Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations
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Date
2024-01
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Journal Article
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Abstract
We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well-known that:
- For n = 2, there exist Morse index 1 solutions whose L∞ norm goes to infinity.
- For n ≥ 3, uniform boundedness holds in the subcritical case for power-type nonlinearities, while for critical nonlinearities the boundedness of the Morse index does not prevent blow-up in L∞.
In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori L∞ bound for finite Morse index solution in the sharp dimensional range 3 ≤ n ≤ 9. As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method.
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published
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Journal / series
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77 (1)
Pages / Article No.
3 - 36
Publisher
Wiley
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Organisational unit
09565 - Figalli, Alessio / Figalli, Alessio
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Funding
721675 - Regularity and Stability in Partial Differential Equations (EC)
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