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Автор: Grafiati

Опубліковано: 14 березня 2023

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1

LIN, LISHAN, and ZHAOLI LIU. "MULTI-BUBBLE SOLUTIONS FOR EQUATIONS OF CAFFARELLI–KOHN–NIRENBERG TYPE." Communications in Contemporary Mathematics 13, no. 06 (December 2011): 945–68. http://dx.doi.org/10.1142/s0219199711004518.

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Анотація:

Multi-bubble solutions are constructed for the elliptic equations of Caffarelli–Kohn–Nirenberg type [Formula: see text] where N ≥ 3, [Formula: see text], [Formula: see text], [Formula: see text], a ≤ b < a + 1, K ∈ C(ℝN), K(x) > 0 for x ∈ ℝN, K(x) → 0 as |x| → 0 and |x| → ∞, and ϵ > 0 is a parameter.

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2

FELLI, VERONICA, and MATTHIAS SCHNEIDER. "COMPACTNESS AND EXISTENCE RESULTS FOR DEGENERATE CRITICAL ELLIPTIC EQUATIONS." Communications in Contemporary Mathematics 07, no. 01 (February 2005): 37–73. http://dx.doi.org/10.1142/s0219199705001623.

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This paper is devoted to the study of degenerate critical elliptic equations of Caffarelli–Kohn–Nirenberg type. By means of blow-up analysis techniques, we prove an a priori estimate in a weighted space of continuous functions. From this compactness result, the existence of a solution to our problem is proved by exploiting the homotopy invariance of the Leray–Schauder degree.

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3

Abdellaoui, B., and I. Peral. "On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities." Communications on Pure & Applied Analysis 2, no. 4 (2003): 539–66. http://dx.doi.org/10.3934/cpaa.2003.2.539.

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4

Dall'Aglio, Andrea, Daniela Giachetti, and Ireneo Peral. "Results on Parabolic Equations Related to Some Caffarelli--Kohn--Nirenberg Inequalities." SIAM Journal on Mathematical Analysis 36, no. 3 (January 2005): 691–716. http://dx.doi.org/10.1137/s0036141003432353.

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5

Felli, Veronica, and Matthias Schneider. "Perturbation results of critical elliptic equations of Caffarelli–Kohn–Nirenberg type." Journal of Differential Equations 191, no. 1 (June 2003): 121–42. http://dx.doi.org/10.1016/s0022-0396(02)00085-2.

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6

Musso, Monica, and Juncheng Wei. "Nonradial Solutions to Critical Elliptic Equations of Caffarelli–Kohn–Nirenberg Type." International Mathematics Research Notices 2012, no. 18 (September 14, 2011): 4120–62. http://dx.doi.org/10.1093/imrn/rnr179.

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7

Bouchekif, M., and A. Matallah. "On singular nonhomogeneous elliptic equations involving critical Caffarelli–Kohn–Nirenberg exponent." Ricerche di Matematica 58, no. 2 (August 5, 2009): 207–18. http://dx.doi.org/10.1007/s11587-009-0056-y.

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8

DENG, YINBIN, LINGYU JIN, and SHUANGJIE PENG. "POSITIVE SOLUTIONS FOR ELLIPTIC EQUATIONS RELATED TO THE CAFFARELLI–KOHN–NIRENBERG INEQUALITIES." Communications in Contemporary Mathematics 11, no. 02 (April 2009): 185–99. http://dx.doi.org/10.1142/s0219199709003338.

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Анотація:

In this paper, we are concerned with the following elliptic problems which are related to the well-known Caffarelli–Kohn–Nirenberg inequalities: [Formula: see text] where a = b < 0, [Formula: see text], a ≤ d ≤ a + 1, a ≤ e ≤ a + 1, [Formula: see text], [Formula: see text], 2 < q < D, λ and η are real constants. We obtain positive solutions for problem (0.1). Moreover, we establish a corresponding Pohozaev identity for problem (0.1), from which, the nonexistence of positive solutions for problem (0.1) is obtained.

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9

Deng, Zhiying, and Yisheng Huang. "On -symmetric solutions of critical elliptic equations of Caffarelli–Kohn–Nirenberg type." Nonlinear Analysis: Real World Applications 12, no. 2 (April 2011): 1089–102. http://dx.doi.org/10.1016/j.nonrwa.2010.09.002.

