# 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 & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
##### References:
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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. 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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