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16
Stable solutions for the bilaplacian with exponential nonlinearity
 SIAM J. Math. Anal
"... Abstract. Let λ ∗> 0 denote the largest possible value of λ such that 8 < ∆ 2 u = λe u ..."
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Cited by 12 (0 self)
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Abstract. Let λ ∗> 0 denote the largest possible value of λ such that 8 < ∆ 2 u = λe u
Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities
, 2004
"... ..."
constants in the HardyRellich inequalities and related improvements
, 2004
"... We consider HardyRellich inequalities and discuss their possible improvement. The procedure is based on decomposition into spherical harmonics, where in addition various new inequalities are obtained (e.g. RellichSobolev inequalities). We discuss also the optimality of these inequalities in the se ..."
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Cited by 7 (4 self)
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We consider HardyRellich inequalities and discuss their possible improvement. The procedure is based on decomposition into spherical harmonics, where in addition various new inequalities are obtained (e.g. RellichSobolev inequalities). We discuss also the optimality of these inequalities in the sense that we establish (in most cases) that the constants appearing there are the best ones. Next, we investigate the polyharmonic operator (Rellich and Higher Order Rellich inequalities); the difficulties arising in this case come from the fact that (generally) minimizing sequences are no longer expected to consist of radial functions. Finally, the successively use of the Rellich inequalities lead to various new Higher Order Rellich inequalities. Keywords: HardyRellich inequalities, RellichSobolev inequalities, Best constants, Optimal inequalities.
ESTIMATES FOR THE OPTIMAL CONSTANTS IN MULTIPOLAR HARDY INEQUALITIES FOR SCHRÖDINGER AND DIRAC OPERATORS
"... Abstract. By expanding squares, we prove several Hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities. These inequalities involve the L 2 norm. Such results are generalized to an arbitrary number of singularities and ..."
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Cited by 2 (0 self)
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Abstract. By expanding squares, we prove several Hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities. These inequalities involve the L 2 norm. Such results are generalized to an arbitrary number of singularities and compared with standard results given by the IMS method. The generalized version of Hardy inequalities with several singularities is equivalent to some spectral information on a Schrödinger operator involving a potential with several inverse square singularities. We also give a generalized Hardy inequality for Dirac operators in the case of a potential having several singularities of Coulomb type, which are critical for Dirac operators.
ON NONEXISTENCE OF BARAS–GOLDSTEIN TYPE FOR HIGHERORDER PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS
, 2010
"... Abstract. The celebrated result by Baras and Goldstein (1984) established that the heat equation with singular inverse square potential in a smooth bounded domain Ω ⊂ RN, N ≥ 3, such that 0 ∈ Ω, ut =Δu+ c x  2 u in Ω × (0,T), u ∣ =0, ..."
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Abstract. The celebrated result by Baras and Goldstein (1984) established that the heat equation with singular inverse square potential in a smooth bounded domain Ω ⊂ RN, N ≥ 3, such that 0 ∈ Ω, ut =Δu+ c x  2 u in Ω × (0,T), u ∣ =0,
Optimal Sobolev and HardyRellich constants under Navier boundary conditions
"... We prove that the best constant for the critical embedding of higher order Sobolev spaces does not depend on all the traces. The proof uses a comparison principle due to Talenti [19] and an extension argument which enables us to extend radial functions from the ball to the whole space with no increa ..."
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We prove that the best constant for the critical embedding of higher order Sobolev spaces does not depend on all the traces. The proof uses a comparison principle due to Talenti [19] and an extension argument which enables us to extend radial functions from the ball to the whole space with no increase of the Dirichlet norm. Similar arguments may also be used to prove the very same result for HardyRellich inequalities. AMS Classification: primary 46E35, secondary 26D10, 35J55 Keywords: optimal constant, Sobolev embedding, HardyRellich inequality
1 Radial entire solutions for supercritical biharmonic equations ∗
"... We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find fini ..."
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We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of regular solutions towards the explicitly known singular solution. It turns out that the convergence is different in space dimensions n ≤ 12 and n ≥ 13. 1
Best constants for higherorder Rellich inequalities in L p (Ω)
, 2008
"... We obtain a series improvement to higherorder L pRellich inequalities on a Riemannian manifold M. The improvement is shown to be sharp as each new term of the series is added. ..."
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We obtain a series improvement to higherorder L pRellich inequalities on a Riemannian manifold M. The improvement is shown to be sharp as each new term of the series is added.
Φ(1/x)f(x)  2 ∫
, 2007
"... Abstract. Sharp L p extensions of Pitt’s inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and SteinWeiss potentials. Optimal constants are obtained for the full SteinWeiss potential as a map from L p to itself which in turn yield semiclassical Rellich ..."
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Abstract. Sharp L p extensions of Pitt’s inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and SteinWeiss potentials. Optimal constants are obtained for the full SteinWeiss potential as a map from L p to itself which in turn yield semiclassical Rellich inequalities on R n. Additional results are obtained for SteinWeiss potentials with gradient estimates and with mixed homogeneity. New proofs are given for the classical Pitt and SteinWeiss estimates. Weighted inequalities for the Fourier transform provide a natural measure to characterize both uncertainty and the balance between functional growth and smoothness. On Rn the question is to determine sharp quantitative comparisons between the relative size of a function and its Fourier transform at infinity. Pitt’s inequality illustrates this principle at the spectral level (see [4], [7]): (1)