Boundary blow-up solutions to fractional elliptic equations in a measure framework

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April 2016, 36(4): 1881-1903. doi: 10.3934/dcds.2016.36.1881

Boundary blow-up solutions to fractional elliptic equations in a measure framework

Huyuan Chen ^1, , Hichem Hajaiej ^2, and Ying Wang ^1,

1.	Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China, China
2.	Department of Mathematics, King Saud University, P.O. Box 2455, 11451 Riyadh

Received January 2015 Revised July 2015 Published September 2015

Abstract
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Let $\alpha\in(0,1)$, $\Omega$ be a bounded open domain in $\mathbb{R}^N$ ($N\ge 2$) with $C^2$ boundary $\partial\Omega$ and $\omega$ be the Hausdorff measure on $\partial\Omega$. We denote by $\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha}$ a measure $$\langle\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha},f\rangle=\int_{\partial\Omega}\frac{\partial^\alpha f(x)}{\partial \vec{n}_x^\alpha} d\omega(x),\quad f\in C^1(\bar\Omega),$$ where $\vec{n}_x$ is the unit outward normal vector at point $x\in\partial\Omega$. In this paper, we prove that problem \begin{equation}\label{0.1} \begin{array}{lll} (-\Delta)^\alpha u+g(u)=k\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha}\quad & {\rm in}\quad \bar\Omega, (1)\\ \phantom{ (-\Delta)^\alpha +g(u)} u=0\quad & {\rm in}\quad \bar\Omega^c \end{array} \end{equation} admits a unique weak solution $u_k$ under the hypotheses that $k>0$, $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$ and $g$ is a nondecreasing function satisfying extra conditions. We prove that the weak solution of (1) is a classical solution of $$ \begin{array}{lll} \ \ \ (-\Delta)^\alpha u+g(u)=0\quad & {\rm in}\quad \Omega,\\ \phantom{------\ } \ \ \ u=0\quad & {\rm in}\quad \mathbb{R}^N\setminus\bar\Omega,\\ \phantom{} \lim_{x\in\Omega,x\to\partial\Omega}u(x)=+\infty. \end{array} $$

Keywords: boundary blow-up solution., Fractional Laplacian, Green kernel.

Mathematics Subject Classification: 35R11, 35J61, 35R0.

Citation: Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881

References:

[1]	C. Bandle and M. Marcus, Asymptotic behaviour of solutions and derivatives for semilinear elliptic problems with blow-up on the boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 155-171.
[2]	Ph. Bénilan, H. Brezis and M. Crandall, A semilinear elliptic equation in $L^1(\mathbbR^N )$, Ann. Sc. Norm. Sup. Pisa Cl. Sci., 2 (1975), 523-555.
[3]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equaitons, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274.
[4]	Z. Chen and R. Song, Estimates on Green functions and poisson kernels for symmetric stable process, Math. Ann., 312 (1998), 465-501. doi: 10.1007/s002080050232.
[5]	H. Chen, P. Felmer and A. Quaas, Large solution to elliptic equations involving fractional Laplacian,, accepted by Ann. Inst. H. Poincaré, (). doi: 10.1016/j.anihpc.2014.08.001.
[6]	H. Chen and H. Hajaiej, Existence, Non-existence, Uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary, arXiv:1410.2672 (2014).
[7]	R. Cignoli and M. Cottlar, An Introduction to Functional Analysis, North-Holland, Amsterdam, 1974.
[8]	H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures, J. Differential equations, 257 (2014), 1457-1486. doi: 10.1016/j.jde.2014.05.012.
[9]	Y. Du and Z. Guo, Uniqueness and layer analysis for boundary blow-up solutions, J. Math. Pures Appl., 83 (2004), 739-763. doi: 10.1016/j.matpur.2004.01.006.
[10]	M. del Pino and R. Letelier, The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear Analysis: Theory, Methods & Applications, 48 (2002), 897-904. doi: 10.1016/S0362-546X(00)00222-4.
[11]	Y. Du, Z. Guo and F. Zhou, Boundary blow-up solutions with interior layers and spikes in a bistable problem, Discrete Contin. Dyn. Syst., 19 (2007), 271-298. doi: 10.3934/dcds.2007.19.271.
[12]	P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Advances in Mathematics, 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023.
[13]	J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.
[14]	J. Garcia-Melián, R. Letelier and J. de Lis, Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc., 129 (2001), 3593-3602. doi: 10.1090/S0002-9939-01-06229-3.
[15]	Z. Guo and F. Zhou, Exact multiplicity for boundary blow-up solutions, J. Differential Equations, 228 (2006), 486-506. doi: 10.1016/j.jde.2006.02.012.
[16]	R. Osserman, On the inequality $\Delta u = f(u)$, Pac. J. Math., 7 (1957), 1641-1647.
[17]	T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems, J. Differential Equations, 146 (1998), 121-156. doi: 10.1006/jdeq.1998.3414.
[18]	M. Pertti, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge University Press, 1995. doi: 10.1017/CBO9780511623813.
[19]	M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 14 (1997), 237-274. doi: 10.1016/S0294-1449(97)80146-1.
[20]	M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equation, J. Evol. Equ., 3 (2003), 637-652. doi: 10.1007/s00028-003-0122-y.

