# American Institute of Mathematical Sciences

November  2018, 17(6): 2261-2281. doi: 10.3934/cpaa.2018108

## The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities

 School of Mathematic and Computer Science, Wuhan Polytechnic University, Wuhan, 430023, China

* Corresponding author

Received  May 2017 Revised  December 2017 Published  June 2018

Fund Project: The first author is supported by NSF grant No.11701439.

We study the combined effect of concave and convex nonlinearities on the numbers of positive solutions for a fractional equation involving critical Sobolev exponents. In this paper, we concerned with the following fractional equation
 $\left\{ \begin{array}{l}{\left( { - \Delta } \right)^s}u = \lambda f\left( x \right){\left| u \right|^{q - 2}}u + g\left( x \right){\left| u \right|^{2_s^* - 2}}u,\;\;\;x \in \Omega ,\\u = 0,\;\;x \in \partial \Omega ,\end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$
where
 $0 , $ λ>0 $, $ 1≤q <2$, $ 2_s^* = \frac{2N}{N-2s} $, $ 0∈ Ω\subset \mathbb{R} ^N(N>4s) $is a bounded domain with smooth boundary $ \partialΩ $, and $ f,\,g $are nonnegative continuous functions on $\bar{Ω} $. Here $ (-Δ)^s $denotes the fractional Laplace operator. Citation: Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108 ##### References:  [1] S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation :$ -Δ u+u = a(x)u^p+f(x) $in$ \mathbb{R} ^N $, Calc. Var. Partial Differential Equations, 11 (2000), 63-95. doi: 10.1007/s005260050003. [2] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078. [3] B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. I. H. Poincaré-AN, 32 (2014), 875-900. doi: 10.1016/j.anihpc.2014.04.003. [4] B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023. [5] J. Bertoin, Lévy Processes, Camb. Tracts Math., 121, Cambridge University Press, Cambridge, 1996. [6] C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A. Math., 142 (2013), 39-71. doi: 10.1017/S0308210511000175. [7] X. Cabre and J. Tan, Positive solutions of nonlinear problems invoving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. [8] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [9] D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in$ \mathbb{R} ^N$, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 443-463. doi: 10.1017/S0308210500022836. [10] A. Capella, J. Davila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954. [11] G. Carboni and D. Mugnai, On some fractional equations with convex-concave and logistic-type nonlinearities, J. Differential Equations, 262 (2017), 2393-2413. doi: 10.1016/j.jde.2016.10.045. [12] W. Chen and S. Deng, The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities, Nonlinear Analysis: Real World Applications, 27 (2016), 80-92. doi: 10.1016/j.nonrwa.2015.07.009. [13] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [14] K. Chen and H. Wang, A necessary and sufficient condition for Palais-Smale conditions, SIAMJ. Math. Anal., 31 (1999), 154-165. doi: 10.1137/S0036141098338016. [15] E. Colorado, A. de Pablo and U. Sánchez, Perturbation of a critical fractional equations, Pacific J. Math., 271 (2014), 65-85. doi: 10.2140/pjm.2014.271.65. [16] E. Di Nezza, G. Palatucci and E. Valdinoci, Hithiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [17] X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involvng critical nonlinearities, Communications on Pure Applied Analysis, 15 (2016), 1285-1308. doi: 10.3934/cpaa.2016.15.1285. [18] N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equations, Nonlinear Anal., 29 (1997), 889-901. doi: 10.1016/S0362-546X(96)00176-9. [19] S. Li, S. Wu and H. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations, 185 (2002), 200-224. doi: 10.1006/jdeq.2001.4167. [20] H. Lin, Positve solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent, Nonlinear Analysis, 75 (2012), 2660-2671. doi: 10.1016/j.na.2011.11.008. [21] J. Serra and X. Ros-Oton, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [22] R. Sevadei and E. Valdinoci, Mountain pass solutions for nonlinear elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032. [23] R. Sevadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalitis driven by (non)local operator, Rev. Mat. Iberoamericana, 29 (2013), 1091-1126. doi: 10.4171/RMI/750. [24] R. Sevadei and E. Valdinoci, The Br$ \acute{e} $zis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4. [25] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. [26] M. Struwe, Variational Methods, 2nd edition, Springer-Verlag, 1996. doi: 10.1007/978-3-662-03212-1. [27] J. Tan, The Br$ \acute{e} $zis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41. doi: 10.1007/s00526-010-0378-3. [28] Y. Wei and X. Su, Multiplicity of solutions for the non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations, 52 (2015), 95-124. doi: 10.1007/s00526-013-0706-5. [29] T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270. doi: 10.1016/j.jmaa.2005.05.057. [30] X. Yu, The Nehari manifold for elliptic equation involving the square root of the Laplacian, J. Differential Equations, 252 (2012), 1283-1308. doi: 10.1016/j.jde.2011.09.015. [31] X. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differential Equations, 92 (331991), 163-178. doi: 10.1016/0022-0396(91)90045-B. show all references ##### References:  [1] S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation :$ -Δ u+u = a(x)u^p+f(x) $in$ \mathbb{R} ^N $, Calc. Var. Partial Differential Equations, 11 (2000), 63-95. doi: 10.1007/s005260050003. [2] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078. [3] B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. I. H. Poincaré-AN, 32 (2014), 875-900. doi: 10.1016/j.anihpc.2014.04.003. [4] B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023. [5] J. Bertoin, Lévy Processes, Camb. Tracts Math., 121, Cambridge University Press, Cambridge, 1996. [6] C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A. Math., 142 (2013), 39-71. doi: 10.1017/S0308210511000175. [7] X. Cabre and J. Tan, Positive solutions of nonlinear problems invoving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. [8] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [9] D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in$ \mathbb{R} ^N$, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 443-463. doi: 10.1017/S0308210500022836. [10] A. Capella, J. Davila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954. [11] G. Carboni and D. Mugnai, On some fractional equations with convex-concave and logistic-type nonlinearities, J. Differential Equations, 262 (2017), 2393-2413. doi: 10.1016/j.jde.2016.10.045. [12] W. Chen and S. Deng, The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities, Nonlinear Analysis: Real World Applications, 27 (2016), 80-92. doi: 10.1016/j.nonrwa.2015.07.009. [13] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [14] K. Chen and H. Wang, A necessary and sufficient condition for Palais-Smale conditions, SIAMJ. Math. Anal., 31 (1999), 154-165. doi: 10.1137/S0036141098338016. [15] E. Colorado, A. de Pablo and U. Sánchez, Perturbation of a critical fractional equations, Pacific J. Math., 271 (2014), 65-85. doi: 10.2140/pjm.2014.271.65. [16] E. Di Nezza, G. Palatucci and E. Valdinoci, Hithiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [17] X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involvng critical nonlinearities, Communications on Pure Applied Analysis, 15 (2016), 1285-1308. doi: 10.3934/cpaa.2016.15.1285. [18] N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equations, Nonlinear Anal., 29 (1997), 889-901. doi: 10.1016/S0362-546X(96)00176-9. [19] S. Li, S. Wu and H. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations, 185 (2002), 200-224. doi: 10.1006/jdeq.2001.4167. [20] H. Lin, Positve solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent, Nonlinear Analysis, 75 (2012), 2660-2671. doi: 10.1016/j.na.2011.11.008. [21] J. Serra and X. Ros-Oton, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [22] R. Sevadei and E. Valdinoci, Mountain pass solutions for nonlinear elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032. [23] R. Sevadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalitis driven by (non)local operator, Rev. Mat. Iberoamericana, 29 (2013), 1091-1126. doi: 10.4171/RMI/750. [24] R. Sevadei and E. Valdinoci, The Br$ \acute{e} $zis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4. [25] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. [26] M. Struwe, Variational Methods, 2nd edition, Springer-Verlag, 1996. doi: 10.1007/978-3-662-03212-1. [27] J. Tan, The Br$ \acute{e} $zis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41. doi: 10.1007/s00526-010-0378-3. [28] Y. Wei and X. 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