\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Elliptic equations having a singular quadratic gradient term and a changing sign datum

Abstract Related Papers Cited by
  • In this paper we study a singular elliptic problem whose model is \begin{eqnarray*} - \Delta u= \frac{|\nabla u|^2}{|u|^\theta}+f(x), in \Omega\\ u = 0, on \partial \Omega; \end{eqnarray*} where $\theta\in (0,1)$ and $f \in L^m (\Omega)$, with $m\geq \frac{N}{2}$. We do not assume any sign condition on the lower order term, nor assume the datum $f$ has a constant sign. We carefully define the meaning of solution to this problem giving sense to the gradient term where $u=0$, and prove the existence of such a solution. We also discuss related questions as the existence of solutions when the datum $f$ is less regular or the boundedness of the solutions when the datum $f \in L^m (\Omega)$ with $m> \frac{N}{2}$.
    Mathematics Subject Classification: 35J75, 35J25.

    Citation:

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

    B. Abdellaoui, D. Giachetti, I. Peral and M. Walias, Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary, Nonlinear Analysis, 74 (2011), 1355-1371.

    [2]

    D. Arcoya, S. Barile and P. J. Martínez-Aparicio, Singular quasilinear equations with quadratic growth in the gradient without sign condition, J. Math. Anal. Appl., 350 (2009), 401-408.doi: 10.1016/j.jmaa.2008.09.073.

    [3]

    D. Arcoya, L. Boccardo, T. Leonori and A. Porretta, Some elliptic problems with singular natural growth lower order terms, J. Differential Equations, 249 (2010), 2771-2795.doi: 10.1016/j.jde.2010.05.009.

    [4]

    D. Arcoya, J. Carmona, T. Leonori, P. J. 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.

    [5]

    D. Arcoya, J. Carmona and P. J. Martínez-Aparicio, Elliptic obstacle problems with natural growth on the gradient and singular nonlinear terms, Adv. Nonlinear Stud., 7 (2007), 299-317.

    [6]

    D. Arcoya and P. J. Martínez-Aparicio, Quasilinear equations with natural growth, Rev. Mat. Iberoamericana, 24 (2008), 597-616.doi: 10.4171/RMI/548.

    [7]

    D. Arcoya and S. Segura de León, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM: Control, Optimization and the Calculus of Variations, 16 (2010), 327-336.doi: 10.1051/cocv:2008072.

    [8]

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

    [9]

    L. Boccardo and T. Gallouët, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal., 19 (1992), 573-579.doi: 10.1016/0362-546X(92)90022-7.

    [10]

    L. Boccardo, T. Leonori, L. Orsina and F. Petitta, Quasilinear elliptic equations with singular quadratic growth terms, Comm. Contemp. Math., 13 (2011), 607-642.doi: 10.1142/S0219199711004300.

    [11]

    L. Boccardo, F. Murat and J. P. Puel, Existence de solutions non bornées pour certaines équations quasi-linéaires, Portugal. Math., 41 (1982), 507-534.

    [12]

    L. Boccardo, F. Murat and J. P. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 213-235.

    [13]

    L. Boccardo, F. Murat and J. P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl., 152 (1988), 183-196.doi: 10.1007/BF01766148.

    [14]

    L. Boccardo, S. Segura de León and C. Trombetti, Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term, J. Math. Pures Appl., 80 (2001), 919-940.doi: 10.1016/S0021-7824(01)01211-9.

    [15]

    F. E. Browder, Existence theorems for nonlinear partial differential equations, "Global Analysis" (Proc. Sympos. Pre Math., vol XVI, Berkeley, California, 1968),

    [16]

    D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behaviour, Boll. Unione Mat. Ital., 2 (2009), 349-370.

    [17]

    D. Giachetti and S. Segura de LeónQuasilinear stationary problems with a quadratic gradient term having singularities, J. London Math. Soc., in press.

    [18]

    A. Porretta and S. Segura de León, Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl., 85 (2006), 465-492.doi: 10.1016/j.matpur.2005.10.009.

    [19]

    S. Segura de León, Existence and uniqueness for $L^1$ data of some elliptic equations with natural growth, Adv. Diff. Eq., 8 (2003), 1377-1408.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(111) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return