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References:

[1]	C. Bandle and M. Marcus, Asymptotic behaviour of solutions and derivatives for semilinear elliptic problems with blow-up on the boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 155-171.
[2]	Ph. Bénilan, H. Brezis and M. Crandall, A semilinear elliptic equation in $L^1(\mathbbR^N )$, Ann. Sc. Norm. Sup. Pisa Cl. Sci., 2 (1975), 523-555.
[3]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equaitons, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274.
[4]	Z. Chen and R. Song, Estimates on Green functions and poisson kernels for symmetric stable process, Math. Ann., 312 (1998), 465-501. doi: 10.1007/s002080050232.
[5]	H. Chen, P. Felmer and A. Quaas, Large solution to elliptic equations involving fractional Laplacian,, accepted by Ann. Inst. H. Poincaré, (). doi: 10.1016/j.anihpc.2014.08.001.
[6]	H. Chen and H. Hajaiej, Existence, Non-existence, Uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary, arXiv:1410.2672 (2014).
[7]	R. Cignoli and M. Cottlar, An Introduction to Functional Analysis, North-Holland, Amsterdam, 1974.
[8]	H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures, J. Differential equations, 257 (2014), 1457-1486. doi: 10.1016/j.jde.2014.05.012.
[9]	Y. Du and Z. Guo, Uniqueness and layer analysis for boundary blow-up solutions, J. Math. Pures Appl., 83 (2004), 739-763. doi: 10.1016/j.matpur.2004.01.006.
[10]	M. del Pino and R. Letelier, The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear Analysis: Theory, Methods & Applications, 48 (2002), 897-904. doi: 10.1016/S0362-546X(00)00222-4.
[11]	Y. Du, Z. Guo and F. Zhou, Boundary blow-up solutions with interior layers and spikes in a bistable problem, Discrete Contin. Dyn. Syst., 19 (2007), 271-298. doi: 10.3934/dcds.2007.19.271.
[12]	P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Advances in Mathematics, 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023.
[13]	J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.
[14]	J. Garcia-Melián, R. Letelier and J. de Lis, Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc., 129 (2001), 3593-3602. doi: 10.1090/S0002-9939-01-06229-3.
[15]	Z. Guo and F. Zhou, Exact multiplicity for boundary blow-up solutions, J. Differential Equations, 228 (2006), 486-506. doi: 10.1016/j.jde.2006.02.012.
[16]	R. Osserman, On the inequality $\Delta u = f(u)$, Pac. J. Math., 7 (1957), 1641-1647.
[17]	T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems, J. Differential Equations, 146 (1998), 121-156. doi: 10.1006/jdeq.1998.3414.
[18]	M. Pertti, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge University Press, 1995. doi: 10.1017/CBO9780511623813.
[19]	M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 14 (1997), 237-274. doi: 10.1016/S0294-1449(97)80146-1.
[20]	M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equation, J. Evol. Equ., 3 (2003), 637-652. doi: 10.1007/s00028-003-0122-y.

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