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10

Benmansour, S., and A. Matallah. "Multiple Solutions for Nonhomogeneous Elliptic Equations Involving Critical Caffarelli–Kohn–Nirenberg Exponent." Mediterranean Journal of Mathematics 13, no. 6 (July 8, 2016): 4679–91. http://dx.doi.org/10.1007/s00009-016-0769-6.

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11

Villavert, John. "Classification of radial solutions to equations related to Caffarelli–Kohn–Nirenberg inequalities." Annali di Matematica Pura ed Applicata (1923 -) 199, no. 1 (July 18, 2019): 299–315. http://dx.doi.org/10.1007/s10231-019-00879-0.

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12

CHUNG, NGUYEN THANH, and HOANG QUOC TOAN. "EXISTENCE RESULT FOR NONUNIFORMLY DEGENERATE SEMILINEAR ELLIPTIC SYSTEMS INN." Glasgow Mathematical Journal 51, no. 3 (September 2009): 561–70. http://dx.doi.org/10.1017/s0017089509005175.

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Анотація:

AbstractWe study the existence of solutions for a class of nonuniformly degenerate elliptic systems inN,N≥ 3, of the formwherehi∈L1loc(N),hi(x) ≧ γ0|x|αwith α ∈ (0, 2) and γ0> 0,i= 1, 2. The proofs rely essentially on a variant of the Mountain pass theorem (D. M. Duc, Nonlinear singular elliptic equations,J. Lond. Math. Soc.40(2) (1989), 420–440) combined with the Caffarelli–Kohn–Nirenberg inequality (First order interpolation inequalities with weights,Composito Math.53(1984), 259–275).

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13

Chabrowski, J., and D. G. Costa. "On existence of positive solutions for a class of Caffarelli–Kohn–Nirenberg type equations." Colloquium Mathematicum 120, no. 1 (2010): 43–62. http://dx.doi.org/10.4064/cm120-1-4.

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14

Volzone, Bruno. "Semilinear elliptic equations with degenerate and singular weights related to Caffarelli–Kohn–Nirenberg inequalities." Journal of Mathematical Analysis and Applications 393, no. 2 (September 2012): 614–31. http://dx.doi.org/10.1016/j.jmaa.2012.04.028.

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15

Liu, Haidong, and Leiga Zhao. "Nontrivial solutions for a class of critical elliptic equations of Caffarelli–Kohn–Nirenberg type." Journal of Mathematical Analysis and Applications 404, no. 2 (August 2013): 317–25. http://dx.doi.org/10.1016/j.jmaa.2013.03.016.

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16

Bouchekif, Mohammed, and Atika Matallah. "On singular elliptic equations involving a concave term and critical Caffarelli-Kohn-Nirenberg exponent." Mathematische Nachrichten 284, no. 2-3 (January 28, 2011): 177–85. http://dx.doi.org/10.1002/mana.200710211.

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17

Abdellaoui, B., and I. Peral Alonso. "Hölder regularity and Harnack inequality for degenerate parabolic equations related to Caffarelli–Kohn–Nirenberg inequalities." Nonlinear Analysis: Theory, Methods & Applications 57, no. 7-8 (June 2004): 971–1003. http://dx.doi.org/10.1016/j.na.2004.03.024.

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18

Bhakta, Mousomi. "Caffarelli–Kohn–Nirenberg type equations of fourth order with the critical exponent and Rellich potential." Journal of Mathematical Analysis and Applications 433, no. 1 (January 2016): 681–700. http://dx.doi.org/10.1016/j.jmaa.2015.07.042.

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19

Bartsch, Thomas, Shuangjie Peng, and Zhitao Zhang. "Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities." Calculus of Variations and Partial Differential Equations 30, no. 1 (January 24, 2007): 113–36. http://dx.doi.org/10.1007/s00526-006-0086-1.

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20

Nhan, Le Cong, Ky Ho, and Le Xuan Truong. "Regularity of solutions for a class of quasilinear elliptic equations related to the Caffarelli-Kohn-Nirenberg inequality." Journal of Mathematical Analysis and Applications 505, no. 1 (January 2022): 125474. http://dx.doi.org/10.1016/j.jmaa.2021.125474.

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21

Balbino Guimarães, Mateus, and Rodrigo da Silva Rodrigues. "Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions." Communications on Pure & Applied Analysis 12, no. 6 (2013): 2697–713. http://dx.doi.org/10.3934/cpaa.2013.12.2697.

