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In this paper, we consider the following variational problem: inf u ∈ D a 1 , 2 (R N) \ Z ‖ u ‖ D a 1 , 2 (R N) 2 - C a , b , N - 1 ‖ u ‖ L p + 1 (| x | - b (p + 1) , R N) 2 inf v ∈ Z ‖ u - v ‖ D a 1 , 2 (R N) 2 : = c BE , where N ≥ 2 , b FS (a) < b < a + 1 , a < 0 , a ≤ b < a + 1 , 0 ≤ a < a c : = N - 2 2 and a + b > 0 with b FS (a) being the Felli–Schneider curve, p = N + 2 (1 + a - b) N - 2 (1 + a - b) , Z = { c τ a c - a W (τ x) ∣ c ∈ R \ { 0 } , τ > 0 } and up to dilations and scalar multiplications, W(x), which is positive and radially symmetric, is the unique extremal function of the following classical Caffarelli–Kohn–Nirenberg (CKN for short) inequality (∫ R N | x | - b (p + 1) | u | p + 1 d x) 2 p + 1 ≤ C a , b , N ∫ R N | x | - 2 a | ∇ u | 2 d x with C a , b , N being the optimal constant. It is known in Wei and Wu (Math Ann 384:1509–1546, 2022) that c BE > 0 . In this paper, we prove that the above variational problem has a minimizer for N ≥ 2 under the following two assumptions: (i) a c ∗ ≤ a < a c and a ≤ b < a + 1 , (ii) a < a c ∗ and b FS ∗ (a) ≤ b < a + 1 , where a c ∗ = (1 - N - 1 2 N ) a c and b FS ∗ (a) = (a c - a) N a c - a + (a c - a) 2 + N - 1 + a - a c. Our results extend that of König (J Eur Math Soc. [Math. AP]) for the Sobolev inequality to the CKN inequality. [ABSTRACT FROM AUTHOR]

In this paper, we establish a number of geometrical inequalities such as Hardy, Sobolev, Rellich, Hardy–Littlewood–Sobolev, Caffarelli–Kohn–Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions for an ample class of sub-elliptic differential operators on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. We also obtain the corresponding uncertainty type principles. [ABSTRACT FROM AUTHOR]

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces. [ABSTRACT FROM AUTHOR]

In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Sobolev, Rellich, Hardy–Littllewood–Sobolev, Gagliardo–Nirenberg, Caffarelli–Kohn–Nirenberg and Heisenberg–Pauli–Weyl type uncertainty inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. The approach developed in this paper relies on establishing integral versions of Hardy inequalities on homogeneous Lie groups, for which we also find necessary and sufficient conditions for the weights for such inequalities to be true. Consequently, we link such integral Hardy inequalities to different hypoelliptic inequalities by using the Riesz and Bessel kernels associated to the described hypoelliptic operators. [ABSTRACT FROM AUTHOR]

We obtain sharp Hardy inequalities on antisymmetric functions, where antisymmetry is understood for multi-dimensional particles. Partially it is an extension of the paper [Th. Hoffmann-Ostenhof and A. Laptev, Hardy inequalities with homogeneous weights, J. Funct. Anal. 268 (2015) 3278–3289], where Hardy's inequalities were considered for the antisymmetric functions in the case of the 1D particles. As a byproduct we obtain some Sobolev and Gagliardo–Nirenberg type inequalities that are applied to the study of spectral properties of Schrödinger operators. [ABSTRACT FROM AUTHOR]

Here, considering - ∞ < a < N - p p , a ≤ e ≤ a + 1 , d = 1 + a - e and p ∗ : = p ∗ (a , e) = Np N - d p , the existence of positive solution of a weighted p-Laplace equation involving vanishing potentials - Δ ap u + V (x) | x | - e p ∗ | u | p - 2 u = | x | - e p ∗ f (u) in R N is proved, where the potential V can vanish at infinity with exponential decay and f is a function with subcritical growth of class C 1 . We use Del Pino & Felmer’s arguments to overcome the lack of compactness and the Moser iteration method with Caffarelli–Kohn–Nirenberg inequality to obtain estimates of the solution in L ∞ (R N). [ABSTRACT FROM AUTHOR]

We obtain sharp Hardy inequalities on antisymmetric functions, where antisymmetry is understood for multi-dimensional particles. Partially it is an extension of the paper [Th. Hoffmann-Ostenhof and A. Laptev, Hardy inequalities with homogeneous weights, J. Funct. Anal. 268 (2015) 3278–3289], where Hardy’s inequalities were considered for the antisymmetric functions in the case of the 1D particles. As a byproduct we obtain some Sobolev and Gagliardo–Nirenberg type inequalities that are applied to the study of spectral properties of Schrödinger operators.

