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Boundary estimates for solutions of weighted semilinear elliptic equations

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  • Let $b(x)$ be a positive function in a bounded smooth domain $\Omega\subset R^N$, and let $f(t)$ be a positive non decreasing function on $(0,\infty)$ such that $\lim_{t\to\infty}f(t)=\infty$. We investigate boundary blow-up solutions of the equation $\Delta u=b(x)f(u)$. Under appropriate conditions on $b(x)$ as $x$ approaches $\partial\Omega$ and on $f(t)$ as $t$ goes to infinity, we find a second order approximation of the solution $u(x)$ as $x$ goes to $\partial\Omega$.
        We also investigate positive solutions of the equation $\Delta u+(\delta(x))^{2\ell}u^{-q}=0$ in $\Omega$ with $u=0$ on $\partial\Omega$, where $\ell\ge 0$, $q>3+2\ell$ and $\delta(x)$ denotes the distance from $x$ to $\partial\Omega$. We find a second order approximation of the solution $u(x)$ as $x$ goes to $\partial\Omega$.
    Mathematics Subject Classification: Primary: 35J25; Secondary: 35B40.

    Citation:

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  • [1]

    C. AneddaSecond-order boundary estimates for solutions to singular elliptic equations, Electronic Journal of Differential Equations, 2009, 15 pp.

    [2]

    C. Anedda, F. Cuccu and G. Porru, Boundary estimates for solutions to singular elliptic equations, Matematiche (Catania), 60 (2005), 339-352.

    [3]

    C. Anedda and G. Porru, Second order estimates forboundary blow-up solutions of elliptic equations, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations,Proceedings of the 6$^th$ AIMS International Conference, Suppl.,54-63.

    [4]

    C. Bandle and M. Marcus, "Large" solutions of semilinear elliptic equations: Existence,uniqueness and asymptotic behaviour, J. Anal. Math., 58 (1992), 9-24.doi: 10.1007/BF02790355.

    [5]

    C. Bandle and M. Marcus, Dependence of blowup rate of large solutions of semilinear ellipticequations on the curvature of the boundary, Complex Var. Theory Appl., 49 (2004), 555-570.doi: 10.1080/02781070410001731729.

    [6]

    C. Bandle and M. Marcus, On second-order effects in the boundary behaviour of large solutionsof semilinear elliptic problems, Differential and Integral Equations, 11 (1998), 23-34.

    [7]

    S. Berhanu, F. Cuccu and G. Porru, On the boundary behaviour, including second order effects, of solutions to singular ellipticproblems, Acta Math. Sin. (Engl. Ser.), 23 (2007), 479-486.doi: 10.1007/s10114-005-0680-8.

    [8]

    S. Berhanu and G. Porru, Qualitative and quantitative estimates for large solutions tosemilinear equations, Communications in Applied Analysis, 4 (2000), 121-131.

    [9]

    L. Bieberback, $\Delta u=e^u$ und die automorphen Functionen, Mat. Ann., 77 (1916), 173-212.doi: 10.1007/BF01456901.

    [10]

    F.-C. Cirstea and V. Rădulescu, Uniqueness of the blow-up boundarysolution of logistic equations with absorbtion, C. R. Acad. Sci. Paris, 335 (2002), 447-452.doi: 10.1016/S1631-073X(02)02503-7.

    [11]

    F.-C. Cirstea and V. Rădulescu, Nonlinear problems with boundaryblow-up: A Karamata regular variation approach, Asymptotic Analysis, 46 (2006), 275-298.

    [12]

    M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Part. Diff. Eq., 2 (1977), 193-222.

    [13]

    M. del Pino and R. Letelier, The influence of domain geometry in boundary blow-up ellipticproblems, Nonlinear Analysis, 48 (2002), 897-904.doi: 10.1016/S0362-546X(00)00222-4.

    [14]

    J. García-Melián, Boundary behavior for large solutions to elliptic equations with singular weights, Nonlinear Anal., 67 (2007), 818-826.doi: 10.1016/j.na.2006.06.041.

    [15]

    J. García-Melián, R. Letelier-Albornoz and J. Sabina de Lis, Uniqueness and asymptotic behaviour for solutions ofsemilinear problems with boundary blow-up, Proc. Amer. Math. Soc., 129 (2001), 3593-3602.doi: 10.1090/S0002-9939-01-06229-3.

    [16]

    D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977.

    [17]

    J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.doi: 10.1002/cpa.3160100402.

    [18]

    A. C. Lazer and P. J. McKenna, Asymptotic behaviour of solutions of boundary blowup problems, Differential and Integral Equations, 7 (1994), 1001-1019.

    [19]

    J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations, 195 (2003), 25-45.doi: 10.1016/j.jde.2003.06.003.

    [20]

    J. López-Gómez, Optimal uniqueness theorems and exactblow-up rates of large solutions, J. Differential Equations, 224 (2006), 385-439.doi: 10.1016/j.jde.2005.08.008.

    [21]

    A. Mohammed, Existence and asymptotic behavior of blow-up solutions to weightedquasilinear equations, J. Math. Anal. Appl., 298 (2004), 621-637.doi: 10.1016/j.jmaa.2004.05.030.

    [22]

    R. Osserman, On the inequality $\Delta u\ge f(u)$, Pacific J. Math., 7 (1957), 1641-1647.

    [23]

    Z. Zhang, The asymptotic behaviour of solutions with blow-upat the boundary for semilinear elliptic problems, J. Math. Anal. Appl., 308 (2005), 532-540.doi: 10.1016/j.jmaa.2004.11.029.

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