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<s<1 $
,
$ 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 & 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.  Google Scholar

[2]

A. AmbrosettiH. 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.  Google Scholar

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B. BarriosE. ColoradoR. 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.  Google Scholar

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B. BarriosE. ColoradoA. 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.  Google Scholar

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J. Bertoin, Lévy Processes, Camb. Tracts Math., 121, Cambridge University Press, Cambridge, 1996.  Google Scholar

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C. BrändleE. 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.  Google Scholar

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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.  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), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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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.  Google Scholar

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A. CapellaJ. DavilaL. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[13]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[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.  Google Scholar

[15]

E. ColoradoA. 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.  Google Scholar

[16]

E. Di NezzaG. 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.  Google Scholar

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X. HeM. 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.  Google Scholar

[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.  Google Scholar

[19]

S. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

M. Struwe, Variational Methods, 2nd edition, Springer-Verlag, 1996. doi: 10.1007/978-3-662-03212-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. AmbrosettiH. 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.  Google Scholar

[3]

B. BarriosE. ColoradoR. 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.  Google Scholar

[4]

B. BarriosE. ColoradoA. 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.  Google Scholar

[5]

J. Bertoin, Lévy Processes, Camb. Tracts Math., 121, Cambridge University Press, Cambridge, 1996.  Google Scholar

[6]

C. BrändleE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. CapellaJ. DavilaL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[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.  Google Scholar

[15]

E. ColoradoA. 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.  Google Scholar

[16]

E. Di NezzaG. 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.  Google Scholar

[17]

X. HeM. 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.  Google Scholar

[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.  Google Scholar

[19]

S. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

M. Struwe, Variational Methods, 2nd edition, Springer-Verlag, 1996. doi: 10.1007/978-3-662-03212-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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