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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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

show all references

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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