# American Institute of Mathematical Sciences

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

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

 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.
Citation: Mayte Pérez-Llanos. Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure and 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, Calc. Var. Partial Differential Equations, 23 (2005), 327-345. doi: 10.1007/s00526-004-0303-8. [2] B. Abdellaoui and I. Peral, Some results for semilinear elliptic equations with critical potential, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1-24. doi: 10.1017/S0308210500001505. [3] B. Abdellaoui and I. Peral, On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities, Commun. Pure Appl. Anal., 2 (2003), 539-566. doi: 10.3934/cpaa.2003.2.539. [4] B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the $p$-Laplacian with a critical potential, Ann. Mat. Pura Appl., 2 (2003), 247-270. doi: 10.1007/s10231-002-0064-y. [5] B. Abdellaoui and I. Peral, The equation $-\Delta u-\lambda\frac{u}{|x|^2}=|\nabla u| ^p+cf(x)$: the optimal power, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 159-183. [6] N. E. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23-35. doi: 10.1137/0524002. [7] W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830. doi: 10.1016/S0362-546X(97)00530-0. [8] L. Boccardo, F. Murat and J. P. Puel, Rèsultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 213-235. [9] L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Contin. Dyn. Syst., 16 (2006), 513-523. doi: 10.3934/dcds.2006.16.513. [10] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 223-262. [11] H. Brezis, L. Dupaigne and A. Tesei, On a semilinear elliptic equation with inverse-square potential, Selecta Math. (N.S.), 11 (2005), 1-7. doi: 10.1007/s00029-005-0003-z. [12] H. Brezis and A. C. Ponce, Kato's inequality when $\Delta u$ is a measure, C. R. Math. Acad. Sci. Paris, 338 (2004), 599-604. doi: 10.1016/j.crma.2003.12.032. [13] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275. [14] F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. [15] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, New York, 1993. [16] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148. [17] A. Kufner, Weighted Sobolev spaces, John Wiley and Sons, Inc., New York, 1985. [18] J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs, 51 American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/051. [19] A. Porretta, Elliptic Equations with First Order Terms, Notes of the course at Alexandria, Ecole Cimpa, 2009. [20] G. Stampacchia, Le problème de Dirichlet pour les èquations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. [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.

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##### References:
 [1] B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. Partial Differential Equations, 23 (2005), 327-345. doi: 10.1007/s00526-004-0303-8. [2] B. Abdellaoui and I. Peral, Some results for semilinear elliptic equations with critical potential, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1-24. doi: 10.1017/S0308210500001505. [3] B. Abdellaoui and I. Peral, On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities, Commun. Pure Appl. Anal., 2 (2003), 539-566. doi: 10.3934/cpaa.2003.2.539. [4] B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the $p$-Laplacian with a critical potential, Ann. Mat. Pura Appl., 2 (2003), 247-270. doi: 10.1007/s10231-002-0064-y. [5] B. Abdellaoui and I. Peral, The equation $-\Delta u-\lambda\frac{u}{|x|^2}=|\nabla u| ^p+cf(x)$: the optimal power, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 159-183. [6] N. E. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23-35. doi: 10.1137/0524002. [7] W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830. doi: 10.1016/S0362-546X(97)00530-0. [8] L. Boccardo, F. Murat and J. P. Puel, Rèsultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 213-235. [9] L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Contin. Dyn. Syst., 16 (2006), 513-523. doi: 10.3934/dcds.2006.16.513. [10] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 223-262. [11] H. Brezis, L. Dupaigne and A. Tesei, On a semilinear elliptic equation with inverse-square potential, Selecta Math. (N.S.), 11 (2005), 1-7. doi: 10.1007/s00029-005-0003-z. [12] H. Brezis and A. C. Ponce, Kato's inequality when $\Delta u$ is a measure, C. R. Math. Acad. Sci. Paris, 338 (2004), 599-604. doi: 10.1016/j.crma.2003.12.032. [13] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275. [14] F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. [15] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, New York, 1993. [16] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148. [17] A. Kufner, Weighted Sobolev spaces, John Wiley and Sons, Inc., New York, 1985. [18] J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs, 51 American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/051. [19] A. Porretta, Elliptic Equations with First Order Terms, Notes of the course at Alexandria, Ecole Cimpa, 2009. [20] G. Stampacchia, Le problème de Dirichlet pour les èquations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. [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.
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