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22

Colorado, Eduardo, and Irened Peral. "Eigenvalues and bifurcation for elliptic equations with mixed Dirichlet-Neumann boundary conditions related to Caffarelli-Kohn-Nirenberg inequalities." Topological Methods in Nonlinear Analysis 23, no. 2 (June 1, 2004): 239. http://dx.doi.org/10.12775/tmna.2004.011.

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23

Bonforte, Matteo, and Nikita Simonov. "Quantitative a priori estimates for fast diffusion equations with Caffarelli–Kohn–Nirenberg weights. Harnack inequalities and Hölder continuity." Advances in Mathematics 345 (March 2019): 1075–161. http://dx.doi.org/10.1016/j.aim.2019.01.018.

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24

Bonforte, Matteo, Jean Dolbeault, Matteo Muratori, and Bruno Nazaret. "Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities." Kinetic & Related Models 10, no. 1 (2017): 33–59. http://dx.doi.org/10.3934/krm.2017002.

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25

Abdellaoui, B., and I. Peral Alonso. "The effect of Harnack inequality on the existence and nonexistence results for quasi-linear parabolic equations related to Caffarelli-Kohn-Nirenberg inequalities." Nonlinear Differential Equations and Applications NoDEA 14, no. 3-4 (September 21, 2007): 335–60. http://dx.doi.org/10.1007/s00030-007-5048-6.

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26

Felli, Veronica, and Matthias Schneider. "A Note on Regularity of Solutions to Degenerate Elliptic Equations of Caffarelli-Kohn-Nirenberg Type." Advanced Nonlinear Studies 3, no. 4 (January 1, 2003). http://dx.doi.org/10.1515/ans-2003-0402.

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27

Binh, Nguyen Dinh, and Cung The Anh. "Attractors for parabolic equations related to Caffarelli-Kohn-Nirenberg inequalities." Boundary Value Problems 2012, no. 1 (March 28, 2012). http://dx.doi.org/10.1186/1687-2770-2012-35.

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28

Kassymov, Aidyn, Michael Ruzhansky, and Durvudkhan Suragan. "Anisotropic Fractional Gagliardo-Nirenberg, Weighted Caffarelli-Kohn-Nirenberg and Lyapunov-type Inequalities, and Applications to Riesz Potentials and p-sub-Laplacian Systems." Potential Analysis, August 18, 2022. http://dx.doi.org/10.1007/s11118-022-10029-6.

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Анотація:

AbstractIn this paper we prove the fractional Gagliardo-Nirenberg inequality on homogeneous Lie groups. Also, we establish weighted fractional Caffarelli-Kohn-Nirenberg inequality and Lyapunov-type inequality for the Riesz potential on homogeneous Lie groups. The obtained Lyapunov inequality for the Riesz potential is new already in the classical setting of $\mathbb {R}^{N}$ ℝ N . As an application, we give two-sided estimate for the first eigenvalue of the Riesz potential. Also, we obtain Lyapunov inequality for the system of the fractional p-sub-Laplacian equations and give an application to estimate its eigenvalues.

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29

Abdellaoui, Boumediene, Veronica Felli, and Ireneo Peral. "A Remark on Perturbed Elliptic Equations of Caffarelli-Kohn-Nirenberg Type." Revista Matemática Complutense 18, no. 2 (September 27, 2005). http://dx.doi.org/10.5209/rev_rema.2005.v18.n2.16677.

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30

Dubrulle, B., and J. D. Gibbon. "A correspondence between the multifractal model of turbulence and the Navier–Stokes equations." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380, no. 2218 (January 17, 2022). http://dx.doi.org/10.1098/rsta.2021.0092.

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The multifractal model of turbulence (MFM) and the three-dimensional Navier–Stokes equations are blended together by applying the probabilistic scaling arguments of the former to a hierarchy of weak solutions of the latter. This process imposes a lower bound on both the multifractal spectrum C ( h ) , which appears naturally in the Large Deviation formulation of the MFM, and on h the standard scaling parameter. These bounds respectively take the form: (i) C ( h ) ≥ 1 − 3 h , which is consistent with Kolmogorov’s four-fifths law ; and (ii) h ≥ − 2 3 . The latter is significant as it prevents solutions from approaching the Navier–Stokes singular set of Caffarelli, Kohn and Nirenberg. This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.

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31

Bonforte, Matteo, Jean Dolbeault, Bruno Nazaret, and Nikita Simonov. "Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations." Discrete and Continuous Dynamical Systems, 2022, 0. http://dx.doi.org/10.3934/dcds.2022093.

