Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations

2023/07/21 by Alessio Figalli, Yi Ru‐Ya Zhang
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.1002/cpa.22132

Abstract

Abstract We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well‐known that: For , there exist Morse index 1 solutions whose norm goes to infinity. For , 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 . In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori bound for finite Morse index solution in the sharp dimensional range . As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method.

Citations

• Solutions of superlinear elliptic equations and their morse indices
• Quasilinear Dirichlet problems driven by positive sources
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

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