Advanced Search
Article Contents
Article Contents

Quasilinear elliptic problem with Hardy potential and singular term

Abstract Related Papers Cited by
  • We consider the following quasilinear elliptic problem \begin{eqnarray*} -\Delta_pu =\lambda\frac{u^{p-1}}{|x|^p}+\frac{h}{u^\gamma} \quad in \quad\Omega, \end{eqnarray*} where $1 < p < N, \Omega\subset R^N$ is a bounded regular domain such that $0\in \Omega, \gamma>0$ and $h$ is a nonnegative measurable function with suitable hypotheses.
    The main goal of this work is to analyze the interaction between the Hardy potential and the singular term $u^{-\gamma}$ in order to get a solution for the largest possible class of the datum $h$. The regularity of the solution is also analyzed.
    Mathematics Subject Classification: Primary: 35K15, 35K55, 35K65; Secondary: 35B05, 35B40.


    \begin{equation} \\ \end{equation}
  • [1]

    B. Abdellaoui, E. Collorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Calc. Var, 23 (2005), 327-345.doi: 10.1007/s00526-004-0303-8.


    B. Abdellaoui, V. Felli and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-laplacian, Boll. Unione Mat. Ital. Sez. B., 2 (2006), 445-484.doi: 10.1007/s10231-002-0064-y.


    B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential, Annal. Math. Pura. Appl, 182 (2003), 247-270.doi: 10.1007/s10231-002-0064-y.


    B. Abdellaoui and I. Peral, A note on a critical problem with natural growth in the gradient, Jour. Euro. Math. Soc, 6 (2006), 119-136doi: 10.4171/JEMS/43.


    B. Abdellaoui and I. Peral, The Equation $-\Delta u-\lambda \fracu{|x|^2} = |\nabla u|^p +cf(x)$, the optimal power, Ann. Scuola Norm. Sup. Pisa, 5 (2007), 159-183.


    C. O. Alves, J. V. Goncalves and L. Maia, Singular nonlinear elliptic equations in $\mathbbR^N$, Abstr. Appl. Anal., 3 (1998), 411-423.doi: 10.1155/S1085337598000633.


    W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications, Nonlinear Ana. T.M.A., 32 (1998), 819-830.doi: 10.1016/S0362-546X(97)00530-0.


    D. Arcoya, J. Carmona, T. Leonori, P. Martínez-Aparicio, L. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations, J. Differential Equations, 246 (2009), 4006-4042.doi: 10.1016/j.jde.2009.01.016.


    P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa. Cl. Sci., 22 (1995), 241-273.


    L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM. Control, Optimisation and Calculus of Variations, 14 (2008), 411-426.doi: 10.1051/cocv:2008031.


    L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var., 37 (2010), 363-380.doi: 10.1007/s00526-009-0266-x.


    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.


    H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223-262.


    H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$, Manuscripta Math., 74 (1992), 87-106.doi: 10.1007/BF02567660.


    J. Cheng and Z. Zhang, Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems, Nonlinear Anal., 57 (2004), 473-484.doi: 10.1016/j.na.2004.02.025.


    J. García Azorero and I. Peral, Hardy Inequalities and some critical elliptic and parabolic problems, J. Diff. Eq., 144 (1998), 441-476.doi: 10.1006/jdeq.1997.3375.


    A. C. Lazer and J. P. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.doi: 10.2307/2048410.


    S. E. MiriQuasilinear elliptic problems with Hardy potential and reaction term, Differ. Equ. Appl. Available from: http://dea.ele-math.com/forthcoming


    F. Murat, L'injection du cone positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl., 60 (1981) 309-322.


    G. Stampacchia, Le problème de Dirichlet pour les équations élliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189-258.doi: 10.5802/aif.204.

  • 加载中

Article Metrics

HTML views() PDF downloads(163) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint