# Existence and uniqueness of weak positive solution for essential singular elliptic problem involving the square root of the Laplacian

Volume 11, Issue 10, pp 1149--1160
Publication Date: July 13, 2018 Submission Date: April 15, 2018 Revision Date: June 15, 2018 Accteptance Date: June 19, 2018
• 1817 Views

### Authors

Xing Wang - School of Science, Xi'an University of Technology, Xi'an, Shaanxi 710054, P. R. China. Li Zhang - School of Science, Chang'an University, Xi'an, Shaanxi 710064, P. R. China.

### Abstract

In this paper we consider the existence and uniqueness of weak positive solution for nonlocal equations of the square root of the Laplacian with singular nonlinearity. The remarkable feature of this paper is the fact that the natural associated functional fails to be Frechet differentiable, critical point theory could not be applied to obtain the existence of weak positive solution. We first establish the priori estimate of weak solution of approximating problems. Then the weak positive solution is constructed by combining sub-and supersolutions method and truncate technology.

### Share and Cite

##### ISRP Style

Xing Wang, Li Zhang, Existence and uniqueness of weak positive solution for essential singular elliptic problem involving the square root of the Laplacian, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 10, 1149--1160

##### AMA Style

Wang Xing, Zhang Li, Existence and uniqueness of weak positive solution for essential singular elliptic problem involving the square root of the Laplacian. J. Nonlinear Sci. Appl. (2018); 11(10):1149--1160

##### Chicago/Turabian Style

Wang, Xing, Zhang, Li. "Existence and uniqueness of weak positive solution for essential singular elliptic problem involving the square root of the Laplacian." Journal of Nonlinear Sciences and Applications, 11, no. 10 (2018): 1149--1160

### Keywords

• Fractional Laplacian
• essential singular nonlinearity
• nondifferentiable functional
• a priori estimate

•  35J75
•  47J05
•  35J25

### References

• [1] D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336–1347.

• [2] B. Barrios, I. De Bonis, M. Medina, I. Peral, Semilinear problems for the fractional Laplacian with a singular nonlinearity, Open Math., 13 (2015), 390–407.

• [3] X. Cabré, J. G. Tan, Positive solutions for nonlinear problems involving the square root of the Laplacian, Adv. Math, 224 (2010), 2052–2093.

• [4] L. Caffarelli, L. Silvestre , An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245–1260.

• [5] M. G. Crandall, P. H. Rabinowitz, L. Tatar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1997), 193–222.

• [6] Y. Q. Fang, Existence, Uniqueness of positive solution to a fractional Laplacians with singular nonlinearity, Analysis of PDEs, 2014 (2014), 11 pages.

• [7] A. Garroni, S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36 (2005), 1943–1964

• [8] N. Hirano, C. Saccon, N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations, 9 (2004), 197–220.

• [9] J. L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, Travaux et Recherches Mathématiques, Paris (1968)

• [10] G. Molica Bisci, V. D. Radulescu, R. Servadei , Variational methods for nonlocal fractional problems, Cambridge University Press, Cambridge (2016)

• [11] T. Mukherjee, K. Sreenadh, Critical growth fractional elliptic problem with singular nonlinearities, Electron. J. Diff. Equ., 2016 (2016), 23 pages.

• [12] M. Struwe, Variational Methods, Springer-Verlag, Berlin (1990)

• [13] C. A. Stuart, Existence and approximation of solutions of non-linear elliptic equations, Math. Z., 147 (1976), 53–63.

• [14] Y. J. Sun, S. J. Li, A nonlinear elliptic equation with critical-exponent: estimates for extremal values , Nonlinear Anal., 69 (2008), 1856–1869.

• [15] Y. J. Sun, S. P. Wu, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 260 (2011), 1257–1284.

• [16] Y. J. Sun, S. P. Wu, Y. M. Long, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations, 176 (2011), 511–531.

• [17] X. Wang, P. H. Zhao, L. Zhang, The existence and multiplicity of classical positive solutions for a singular nonlinear elliptic problem with any growth exponents, Nonlinear Anal., 101 (2014), 37–46.

• [18] X. Wang, L. Zhao, P. H. Zhao, Combined effects of singular and critical nonlinearities in elliptic problems, Nonlinear Anal., 87 (2013), 1–10.

• [19] A. L. Xia, J. F. Yang, Regularity of nonlinear equations for fractional Laplacian, Proc. Amer. Math. Soc., 141 (2013), 2665–2672.

• [20] Z. T. Zhang, Critical points and positive solutions of singular elliptic boundary value problems, J. Math. Anal. Appl., 302 (2005), 476–483.