# American Institute of Mathematical Sciences

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. [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. [3] J. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", Pure and Applied Mathematics, (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. 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. 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. 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. 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, 74 (2011), 1555. 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. 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. doi: 10.1016/j.aim.2010.01.025. [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. [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. [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. [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. [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. [16] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent,, Ann. Inst. H. Poincare Anal. Non Linearire, 9 (1992), 281. [17] V. Varlamov, Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball,, Studia Math., 142 (2000), 71. [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. [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. [20] L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: a tutorial,, In, (2008). [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.

show all references

##### References:
 [1] D. Applebaum, Lévy processes - from probability to finance and quantum groups,, Notices Amer. Math. Soc., 51 (2004), 1336. [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. [3] J. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", Pure and Applied Mathematics, (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. 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. 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. 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. 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, 74 (2011), 1555. 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. 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. doi: 10.1016/j.aim.2010.01.025. [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. [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. [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. [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. [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. [16] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent,, Ann. Inst. H. Poincare Anal. Non Linearire, 9 (1992), 281. [17] V. Varlamov, Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball,, Studia Math., 142 (2000), 71. [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. [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. [20] L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: a tutorial,, In, (2008). [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.
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] 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 [10] De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431 [11] Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393 [12] Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154 [13] 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 [14] 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 [15] Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921 [16] Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795 [17] 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 [18] 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 [19] Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 [20] Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

2016 Impact Factor: 0.801