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Title of Journal: Arch Rational Mech Anal

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Abbravation: Archive for Rational Mechanics and Analysis

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Springer-Verlag

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DOI

10.1007/s40278-015-6147-5

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1432-0673

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The Gel’fand Problem for the Biharmonic Operator

Authors: Louis Dupaigne Marius Ghergu Olivier Goubet Guillaume Warnault
Publish Date: 2013/04/12
Volume: 208, Issue: 3, Pages: 725-752
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Abstract

We study stable and finite Morse index solutions of the equation Delta2 u = eu If the equation is posed in mathbbRN we classify radial stable solutions We then construct nonradial stable solutions and we prove that unlike the corresponding second order problem no Liouvilletype theorem holds unless additional information is available on the asymptotics of solutions at infinity Thanks to this analysis we prove that stable solutions of the equation on a smoothly bounded domain supplemented with Navier boundary conditions are smooth if and only if N leqq 12 We find an upper bound for the Hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded Morse index provided they are controlled in a suitable Morrey norm

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Other Papers In This Journal:
  1. Optimal Design in Conductivity Under Locally Constrained Heat Flux
  2. Regularity Issues in the Problem of Fluid Structure Interaction
  3. Negative Eigenvalues of Two-Dimensional Schrödinger Operators
  4. Dobrushin States in the $${\phi^4_1}$$ Model
  5. The Liouville Theorem and the L 2 Decay for the FENE Dumbbell Model of Polymeric Flows
  6. The Liouville Theorem and the L 2 Decay for the FENE Dumbbell Model of Polymeric Flows
  7. Counterexamples to the Sonic Criterion
  8. Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution
  9. On Nonlinear Stochastic Balance Laws
  10. Weak Solutions of the Navier–Stokes Equations for Compressible Flows in a Half-Space with No-Slip Boundary Conditions
  11. Generalized Transition Fronts for One-Dimensional Almost Periodic Fisher-KPP Equations
  12. The Existence Problem for Steiner Networks in Strictly Convex Domains
  13. Stability of Detonation Profiles in the ZND Limit
  14. Uniqueness of Absolute Minimizers for $${L^\infty}$$-Functionals Involving Hamiltonians $${H(x,p)}$$H(x,p)
  15. A Condition for the Hölder Regularity of Local Minimizers of a Nonlinear Elastic Energy in Two Dimensions
  16. Kinetic Stabilization¶of a Nonlinear Sonic Phase Boundary
  17. The Toda System and Clustering Interfaces in the Allen–Cahn equation
  18. A Stability Criterion for Two-Fluid Interfaces and Applications
  19. Long-Time Convergence of an Adaptive Biasing Force Method: The Bi-Channel Case
  20. Displacement Convexity and Minimal Fronts at Phase Boundaries
  21. Gradient Bounds for Elliptic Problems Singular at the Boundary
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