Large solutions to elliptic equations involving fractional Laplacian

Chen, Huyuan; Felmer, Patricio; Quaas, Alexander

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 6, pp. 1199-1228.

The purpose of this paper is to study boundary blow up solutions for semi-linear fractional elliptic equations of the form

$$\begin{array}{cc}\{\begin{array}{cc}{(-\Delta )}^{\alpha }u\left(x\right)+{\left|u\right|}^{p-1}u\left(x\right)=f\left(x\right),\hfill & x\in \Omega ,\hfill \\ u\left(x\right)=0,\hfill & x\in {\overline{\Omega }}^c,\hfill \\ \underset{x\in \Omega ,x\to \partial \Omega }{\lim }u\left(x\right)=+\infty ,\hfill \end{array}& \text{(0.1)}\end{array}$$

where $p>1$, Ω is an open bounded $C^2$ domain of ${ℝ}^N$, $N\ge 2$, the operator ${(-\Delta )}^{\alpha }$ with $\alpha \in (0,1)$ is the fractional Laplacian and $f:\Omega \to ℝ$ is a continuous function which satisfies some appropriate conditions. We obtain that problem (0.1) admits a solution with boundary behavior like $d{\left(x\right)}^{-\frac{2\alpha }{p-1}}$, when $1+2\alpha <p<1-\frac{2\alpha }{{\tau }_0\left(\alpha \right)}$, for some ${\tau }_0\left(\alpha \right)\in (-1,0)$, and has infinitely many solutions with boundary behavior like $d{\left(x\right)}^{{\tau }_0\left(\alpha \right)}$, when $\max \{1-\frac{2\alpha }{{\tau }_0}+\frac{{\tau }_0\left(\alpha \right)+1}{{\tau }_0},1\}<p<1-\frac{2\alpha }{{\tau }_0}$. Moreover, we also obtained some uniqueness and non-existence results for problem (0.1).

[1] N. Abatangelo, Large s-harmonic functions and boundary Blow-up solutions for the fractional Laplacian, arXiv:1310.3193 [math.AP] | MR | Zbl

[2] J.M. Arrieta, A. Rodríguez-Bernal, Localization on the boundary of blow-up for reaction–diffusion equations with nonlinear boundary conditions, Commun. Partial Differ. Equ. 29 (2004), 1127 -1148 | MR | Zbl

[3] C. Bandle, M. Marcus, Large solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour, J. Anal. Math. 58 (1992), 9 -24 | MR | Zbl

[4] C. Bandle, M. Marcus, Dependence of blowup rate of large solutions of semilinear elliptic equations on the curvature of the boundary, Complex Var. Theory Appl. 49 (2004), 555 -570 | MR | Zbl

[5] X. Cabré, L. Caffarelli, Fully Nonlinear Elliptic Equation, Colloq. Publ. – Am. Math. Soc. vol. 43 (1995) | MR

[6] L. Caffarelli, L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Commun. Pure Appl. Math. 62 no. 5 (2009), 597 -638 | MR | Zbl

[7] L. Caffarelli, L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal. 200 no. 1 (2011), 59 -88 | MR | Zbl

[8] L. Caffarelli, L. Silvestre, The Evans–Krylov theorem for nonlocal fully non linear equations, Ann. Math. 174 no. 2 (2011), 1163 -1187 | MR | Zbl

[9] G. Barles, C. Imbert, Second-order elliptic integro-differential equations: viscosity solutions theory revisited, Ann. Henri Poincaré 25 (2008), 567 -585 | EuDML | Numdam | MR | Zbl

[10] H. Chen, P. Felmer, Liouville Property for fully nonlinear integral equation in exterior domain, preprint.

[11] M. Chuaqui, C. Cortázar, M. Elgueta, J. García-Melián, Uniqueness and boundary behaviour of large solutions to elliptic problems with singular weights, Commun. Pure Appl. Anal. 3 (2004), 653 -662 | MR | Zbl

[12] M. Del Pino, R. Letelier, The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear Anal. 48 no. 6 (2002), 897 -904 | MR | Zbl

[13] G. Díaz, R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonlinear Anal. 20 no. 2 (1993), 97 -125 | MR | Zbl

[14] Y. Du, Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal. 31 (1999), 1 -18 | EuDML | MR | Zbl

[15] P. Felmer, A. Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal. 78 no. 3 (2012), 123 -144 | MR | Zbl

[16] P. Felmer, A. Quaas, J. Tan, Positive solutions of nonlinear Schrodinger equation with the fractional Laplacian, Proc. R. Soc. Edinb., Sect. A, Math. 142 (2012), 1 -26 | MR

[17] P. Felmer, A. Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Am. Math. Soc. 361 no. 11 (2009), 5721 -5736 | MR | Zbl

[18] P. Felmer, A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math. 226 (2011), 2712 -2738 | MR | Zbl

[19] P. Felmer, A. Quaas, Fundamental solutions for a class of Isaacs integral operators, Discrete Contin. Dyn. Syst. 30 no. 2 (2011), 493 -508 | MR | Zbl

[20] J. García-Melián, Nondegeneracy and uniqueness for boundary blow-up elliptic problems, J. Differ. Equ. 223 no. 1 (2006), 208 -227 | MR | Zbl

[21] J. García-Meliían, R. Gómez-Reñasco, J. López-Gómez, J. Sabina De Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Ration. Mech. Anal. 145 no. 3 (1998), 261 -289 | MR | Zbl

[22] J.B. Keller, On solutions of $\Delta u=f\left(u\right)$ , Commun. Pure Appl. Math. 10 (1957), 503 -510 | MR | Zbl

[23] S. Kim, A note on boundary blow-up problem of $\Delta u=u^p$, IMA preprint, No. 1872, 2002.

[24] C. Loewner, L. Nirenberg, Partial differential equations invariant under conformal projective transformations, Contributions to Analysis, Academic Press, New York (1974), 245 -272 | MR

[25] M. Marcus, L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equation, J. Evol. Equ. 3 (2003), 637 -652 | MR | Zbl

[26] M. Marcus, L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. Henri Poincaré 14 no. 2 (1997), 237 -274 | EuDML | Numdam | MR | Zbl

[27] R. Osserman, On the inequality $\Delta u=f\left(u\right)$ , Pac. J. Math. 7 (1957), 1641 -1647 | MR | Zbl

[28] G. Palatucci, O. Savin, E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl. (4) 192 (2013), 673 -718 | MR | Zbl

[29] V. Rǎdulescu, Singular phenomena in nonlinear elliptic problems: from blow-up boundary solutions to equations with singular nonlinearities, Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 4 (2007), 485 -593 | MR | Zbl

[30] X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional Laplacian, regularity up to the boundary, J. Math. Pures Appl. 101 no. 3 (2014), 275 -302 | MR | Zbl

[31] R. Servadei, E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat. 58 no. 1 (2014), 133 -154 | MR | Zbl

[32] L. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace, Indiana Univ. Math. J. 55 (2006), 1155 -1174 | MR | Zbl

[33] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press (1970) | MR | Zbl