    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 & Applied Analysis, 2016, 15 (6) : 1975-2005. doi: 10.3934/cpaa.2016024
##### References:
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##### References:
  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275. Google Scholar  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.  Google Scholar  J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, New York, 1993. Google Scholar  T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148. Google Scholar  A. Kufner, Weighted Sobolev spaces, John Wiley and Sons, Inc., New York, 1985. Google Scholar  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.  Google Scholar  A. Porretta, Elliptic Equations with First Order Terms, Notes of the course at Alexandria, Ecole Cimpa, 2009. Google Scholar  G. Stampacchia, Le problème de Dirichlet pour les èquations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. Google Scholar  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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