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 and Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567
References:
[1]

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

[2]

T. An, Non-existence of positive solutions of some elliptic equations in positive-type domains, Appl. Math. Letters., 20 (2007), 681-685. doi: 10.1016/j.aml.2006.07.008.

[3]

J. Aubin and I. Ekeland, "Applied Nonlinear Analysis," Pure and Applied Mathematics, Wiley Interscience Publications, 1984.

[4]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.

[5]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[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), 39-71. doi: 10.1017/S0308210511000175.

[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), 6133-6162. doi: 10.1016/j.jde.2012.02.023.

[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, Methods Applications, 74 (2011), 1555-1576. doi: 10.1016/j.na.2010.10.027.

[9]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[10]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.

[11]

D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb{R}^{N}$, Proc. Roy. Soc. Edinburgh. Sect. A, 126 (1996), 443-463. doi: 10.1017/S0308210500022836.

[12]

P. Drábek and Y. Huang, Multiplicity of positive solutions for some quasilinear elliptic equation in $\mathbb{R}^{N}$ with critical sobolev exponent, J. Differential Equations, 140 (1997), 106-132. doi: 10.1006/jdeq.1997.3306.

[13]

I. Kim and K. Kim, A generalization of the Littlewood-Paley inequality for the fractional Laplacian, J. Math. Anal. Appl, 388 (2012), 175-190. doi: 10.1016/j.jmaa.2011.11.031.

[14]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math, 60 (2007), 67-112. doi: 10.1002/cpa.20153.

[15]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. var. Partial Differential Equations, 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.

[16]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincare Anal. Non Linearire, 9 (1992), 281-304.

[17]

V. Varlamov, Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball, Studia Math., 142 (2000), 71-99.

[18]

X. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differential Equations, 92 (1991), 163-178. doi: 10.1016/0022-0396(91)90045-B.

[19]

X. Yu, The Nehari manifold for elliptic equation involving the square root of the Laplacian, J. Differential Equations, 252 (2012), 145-190 doi: 10.1016/j.jde.2011.09.015.

[20]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: a tutorial, In "Order and Chaos," 10th volume, T. Bountis, Patras University Press, 2008.

[21]

Z. Wang and H. Zhou, Positive solutions for a nonhomogeneous elliptic equation on $\mathbb{R}^{N}$ without (AR) condition, J. Math. Anal. Appl., 353 (2009), 470-479. doi: 10.1016/j.jmaa.2008.11.080.

show all references

References:
[1]

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

[2]

T. An, Non-existence of positive solutions of some elliptic equations in positive-type domains, Appl. Math. Letters., 20 (2007), 681-685. doi: 10.1016/j.aml.2006.07.008.

[3]

J. Aubin and I. Ekeland, "Applied Nonlinear Analysis," Pure and Applied Mathematics, Wiley Interscience Publications, 1984.

[4]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.

[5]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[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), 39-71. doi: 10.1017/S0308210511000175.

[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), 6133-6162. doi: 10.1016/j.jde.2012.02.023.

[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, Methods Applications, 74 (2011), 1555-1576. doi: 10.1016/j.na.2010.10.027.

[9]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[10]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.

[11]

D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb{R}^{N}$, Proc. Roy. Soc. Edinburgh. Sect. A, 126 (1996), 443-463. doi: 10.1017/S0308210500022836.

[12]

P. Drábek and Y. Huang, Multiplicity of positive solutions for some quasilinear elliptic equation in $\mathbb{R}^{N}$ with critical sobolev exponent, J. Differential Equations, 140 (1997), 106-132. doi: 10.1006/jdeq.1997.3306.

[13]

I. Kim and K. Kim, A generalization of the Littlewood-Paley inequality for the fractional Laplacian, J. Math. Anal. Appl, 388 (2012), 175-190. doi: 10.1016/j.jmaa.2011.11.031.

[14]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math, 60 (2007), 67-112. doi: 10.1002/cpa.20153.

[15]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. var. Partial Differential Equations, 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.

[16]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincare Anal. Non Linearire, 9 (1992), 281-304.

[17]

V. Varlamov, Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball, Studia Math., 142 (2000), 71-99.

[18]

X. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differential Equations, 92 (1991), 163-178. doi: 10.1016/0022-0396(91)90045-B.

[19]

X. Yu, The Nehari manifold for elliptic equation involving the square root of the Laplacian, J. Differential Equations, 252 (2012), 145-190 doi: 10.1016/j.jde.2011.09.015.

[20]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: a tutorial, In "Order and Chaos," 10th volume, T. Bountis, Patras University Press, 2008.

[21]

Z. Wang and H. Zhou, Positive solutions for a nonhomogeneous elliptic equation on $\mathbb{R}^{N}$ without (AR) condition, J. Math. Anal. Appl., 353 (2009), 470-479. doi: 10.1016/j.jmaa.2008.11.080.

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