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Title of Journal: Calc Var

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Abbravation: Calculus of Variations and Partial Differential Equations

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Springer-Verlag

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1432-0835

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Singlepoint condensation phenomena for a fourdim

Authors: Futoshi Takahashi
Publish Date: 2007/03/15
Volume: 29, Issue: 4, Pages: 509-520
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Abstract

We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem E p E p left beginarrayll Delta2 u = up hboxin Omega u 0 hboxin Omega u partialOmega = Delta u partialOmega = 0 endarray right as p gets large where Ω is a smooth bounded domain in R 4 In our earlier paper Takahashi in Osaka J Math 2006 we have shown that the least energy solutions remain bounded uniformly in p and they have one or two “peaks” away form the boundary In this note following the arguments in Adimurthi and Grossi Proc AMS 13241013–1019 2003 and Lin and Wei Comm Pure Appl Math 56784–809 2003 we will obtain more sharper estimates of the upper bound of the least energy solutions and prove that the least energy solutions must develop singlepoint spiky pattern under the assumption that the domain is convex

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  1. Asymptotic behavior at isolated singularities for solutions of nonlocal semilinear elliptic systems of inequalities
  2. Fractional Poincaré inequality with finite total Q -curvature
  3. Solitary waves for a class of quasilinear Schrödinger equations in dimension two
  4. Thick clusters for the radially symmetric nonlinear Schrödinger equation
  5. On elliptic systems with Sobolev critical growth
  6. Critical points for functionals with quasilinear singular Euler–Lagrange equations
  7. Duality for rectified cost functions
  8. An approach to minimization under a constraint: the added mass technique
  9. Uniqueness and nondegeneracy of positive radial solutions of $$\mathbf {div\,{\varvec{(}}{\varvec{\rho }} \nabla u{\varvec{)}} +{\varvec{\rho }}{\varvec{(}}-gu+hu^p{\varvec{)}}=0}$$
  10. Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds
  11. Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators
  12. Sharp energy estimates for nonlinear fractional diffusion equations
  13. On Newton equations which are totally integrable at infinity
  14. Energy asymptotics for Type II superconductors
  15. Regularity for fully nonlinear nonlocal parabolic equations with rough kernels
  16. Singular points of Hölder asymptotically optimally doubling measures
  17. Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations
  18. Cahn–Hilliard–Navier–Stokes systems with moving contact lines
  19. A mean field type flow part I: compactness of solutions to a perturbed mean field type equation
  20. Boundary value problems for Willmore curves in $$\mathbb {R}^2$$
  21. Fractional eigenvalues
  22. On a quasilinear system involving the operator curl
  23. A Penrose inequality for graphs over Kottler space
  24. Moser–Trudinger inequalities of vector bundle over a compact Riemannian manifold of dimension 2
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