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<p style='text-indent:20px;'>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.</p>

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32

Berselli, Luigi C., and Stefano Spirito. "Convergence of second-order in time numerical discretizations for the evolution Navier-Stokes equations." Advances in Continuous and Discrete Models 2022, no. 1 (November 28, 2022). http://dx.doi.org/10.1186/s13662-022-03736-2.

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AbstractWe prove the convergence of certain second-order numerical methods to weak solutions of the Navier–Stokes equations satisfying, in addition, the local energy inequality, and therefore suitable in the sense of Scheffer and Caffarelli–Kohn–Nirenberg. More precisely, we treat the space-periodic case in three space dimensions and consider a full discretization in which the classical Crank–Nicolson method (θ-method with $\theta =1/2$ θ = 1 / 2 ) is used to discretize the time variable. In contrast, in the space variables, we consider finite elements. The convective term is discretized in several implicit, semi-implicit, and explicit ways. In particular, we focus on proving (possibly conditional) convergence of the discrete solutions toward weak solutions (satisfying a precise local energy balance) without extra regularity assumptions on the limit problem. We do not prove orders of convergence, but our analysis identifies some numerical schemes, providing alternate proofs of the existence of “physically relevant” solutions in three space dimensions.

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33

Abdellaoui, Boumediene, Eduardo Colorado, and Ireneo Peral. "Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities." Journal of the European Mathematical Society, 2004, 119–48. http://dx.doi.org/10.4171/jems/4.

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34

Pardo, Rosa. "$$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity." Journal of Fixed Point Theory and Applications 25, no. 2 (February 6, 2023). http://dx.doi.org/10.1007/s11784-023-01048-w.

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AbstractWe consider a semilinear boundary value problem $$ -\Delta u= f(x,u),$$ - Δ u = f ( x , u ) , in $$\Omega ,$$ Ω , with Dirichlet boundary conditions, where $$\Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N with $$N> 2,$$ N > 2 , is a bounded smooth domain, and f is a Carathéodory function, superlinear and subcritical at infinity. We provide $$L^\infty (\Omega )$$ L ∞ ( Ω ) a priori estimates for weak solutions in terms of their $$L^{2^*}(\Omega )$$ L 2 ∗ ( Ω ) -norm, where $$2^*=\frac{2N}{N-2}\ $$ 2 ∗ = 2 N N - 2 is the critical Sobolev exponent. In particular, our results also apply to $$f(x,s)=a(x)\,\frac{|s|^{2^*_{N/r}-2}s}{\big [\log (e+|s|)\big ]^\beta }\,$$ f ( x , s ) = a ( x ) | s | 2 N / r ∗ - 2 s [ log ( e + | s | ) ] β , where $$a\in L^r(\Omega )$$ a ∈ L r ( Ω ) with $$N/2<r\le \infty $$ N / 2 < r ≤ ∞ , and $$2_{N/r}^*:=2^*\left( 1-\frac{1}{r}\right) $$ 2 N / r ∗ : = 2 ∗ 1 - 1 r . Assume $$N/2<r\le N$$ N / 2 < r ≤ N . We show that for any $$\varepsilon >0$$ ε > 0 there exists a constant $$C_\varepsilon >0$$ C ε > 0 such that for any solution $$u\in H^1_0(\Omega )$$ u ∈ H 0 1 ( Ω ) , the following holds: $$\begin{aligned} \Big [\log \big (e+\Vert u\Vert _{\infty }\big )\Big ]^\beta \le C _\varepsilon \, \Big (1+\Vert u\Vert _{2^*}\Big )^{\, (2^*_{N/r}-2)(1+\varepsilon )}. \end{aligned}$$ [ log ( e + ‖ u ‖ ∞ ) ] β ≤ C ε ( 1 + ‖ u ‖ 2 ∗ ) ( 2 N / r ∗ - 2 ) ( 1 + ε ) . To establish our results, we do not assume any restrictions on the sign of the solutions, or on the non-linearity. Our approach is based on Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg interpolation inequalities. Finally, we state sufficient conditions for having $$H_0^1(\Omega )$$ H 0 1 ( Ω ) uniform a priori bounds for non-negative solutions, so finally we provide suficient conditions for having $$L^\infty (\Omega )$$ L ∞ ( Ω ) uniform a priori bounds, which holds roughly speaking for superlinear and subcritical non-linearities.

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