We establish the full range of the Caffarelli–Kohn–Nirenberg inequalities for radial functions in the Sobolev and the fractional Sobolev spaces of order $0 < s \le 1$. In particular, we show that the range of the parameters for radial functions is strictly larger than the one without symmetric assumption. Previous known results reveal only some special ranges of parameters even in the case $s=1$. The known proofs used the Riesz potential and inequalities for fractional integrations. Our proof is new, elementary, and is based on one-dimensional case. Applications on compact embeddings are also mentioned.

In this article, multiplicity of nontrivial solutions for an inhomogeneous singular biharmonic equation with Rellich potential are studied. Firstly, a negative energy solution of the studied equations is achieved via the Ekeland's variational principle and Caffarelli–Kohn–Nirenberg inequality. Then by applying Mountain pass theorem lack of Palais–Smale conditions, the second solution with positive energy is also obtained.

We consider the punctured plane with volume density | x | α and perimeter density | x | β . We show that centred balls are uniquely isoperimetric for indices (α , β) which satisfy the conditions α - β + 1 > 0 , α ≤ 2 β and α (β + 1) ≤ β 2 except in the case α = β = 0 which corresponds to the classical isoperimetric inequality. As an application, we verify a conjecture due to Caldiroli and Musina relating to the best constant in the Caffarelli–Kohn–Nirenberg inequality. [ABSTRACT FROM AUTHOR]

We provide a scheme of a recent stability result for a family of Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved entropy – entropy production inequality associated with an appropriate fast diffusion equation (FDE) written in self-similar variables. This result can be rephrased as an improved decay rate of the entropy of the solution of (FDE) for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar structure. When the exponents are in a range for which the optimal functions for (CKN) are radially symmetric, we investigate how the methods for (GNS) can be extended to (CKN). In particular, we prove that the solutions of the evolution equation associated to (CKN) also satisfy an improved decay rate of the entropy, after an explicit delay. However, the improved rate is obtained without assuming that initial data are well prepared, which is a major difference with the (GNS) case. [ABSTRACT FROM AUTHOR]

We introduce Herz-type Sobolev spaces on domains, which unify and generalize the classical Sobolev spaces. We will give a proof of the Sobolev-type embedding for these function spaces. All these results generalize the classical results on Sobolev spaces. Moreover, some remarks on Caffarelli-Kohn-Nirenberg inequality are given. [ABSTRACT FROM AUTHOR]

In this work we prove the existence of nontrivial bounded variation solutions to quasilinear elliptic problems involving a weighted 1-Laplacian operator. A key feature of these problems is that weights are unbounded. One of our main tools is the well-known Caffarelli-Kohn-Nirenberg's inequality, which is established in the framework of weighted spaces of functions of bounded variation (and that provides us the necessary embeddings between weighted spaces). Additional tools are suitable variants of the Mountain Pass Theorem as well as an extension of the pairing theory by Anzellotti to this new setting. [ABSTRACT FROM AUTHOR]

We study the Hardy type inequalities and Caffarelli-Kohn-Nirenberg type inequalities with nonradial weights of the form $| x_1| ^{A_1}\cdots| x_N| ^{A_N}/ |x| ^{\lambda}$.

