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
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

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

[1]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[2]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[3]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[4]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[5]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[6]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

[7]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[8]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[9]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73

[10]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053

[11]

Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391

[12]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[13]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113

[14]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035

[15]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[16]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475

[17]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[18]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617

[19]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[20]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]