Florin Catrina

On the Caffarelli‐Kohn‐Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions

Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6] where, for N ≥ 3, −∞ < a < (N − 2)/2, a ≤ b ≤ a + 1, and p = 2N/(N − 2 + 2(b − a)). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. While the case a ≥ 0 has been studied extensively and a complete solution is known, little has been known for the case a < 0. Our results for the case a < 0 reveal some new phenomena which are in striking contrast with those for the case a ≥ 0. Results for N = 1 and N = 2 are also given.

On the Caffarelli–Kohn–Nirenberg inequalities

Consider the following inequalities due to Caffarelli, Kohn and Nirenberg (3): Z RN jxj bp juj p dx 2=p 6Ca;b Z RN jxj 2a jruj 2 dx; where forN> 3:1 <a< N 2 2 ,a6b6a +1 ,a ndp = 2N N 2+2(b a) . We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. We also study the bound state solutions of the corresponding Euler equations and construct positive solutions having prescribed symmetry for certain parameter region.

Nonexistence of positive radial solutions for a problem with singular potential

Abstract. This article completes the picture in the study of positive radial solutions in the function space 𝒟 1,2 (ℝ N )∩L 2 (ℝ N ,|x| -α dx)∩L p (ℝ N )

Positive solutions for nonlinear elliptic equations with fast increasing weights

{{\mathcal {D}^{1,2}({\mathbb {R}^N}) \cap L^2({{\mathbb {R}^N}, | x |^{-\alpha } dx})\cap L^p({\mathbb {R}^N})}}

RADIAL SOLUTIONS FOR WEIGHTED SEMILINEAR EQUATIONS

for the equation -Δu+A |x| α u=u p-1 inℝ N ∖{0}withN≥3,A>0,α>0,p>2.

A note on a result of M. Grossi

- \Delta u + \frac{A}{| x |^\alpha } u = u^{p-1} \quad \mbox{in } {\mathbb {R}^N}\setminus \lbrace 0\rbrace \mbox{ with } N\ge 3, A> 0, \alpha > 0, p>2.

Nonlinear Elliptic Equations on Expanding Symmetric Domains

An energy balance identity is employed to prove nonexistence of such solutions in the last remaining open region in the (α,p)

Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in RN

{{(\alpha , p)}}

Sharp weighted-norm inequalities for functions with compact support in RN∖{0}

plane.

Symmetric solutions for the prescribed scalar curvature problem

We study the equation − div(K(x)∇u) = K(x)u2−1 + λK(x)|x|α−2u, u > 0 ∈ R , (1.1) where N 3, the nonlinearity is given by the critical Sobolev exponent 2∗ = 2N/(N−2), the weight is K(x) = exp(4 |x|), α 2 and λ is a parameter. According to the function space in which we seek solutions, u is forced to decrease sufficiently fast to infinity. As in [12], for α = 2 and λ = (N − 2)/(N + 2), equation (1.1) occurs when one tries to find self-similar solutions