We observe, utilize dualities in differential equations and differential inequalities (see Theorem 2.1), dualities between comparison theorems in differential equations (see Theorems E and 2.2), and obtain dualities in 'swapping' comparison theorems in differential equations. These dualities generate comparison theorems on differential equations of mixed types I and II (see Theorems 2.3 and 2.4) and lead to comparison theorems in Riemannian geometry (see Theorems 2.5 and 2.8) with analytic, geometric, PDE's and physical applications. In particular, we prove Hessian comparison theorems (see Theorems 3.1–3.5) and Laplacian comparison theorems (see Theorems 2.6, 2.7 and 3.1–3.5) under varied radial Ricci curvature, radial curvature, Ricci curvature and sectional curvature assumptions, generalizing and extending the work of Han-Li-Ren-Wei (2014) and Wei (2016). We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship (see Theorem 5.4). These provide tools in extending the notion, integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms, introducing Condition W for bundle-valued differential forms, and proving the duality theorem and the unity theorem, generalizing the work of Andreotti and Vesentini (1965) and Wei (2020). We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature, generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds, the embedding theorem for weighted Sobolev spaces of functions on manifolds, geometric differential-integral inequalities, generalized sharp Hardy type inequalities on Riemannian manifolds, monotonicity formulas and vanishing theorems for differential forms of degree k with values in vector bundles, such as F-Yang-Mills fields (when F is the identity map, they are Yang-Mills fields), generalized Yang-Mills-Born-Infeld fields on manifolds, Liouville type theorems for F - harmonic maps (when F (t) = 1 p (2 t) p 2 , p > 1 , they become p-harmonic maps or harmonic maps if p = 2), and Dirichlet problems on starlike domains for vector bundle valued differential 1-forms and {tF}-harmonic maps (see Theorems 4.1, 7.3–7.7, 8.1, 9.1–9.3, 10.1, 11.2, 12.1 and 12.2), generalizing the work of Caffarelli et al. (1984) and Costa (2008), in which M = ℝn and its radial curvature K(r) = 0, the work of Wei and Li (2009), Chen et al. (2011, 2014), Dong and Wei (2011), Wei (2020) and Karcher and Wood (1984), etc. The boundary value problem for bundle-valued differential 1-forms is in contrast to the Dirichlet problem for p-harmonic maps to which the solution is due to Hamilton (1975) for the case p = 2 and RiemN ⩽ 0, and Wei (1998) for 1 < p < ∞. [ABSTRACT FROM AUTHOR]

We consider Hardy inequalities on antisymmetric functions. Such inequalities have substantially better constants. We show that they depend on the lowest degree of an antisymmetric harmonic polynomial. This allows us to obtain some Caffarelli–Kohn–Nirenberg-type inequalities that are useful for studying spectral properties of Schrödinger operators. [ABSTRACT FROM AUTHOR]

We analyze the general form of Caffarelli-Kohn-Nirenberg inequality. Due to a new introduced parameter, this inequality presents two distinguishable ranges. One of them, the inequality is shown to be the interpolation between Hardy and weighted Sobolev inequalities. The other range, which is no more an interpolation, the positive constant in the inequality is not necessarily bounded for all value of the parameters. In both cases, the precise value of the constants were given. [ABSTRACT FROM AUTHOR]

In this paper we derive both local and global geometric inequalities on general Riemannnian and Finsler manifolds and prove generalized Caffarelli–Kohn–Nirenberg type and Hardy type inequalities on Finsler manifolds, illuminating curvatures of both Riemannian and Finsler manifolds influence geometric inequalities. [ABSTRACT FROM AUTHOR]

Though there has been an extensive study on first-order Caffarelli–Kohn–Nirenberg inequalities, not much is known for the existence of extremal functions for higher-order ones. The higher-order derivative of the Caffarelli–Kohn–Nirenberg inequality established by Lin [14] states

In this paper, we introduce and study a new functional which was motivated by the work of Bahrouni et al. (Nonlinearity 31:1518–1534, 2018) on the Caffarelli–Kohn–Nirenberg inequality with variable exponent. We also study the eigenvalue problem for equations involving this new functional. [ABSTRACT FROM AUTHOR]

In this paper, we use a suitable transform of quasi-conformal mapping type to investigate the sharp constants and optimizers for the following Caffarelli–Kohn–Nirenberg inequalities for a large class of parameters (r,p,q,s,μ,σ){(r,p,q,s,\mu,\sigma)} and 0≤a≤1{0\leq a\leq 1}:

