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

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

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 $$ \begin{array}{lll} \ \ \ (-\Delta)^\alpha u+g(u)=0\quad & {\rm in}\quad \Omega,\\ \phantom{------\ } \ \ \ u=0\quad & {\rm in}\quad \mathbb{R}^N\setminus\bar\Omega,\\ \phantom{} \lim_{x\in\Omega,x\to\partial\Omega}u(x)=+\infty. \end{array} $$
Citation: Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.   Google Scholar

[3]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equaitons,, Comm. Pure Appl. Math., 62 (2009), 597.  doi: 10.1002/cpa.20274.  Google Scholar

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Z. Chen and R. Song, Estimates on Green functions and poisson kernels for symmetric stable process,, Math. Ann., 312 (1998), 465.  doi: 10.1007/s002080050232.  Google Scholar

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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.  Google Scholar

[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)., (2014).   Google Scholar

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R. Cignoli and M. Cottlar, An Introduction to Functional Analysis,, North-Holland, (1974).   Google Scholar

[8]

H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures,, J. Differential equations, 257 (2014), 1457.  doi: 10.1016/j.jde.2014.05.012.  Google Scholar

[9]

Y. Du and Z. Guo, Uniqueness and layer analysis for boundary blow-up solutions,, J. Math. Pures Appl., 83 (2004), 739.  doi: 10.1016/j.matpur.2004.01.006.  Google Scholar

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

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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.  doi: 10.3934/dcds.2007.19.271.  Google Scholar

[12]

P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators,, Advances in Mathematics, 226 (2011), 2712.  doi: 10.1016/j.aim.2010.09.023.  Google Scholar

[13]

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

[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.  doi: 10.1090/S0002-9939-01-06229-3.  Google Scholar

[15]

Z. Guo and F. Zhou, Exact multiplicity for boundary blow-up solutions,, J. Differential Equations, 228 (2006), 486.  doi: 10.1016/j.jde.2006.02.012.  Google Scholar

[16]

R. Osserman, On the inequality $\Delta u = f(u)$,, Pac. J. Math., 7 (1957), 1641.   Google Scholar

[17]

T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems,, J. Differential Equations, 146 (1998), 121.  doi: 10.1006/jdeq.1998.3414.  Google Scholar

[18]

M. Pertti, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511623813.  Google Scholar

[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, 14 (1997), 237.  doi: 10.1016/S0294-1449(97)80146-1.  Google Scholar

[20]

M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equation,, J. Evol. Equ., 3 (2003), 637.  doi: 10.1007/s00028-003-0122-y.  Google Scholar

show all references

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.   Google Scholar

[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.   Google Scholar

[3]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equaitons,, Comm. Pure Appl. Math., 62 (2009), 597.  doi: 10.1002/cpa.20274.  Google Scholar

[4]

Z. Chen and R. Song, Estimates on Green functions and poisson kernels for symmetric stable process,, Math. Ann., 312 (1998), 465.  doi: 10.1007/s002080050232.  Google Scholar

[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.  Google Scholar

[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)., (2014).   Google Scholar

[7]

R. Cignoli and M. Cottlar, An Introduction to Functional Analysis,, North-Holland, (1974).   Google Scholar

[8]

H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures,, J. Differential equations, 257 (2014), 1457.  doi: 10.1016/j.jde.2014.05.012.  Google Scholar

[9]

Y. Du and Z. Guo, Uniqueness and layer analysis for boundary blow-up solutions,, J. Math. Pures Appl., 83 (2004), 739.  doi: 10.1016/j.matpur.2004.01.006.  Google Scholar

[10]

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

[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.  doi: 10.3934/dcds.2007.19.271.  Google Scholar

[12]

P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators,, Advances in Mathematics, 226 (2011), 2712.  doi: 10.1016/j.aim.2010.09.023.  Google Scholar

[13]

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

[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.  doi: 10.1090/S0002-9939-01-06229-3.  Google Scholar

[15]

Z. Guo and F. Zhou, Exact multiplicity for boundary blow-up solutions,, J. Differential Equations, 228 (2006), 486.  doi: 10.1016/j.jde.2006.02.012.  Google Scholar

[16]

R. Osserman, On the inequality $\Delta u = f(u)$,, Pac. J. Math., 7 (1957), 1641.   Google Scholar

[17]

T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems,, J. Differential Equations, 146 (1998), 121.  doi: 10.1006/jdeq.1998.3414.  Google Scholar

[18]

M. Pertti, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511623813.  Google Scholar

[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, 14 (1997), 237.  doi: 10.1016/S0294-1449(97)80146-1.  Google Scholar

[20]

M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equation,, J. Evol. Equ., 3 (2003), 637.  doi: 10.1007/s00028-003-0122-y.  Google Scholar

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