# American Institute of Mathematical Sciences

March  2014, 13(2): 567-584. doi: 10.3934/cpaa.2014.13.567

## Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent

 1 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China 2 Jiangsu Key Laboratory for NSLSCS, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu 3 School of Science, Jiangnan University, Wuxi, 214122, China

Received  October 2012 Revised  June 2013 Published  October 2013

In this paper, we study the following problem \begin{eqnarray} (-\Delta)^{\frac{\alpha}{2}}u = K(x)|u|^{2_{\alpha}^{*}-2}u + f(x) \quad in \ \Omega,\\ u=0 \quad on \ \partial \Omega, \end{eqnarray} where $\Omega\subset R^N$ is a smooth bounded domain, $0< \alpha < 2$, $N>\alpha$, $2_{\alpha}^{*}= \frac{2N}{N-\alpha}$, $f\in H^{-\frac{\alpha}{2}}(\Omega)$ and $K(x)\in L^\infty(\Omega)$. Under appropriate assumptions on $K$ and $f$, we prove that this problem has at least two positive solutions. When $\alpha = 1$, we also establish a nonexistence result for a positive solution in a class of linear positive-type domains are more general than star-shaped ones.
Citation: Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567
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##### References:
 [1] D. Applebaum, Lévy processes - from probability to finance and quantum groups,, Notices Amer. Math. Soc., 51 (2004), 1336.   Google Scholar [2] T. An, Non-existence of positive solutions of some elliptic equations in positive-type domains,, Appl. Math. Letters., 20 (2007), 681.  doi: 10.1016/j.aml.2006.07.008.  Google Scholar [3] J. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", Pure and Applied Mathematics, (1984).   Google Scholar [4] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Am. Math. Soc., 88 (1983), 486.  doi: 10.2307/2044999.  Google Scholar [5] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar [6] C. Brändle, E. Colorado, A. de pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Lplacian,, Proc. Roy. Soc. Edinburgh. Sect. A, 143 (2013), 39.  doi: 10.1017/S0308210511000175.  Google Scholar [7] B. Barrios, E. Colorado, A. de pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator,, J. Differential Equations, 252 (2012), 6133.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar [8] R. Chemmam, H. Maagli and S. Masmoudi, On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains,, Non. Analysis: Theory, 74 (2011), 1555.  doi: 10.1016/j.na.2010.10.027.  Google Scholar [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar [10] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar [11] D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\R^N$,, Proc. Roy. Soc. Edinburgh. Sect. A, 126 (1996), 443.  doi: 10.1017/S0308210500022836.  Google Scholar [12] P. Drábek and Y. Huang, Multiplicity of positive solutions for some quasilinear elliptic equation in $\R^N$ with critical sobolev exponent,, J. Differential Equations, 140 (1997), 106.  doi: 10.1006/jdeq.1997.3306.  Google Scholar [13] I. Kim and K. Kim, A generalization of the Littlewood-Paley inequality for the fractional Laplacian,, J. Math. Anal. Appl, 388 (2012), 175.  doi: 10.1016/j.jmaa.2011.11.031.  Google Scholar [14] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math, 60 (2007), 67.  doi: 10.1002/cpa.20153.  Google Scholar [15] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. var. Partial Differential Equations, 42 (2011), 21.  doi: 10.1007/s00526-010-0378-3.  Google Scholar [16] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent,, Ann. Inst. H. Poincare Anal. Non Linearire, 9 (1992), 281.   Google Scholar [17] V. Varlamov, Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball,, Studia Math., 142 (2000), 71.   Google Scholar [18] X. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation,, J. Differential Equations, 92 (1991), 163.  doi: 10.1016/0022-0396(91)90045-B.  Google Scholar [19] X. Yu, The Nehari manifold for elliptic equation involving the square root of the Laplacian,, J. Differential Equations, 252 (2012), 145.  doi: 10.1016/j.jde.2011.09.015.  Google Scholar [20] L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: a tutorial,, In, (2008).   Google Scholar [21] Z. Wang and H. Zhou, Positive solutions for a nonhomogeneous elliptic equation on $\R^N$ without (AR) condition,, J. Math. Anal. Appl., 353 (2009), 470.  doi: 10.1016/j.jmaa.2008.11.080.  Google Scholar
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