R N , N ≥ 3 . This can be seen as a perturbation of the problem with f = 0 and u 0 = 0 , which is odd in u . If Ω is invariant with respect to a closed strict subgroup of O ( N ), then we prove infinite existence for all functions f and u 0 in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on Ω , employing improved Sobolev embeddings for spaces of invariant functions.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-016-1107-3