We are concerned with a class of nonlocal elliptic equations as follows: { − M (∫RN|∇u|2dx)∆u + λV(x)u = f(x,u) in RN, u ∈ H1(RN), where N ≥ 1, λ > 0 is a parameter, M(t) = am(t) + b with a,b > 0 and m ∈ C(R+,R+), V ∈ C (RN,R+) and f ∈ C (RN × R,R) satisfying lim|u|→∞ f(x,u)/|u|k−1 = q(x) uniformly in x ∈ RN for any 2 < k < 2∗(2∗ = ∞ for N = 1,2 and 2∗ = 2N/(N − 2) for N ≥ 3). Unlike most other papers on this problem, we are more interested in the effects of the functions m and q on the number and behavior of solutions. By using minimax method as well as Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicityof positive solutions for the above problem. [ABSTRACT FROM AUTHOR]

The two related one space dimensional singular linear parabolic equations (1), (2) studied by H. Brezis et al. [Comm. Pure Appl. Math. 24 (1971), pp. 395-416] have different scaling properties. These scaling properties lead to new variants of the Hardy and Caffarelli-Kohn-Nirenberg inequalities. These results are proved, and they imply some non-wellposedness results when the constant in the singular potential term is large enough. [ABSTRACT FROM AUTHOR]

In this paper, generalised weighted Lp-Hardy, Lp-Caffarelli-Kohn-Nirenberg, and Lp-Rellich inequalities with boundary terms are obtained on stratified Lie groups. As consequences, most of the Hardy type inequalities and Heisenberg-Pauli-Weyl type uncertainty principles on stratified groups are recovered. Moreover, a weighted L2-Rellich type inequality with the boundary term is obtained. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the existence and multiplicity of symmetric solutions for a weighted semilinear elliptic system with multiple critical Hardy-Sobolev exponents and singular potentials in R N $\mathbb{R}^{N}$ . Applying the symmetric criticality principle and Caffarelli-Kohn-Nirenberg inequality, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions.

The main purpose of this paper is to establish sharp weighted Trudinger–Moser inequalities (Theorems 1.1, 1.2 and 1.3) and Caffarelli–Kohn–Nirenberg inequalities in the borderline case p = N (Theorems 1.5, 1.6 and 1.7) with best constants. Existence of extremal functions is also investigated for both the weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities. Radial symmetry of extremal functions for the weighted Trudinger–Moser inequalities are established (Theorem 1.4). Moreover, the sharp constants and the forms of the optimizers for the Caffarelli–Kohn–Nirenberg inequalities in some particular families of parameters in the borderline case p = N will be computed explicitly. Symmetrization arguments do not work in dealing with these weighted inequalities because of the presence of weights and the failure of the Polyá – Szegö inequality with weights. We will thus use a quasi-conformal mapping type transform and the corresponding symmetrization lemma to overcome this difficulty and carry out proofs of these results. As an application of the Caffarelli–Kohn–Nirenberg inequality, we also establish a weighted Moser–Onofri type inequality on the entire Euclidean space R 2 (see Theorem 1.8). [ABSTRACT FROM AUTHOR]

This paper is devoted to present a version of horizontal weighted Hardy-Rellich type inequality on stratified Lie groups and study some of its consequences. In particular, Sobolev type spaces are defined on stratified Lie groups and proved embedding theorems for these functional spaces. [ABSTRACT FROM AUTHOR]

We establish sharp remainder terms of the -Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups, yielding the inequalities with best constants. Our methods also give new sharp Caffarelli-Kohn-Nirenberg-type inequalities in with arbitrary quasi-norms. We also present explicit examples to illustrate our results for different weights and in abelian cases. [ABSTRACT FROM AUTHOR]

New embeddings of some weighted Sobolev spaces with weights a( x) and b( x) are established. The weights a( x) and b( x) can be singular. Some applications of these embeddings to a class of degenerate elliptic problems of the form −div( a( x)∇ u) = b( x) f( x, u) in Ω, u = 0 on ∂Ω, where Ω is a bounded or unbounded domain in R, N > 2, are presented. The main results of this paper also give some generalizations of the well-known Caffarelli-Kohn-Nirenberg inequality. [ABSTRACT FROM AUTHOR]

