# American Institute of Mathematical Sciences

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 $$\label{0.1} \begin{array}{lll} (-\Delta)^\alpha u+g(u)=k\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha}\quad & {\rm in}\quad \bar\Omega, (1)\\ \phantom{ (-\Delta)^\alpha +g(u)} u=0\quad & {\rm in}\quad \bar\Omega^c \end{array}$$ 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 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, Analyse Non Linéaire. 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.

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), 155-171. [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, Analyse Non Linéaire. 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.
 [1] Nicola Abatangelo. Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5555-5607. doi: 10.3934/dcds.2015.35.5555 [2] Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267 [3] Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022106 [4] Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271 [5] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [6] Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671 [7] Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147 [8] Yihong Du, Zongming Guo. The degenerate logistic model and a singularly mixed boundary blow-up problem. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 1-29. doi: 10.3934/dcds.2006.14.1 [9] 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 [10] Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 [11] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control and Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 [12] Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225 [13] Tetsuya Ishiwata, Shigetoshi Yazaki. A fast blow-up solution and degenerate pinching arising in an anisotropic crystalline motion. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2069-2090. doi: 10.3934/dcds.2014.34.2069 [14] Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809 [15] Juntang Ding, Chenyu Dong. Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4173-4183. doi: 10.3934/dcdsb.2021222 [16] Qiang Lin, Xueteng Tian, Runzhang Xu, Meina Zhang. Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 2095-2107. doi: 10.3934/dcdss.2020160 [17] Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure and Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034 [18] Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085 [19] Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, 2021, 29 (6) : 3581-3607. doi: 10.3934/era.2021052 [20] Mingqi Xiang, Die Hu. Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4609-4629. doi: 10.3934/dcdss.2021125

2021 Impact Factor: 1.588