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November  2016, 15(6): 1975-2005. doi: 10.3934/cpaa.2016024

Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities

Mayte Pérez-Llanos 1,

1. 

Instituto de Investigaciones Matemáticas Luis Santaló and CONICET, Facultad de Ciencias Exactas y Naturales, Pabellón I, Ciudad Universitaria, 1428, Argentina

Received  November 2012 Revised  May 2013 Published  September 2016

In this paper we analyze the following elliptic problem related to some Caffarelli-Kohn-Nirenberg inequalities: \begin{eqnarray} -div(|x|^{-2\gamma}\nabla u)-\lambda \frac{u}{|x|^{2(\gamma+1)}}=|\nabla u|^p|x|^{-\gamma p}+cf,\; u>0\; \mbox{ in }\; \Omega, \qquad u_{|\partial \Omega}\equiv0, \end{eqnarray} where $\Omega \subset R^N$ is a domain such that $0\in\Omega$, $N\geq 3$, and $c, \lambda, \gamma, p $ are positive constants verifying $0 < \lambda \leq \Lambda_{N,\gamma}=\left(\frac{N-2(\gamma+1)}{2}\right)^{2}$, $-\infty<\gamma<\frac{N-2} 2$ and $p>0$. Our study concerns to existence of solutions to the former problem. More precisely, first we determine a critical thereshold for the power $p$, in the sense that, beyond this value it does not exist any positive supersolution to our problem, not even in a very weak sense. In addition, we show existence of solutions for all the values $p>0$ below this threshold, with the restriction $\gamma>-\frac{N(1-p)+2}{2}$, whenever the righthand side verifies $f(x)\leq |x|^{-2(\gamma+1)}$ if $\gamma>-1$. When $-\frac{N(1-p)+2}{2}<\gamma\leq -1$ it suffices that $f\in L^{2/p}(\Omega)$. The existence of solutions for $0 < p < 1$ and $\gamma\leq -\frac{N(1-p)+2}{2}$ is an open question.
Keywords: Caffarelli-Kohn-Nirenberg inequality, Hardy potential, weighted Sobolev Spaces..
Mathematics Subject Classification: Primary: 35D05, 35J10; Secondary: 35J6.
Citation: Mayte Pérez-Llanos. Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure & Applied Analysis, 2016, 15 (6) : 1975-2005. doi: 10.3934/cpaa.2016024
References:
[1]

B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities,, \emph{Calc. Var. Partial Differential Equations}, 23 (2005), 327.  doi: 10.1007/s00526-004-0303-8.  Google Scholar

[2]

B. Abdellaoui and I. Peral, Some results for semilinear elliptic equations with critical potential,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 132 (2002), 1.  doi: 10.1017/S0308210500001505.  Google Scholar

[3]

B. Abdellaoui and I. Peral, On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities,, \emph{Commun. Pure Appl. Anal.}, 2 (2003), 539.  doi: 10.3934/cpaa.2003.2.539.  Google Scholar

[4]

B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the $p$-Laplacian with a critical potential,, \emph{Ann. Mat. Pura Appl.}, 2 (2003), 247.  doi: 10.1007/s10231-002-0064-y.  Google Scholar

[5]

B. Abdellaoui and I. Peral, The equation $-\Delta u-\lambda\frac{u}{|x|^2}=|\nabla u| ^p+cf(x)$: the optimal power,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 6 (2007), 159.   Google Scholar

[6]

N. E. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures,, \emph{SIAM J. Math. Anal.}, 24 (1993), 23.  doi: 10.1137/0524002.  Google Scholar

[7]

W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications,, \emph{Nonlinear Anal.}, 32 (1998), 819.  doi: 10.1016/S0362-546X(97)00530-0.  Google Scholar

[8]

L. Boccardo, F. Murat and J. P. Puel, Rèsultats d'existence pour certains problèmes elliptiques quasilinéaires,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 11 (1984), 213.   Google Scholar

[9]

L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential,, \emph{Discrete Contin. Dyn. Syst.}, 16 (2006), 513.  doi: 10.3934/dcds.2006.16.513.  Google Scholar

[10]

H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions,, \emph{Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat.}, 1 (1998), 223.   Google Scholar

[11]

H. Brezis, L. Dupaigne and A. Tesei, On a semilinear elliptic equation with inverse-square potential,, \emph{Selecta Math. (N.S.)}, 11 (2005), 1.  doi: 10.1007/s00029-005-0003-z.  Google Scholar

[12]

H. Brezis and A. C. Ponce, Kato's inequality when $\Delta u$ is a measure,, \emph{C. R. Math. Acad. Sci. Paris}, 338 (2004), 599.  doi: 10.1016/j.crma.2003.12.032.  Google Scholar