This work focuses on the symmetric solutions for a weighted quasilinear elliptic system involving multiple critical exponents in $\mathbb{R}^{N}$ . Based upon the Caffarelli-Kohn-Nirenberg inequality and the symmetric criticality principle due to Palais, we prove a variety of symmetric results under certain appropriate hypotheses on the singular potentials and the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we present a version of horizontal weighted Hardy–Rellich type and Caffarelli–Kohn–Nirenberg type inequalities on stratified groups and study some of their consequences. Our results reflect on many results previously known in special cases. Moreover, a new simple proof of the Badiale–Tarantello conjecture [2] on the best constant of a Hardy type inequality is provided. We also show a family of Poincaré inequalities as well as inequalities involving the weighted and unweighted p -sub-Laplacians. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and multiplicity of symmetric solutions for a weighted semilinear elliptic system with multiple critical Hardy-Sobolev exponents and singular potentials in $\mathbb{R}^{N}$ . Applying the symmetric criticality principle and Caffarelli-Kohn-Nirenberg inequality, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions. [ABSTRACT FROM AUTHOR]

We establish the full range Gagliardo-Nirenberg and the Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Coulomb-Sobolev spaces for the (fractional) derivative 0 ≤ s ≤ 1. As a result, we rediscover known Gagliardo-Nirenberg interpolation type inequalities associated with Coulomb-Sobolev spaces which were previously established in the scale of H s with 0 < s ≤ 1 and extend them for the full range W s , p with 0 ≤ s ≤ 1 and 1 < p < + ∞. Using these newly established weighted inequalities, we derive a new family of one body Hardy-Lieb-Thirring inequalities and use it to establish a new family of many body Hardy-Lieb-Thirring inequalities with a strong repulsive interaction term in L p scale. [ABSTRACT FROM AUTHOR]

In this paper, by using the idea of category, we investigate how the shape of the graph of h( x) affects the number of positive solutions to the following weighted nonlinear elliptic system: where 0 ∈ Ω is a smooth bounded domain in ℝ ( N ⩾ 3), λ, σ > 0 are parameters, $0 \leqslant \mu < \bar \mu _a \triangleq (\frac{{N - 2 - 2a}} {2})2 $; h( x), K( x) and K( x) are positive continuous functions in $\bar \Omega $, 1 ⩽ q < 2, α, β > 1 and α + β = 2*( a, b) $(2*(a,b) \triangleq \frac{{2N}} {{N - 2(1 + a - b)}}) $, is critical Sobolev-Hardy exponent). We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters ( λ, σ) belongs to a certain subset of ℝ. [ABSTRACT FROM AUTHOR]

We prove dilation invariant inequalities involving radial functions, poliharmonic operators and weights that are powers of the distance from the origin. Then, we discuss the existence of extremals, and in some cases, we compute the best constants. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and multiplicity of nontrivial solutions to a weighted nonlinear elliptic system with nonlinear homogeneous boundary condition in a bounded domain. By using the Caffarelli-Kohn-Nirenberg inequality and variational method, we prove that the system has at least two nontrivial solutions when the parameter λ belongs to a certain subset of ℝ. [ABSTRACT FROM AUTHOR]

-The purpose of this paper is to explain the phenomenon of symmetry breaking for optimal functions in functional inequalities by the numerical computations of some well chosen solutions of the corresponding Euler-Lagrange equations. For many of those inequalities it was believed that the only source of symmetry breaking would be the instability of the symmetric optimizer in the class of all admissible functions. But recently, it was shown by an indirect argument that for some Caffarelli-Kohn-Nirenberg inequalities this conjecture was not true. In order to understand this new symmetry breaking mechanism we have computed the branch of minimal solutions for a simple problem. A reparametrization of this branch allows us to build a scenario for the new phenomenon of symmetry breaking. The computations have been performed using freefem++. [ABSTRACT FROM AUTHOR]

A generalization of the classical Caffarelli–Kohn–Nirenberg inequality is obtained in the setting of Orlicz–Sobolev spaces. As applications, we prove a compact embedding result, and we establish the existence of weak solutions of the Dirichlet problem for a nonhomogeneous and degenerate/singular elliptic PDE. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the following elliptic equation where , , is differentiable in and is a given nonnegative Hölder continuous function in . The asymptotic behavior at infinity and structure of separation property of positive radial solutions with different initial data for (0.1) are discussed. Moreover, the existence and separation property of infinitely many positive solutions for Hardy equation and an equation related to Caffarelli–Kohn–Nirenberg inequality are obtained respectively, as special cases. [Copyright &y& Elsevier]