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]

Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283

[2]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179

[3]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527

[4]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991

[5]

Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141

[6]

Rongrong Yang, Zhongxue Lü. The properties of positive solutions to semilinear equations involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1073-1089. doi: 10.3934/cpaa.2019052

[7]

Lishan Lin. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1517-1531. doi: 10.3934/dcds.2019065

[8]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069

[9]

Antonio Capella. Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1645-1662. doi: 10.3934/cpaa.2011.10.1645

[10]

Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013

[11]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033

[12]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168

[13]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076

[14]

Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113

[15]

Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143

[16]

Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108

[17]

Nikolaos S. Papageorgiou, George Smyrlis. Positive solutions for parametric $p$-Laplacian equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1545-1570. doi: 10.3934/cpaa.2016002

[18]

Friedemann Brock, Leonelo Iturriaga, Justino Sánchez, Pedro Ubilla. Existence of positive solutions for $p$--Laplacian problems with weights. Communications on Pure & Applied Analysis, 2006, 5 (4) : 941-952. doi: 10.3934/cpaa.2006.5.941

[19]

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

[20]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]