# American Institute of Mathematical Sciences

November  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, Universitá 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, 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 ().   Google Scholar [2] C. Anedda, F. Cuccu and G. Porru, Boundary estimates for solutions to singular elliptic equations, Matematiche (Catania), 60 (2005), 339-352.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] S. Berhanu and G. Porru, Qualitative and quantitative estimates for large solutions tosemilinear equations, Communications in Applied Analysis, 4 (2000), 121-131.  Google Scholar [9] L. Bieberback, $\Delta u=e^u$ und die automorphen Functionen, Mat. Ann., 77 (1916), 173-212. doi: 10.1007/BF01456901.  Google Scholar [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.  Google Scholar [11] F.-C. Cirstea and V. Rădulescu, Nonlinear problems with boundaryblow-up: A Karamata regular variation approach, Asymptotic Analysis, 46 (2006), 275-298.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.  Google Scholar [18] A. C. Lazer and P. J. McKenna, Asymptotic behaviour of solutions of boundary blowup problems, Differential and Integral Equations, 7 (1994), 1001-1019.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] R. Osserman, On the inequality $\Delta u\ge f(u)$, Pacific J. Math., 7 (1957), 1641-1647.  Google Scholar [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.  Google Scholar

show all references

##### References:
 [1] C. Anedda, Second-order boundary estimates for solutions to singular elliptic equations,, Electronic Journal of Differential Equations, 2009 ().   Google Scholar [2] C. Anedda, F. Cuccu and G. Porru, Boundary estimates for solutions to singular elliptic equations, Matematiche (Catania), 60 (2005), 339-352.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] S. Berhanu and G. Porru, Qualitative and quantitative estimates for large solutions tosemilinear equations, Communications in Applied Analysis, 4 (2000), 121-131.  Google Scholar [9] L. Bieberback, $\Delta u=e^u$ und die automorphen Functionen, Mat. Ann., 77 (1916), 173-212. doi: 10.1007/BF01456901.  Google Scholar [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.  Google Scholar [11] F.-C. Cirstea and V. Rădulescu, Nonlinear problems with boundaryblow-up: A Karamata regular variation approach, Asymptotic Analysis, 46 (2006), 275-298.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.  Google Scholar [18] A. C. Lazer and P. J. McKenna, Asymptotic behaviour of solutions of boundary blowup problems, Differential and Integral Equations, 7 (1994), 1001-1019.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] R. Osserman, On the inequality $\Delta u\ge f(u)$, Pacific J. Math., 7 (1957), 1641-1647.  Google Scholar [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.  Google Scholar
 [1] Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54 [2] Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881 [3] Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023 [4] Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25. [5] Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828 [6] Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771 [7] Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733 [8] Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure & Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465 [9] Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182 [10] Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621 [11] Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661 [12] Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure & Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043 [13] Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677 [14] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [15] Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601 [16] Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521 [17] Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377 [18] Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126 [19] Jens Lorenz, Wilberclay G. Melo, Natã Firmino Rocha. The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3819-3841. doi: 10.3934/dcdsb.2018332 [20] Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027

2020 Impact Factor: 1.392