[13]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, \emph{Compositio Math.}, 53 (1984), 259.   Google Scholar

[14]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions,, \emph{Comm. Pure Appl. Math.}, 54 (2001), 229.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[15]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford University Press, (1993).   Google Scholar

[16]

T. Kato, Schrödinger operators with singular potentials,, \emph{Israel J. Math.}, 13 (1972), 135.   Google Scholar

[17]

A. Kufner, Weighted Sobolev spaces,, John Wiley and Sons, (1985).   Google Scholar

[18]

J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations,, Mathematical Surveys and Monographs, 51 (1997).  doi: 10.1090/surv/051.  Google Scholar

[19]

A. Porretta, Elliptic Equations with First Order Terms,, Notes of the course at Alexandria, (2009).   Google Scholar

[20]

G. Stampacchia, Le problème de Dirichlet pour les èquations elliptiques du second ordre à coefficients discontinus,, \emph{Ann. Inst. Fourier (Grenoble)}, 15 (1965), 189.   Google Scholar

[21]

C. A. Swanson, Remarks on Picone's identity and related identities,, \emph{Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia}, 11 (): 3.   Google Scholar

show all references

References:
[1]

B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities,, \emph{Calc. Var. Partial Differential Equations}, 23 (2005), 327.  doi: 10.1007/s00526-004-0303-8.  Google Scholar

[2]

B. Abdellaoui and I. Peral, Some results for semilinear elliptic equations with critical potential,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 132 (2002), 1.  doi: 10.1017/S0308210500001505.  Google Scholar

[3]

B. Abdellaoui and I. Peral, On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities,, \emph{Commun. Pure Appl. Anal.}, 2 (2003), 539.  doi: 10.3934/cpaa.2003.2.539.  Google Scholar

[4]

B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the $p$-Laplacian with a critical potential,, \emph{Ann. Mat. Pura Appl.}, 2 (2003), 247.  doi: 10.1007/s10231-002-0064-y.  Google Scholar

[5]

B. Abdellaoui and I. Peral, The equation $-\Delta u-\lambda\frac{u}{|x|^2}=|\nabla u| ^p+cf(x)$: the optimal power,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 6 (2007), 159.   Google Scholar

[6]

N. E. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures,, \emph{SIAM J. Math. Anal.}, 24 (1993), 23.  doi: 10.1137/0524002.  Google Scholar

[7]

W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications,, \emph{Nonlinear Anal.}, 32 (1998), 819.  doi: 10.1016/S0362-546X(97)00530-0.  Google Scholar

[8]

L. Boccardo, F. Murat and J. P. Puel, Rèsultats d'existence pour certains problèmes elliptiques quasilinéaires,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 11 (1984), 213.   Google Scholar

[9]

L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential,, \emph{Discrete Contin. Dyn. Syst.}, 16 (2006), 513.  doi: 10.3934/dcds.2006.16.513.  Google Scholar

[10]

H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions,, \emph{Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat.}, 1 (1998), 223.   Google Scholar

[11]

H. Brezis, L. Dupaigne and A. Tesei, On a semilinear elliptic equation with inverse-square potential,, \emph{Selecta Math. (N.S.)}, 11 (2005), 1.  doi: 10.1007/s00029-005-0003-z.  Google Scholar

[12]

H. Brezis and A. C. Ponce, Kato's inequality when $\Delta u$ is a measure,, \emph{C. R. Math. Acad. Sci. Paris}, 338 (2004), 599.  doi: 10.1016/j.crma.2003.12.032.  Google Scholar

[13]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, \emph{Compositio Math.}, 53 (1984), 259.   Google Scholar

[14]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions,, \emph{Comm. Pure Appl. Math.}, 54 (2001), 229.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[15]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford University Press, (1993).   Google Scholar

[16]

T. Kato, Schrödinger operators with singular potentials,, \emph{Israel J. Math.}, 13 (1972), 135.   Google Scholar

[17]

A. Kufner, Weighted Sobolev spaces,, John Wiley and Sons, (1985).   Google Scholar

[18]

J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations,, Mathematical Surveys and Monographs, 51 (1997).  doi: 10.1090/surv/051.  Google Scholar

[19]

A. Porretta, Elliptic Equations with First Order Terms,, Notes of the course at Alexandria, (2009).   Google Scholar

[20]

G. Stampacchia, Le problème de Dirichlet pour les èquations elliptiques du second ordre à coefficients discontinus,, \emph{Ann. Inst. Fourier (Grenoble)}, 15 (1965), 189.   Google Scholar

[21]

C. A. Swanson, Remarks on Picone's identity and related identities,, \emph{Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia}, 11 (): 3.   Google Scholar

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