2012, 32(11): 3801-3817. doi: 10.3934/dcds.2012.32.3801

Boundary estimates for solutions of weighted semilinear elliptic equations

1. 

Dipartimento di Matematica e Informatica, Universitá degli studi di Cagliari, 09124, Cagliari, Italy, Italy

Received  January 2011 Revised  September 2011 Published  June 2012

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 $\partial\Omega$.
Citation: Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801
References:
[1]

C. Anedda, Second-order boundary estimates for solutions to singular elliptic equations,, Electronic Journal of Differential Equations, 2009 ().

[2]

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

[3]

C. Anedda and G. Porru, Second order estimates forboundary blow-up solutions of elliptic equations,, Discrete Contin. Dyn. Syst., (2007), 54.

[4]

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

[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. 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.

[9]

L. Bieberback, $\Delta u=e^u$ und die automorphen Functionen,, Mat. Ann., 77 (1916), 173. 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. 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.

[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.

[13]

M. del Pino and R. Letelier, The influence of domain geometry in boundary blow-up ellipticproblems,, Nonlinear Analysis, 48 (2002), 897. 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. 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. 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, (1977).

[17]

J. B. Keller, On solutions of $\Delta u=f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503. 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.

[19]

J. López-Gómez, The boundary blow-up rate of large solutions,, J. Differential Equations, 195 (2003), 25. 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. 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. 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.

[23]

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

show all references

References:
[1]

C. Anedda, Second-order boundary estimates for solutions to singular elliptic equations,, Electronic Journal of Differential Equations, 2009 ().

[2]

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

[3]

C. Anedda and G. Porru, Second order estimates forboundary blow-up solutions of elliptic equations,, Discrete Contin. Dyn. Syst., (2007), 54.

[4]

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

[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. 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.

[9]

L. Bieberback, $\Delta u=e^u$ und die automorphen Functionen,, Mat. Ann., 77 (1916), 173. 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. 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.

[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.

[13]

M. del Pino and R. Letelier, The influence of domain geometry in boundary blow-up ellipticproblems,, Nonlinear Analysis, 48 (2002), 897. 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. 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. 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, (1977).

[17]

J. B. Keller, On solutions of $\Delta u=f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503. 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.

[19]

J. López-Gómez, The boundary blow-up rate of large solutions,, J. Differential Equations, 195 (2003), 25. 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. 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. 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.

[23]

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

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