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Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation
The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities
School of Mathematic and Computer Science, Wuhan Polytechnic University, Wuhan, 430023, China |
$ \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) $ |
$ 0<s<1 $ |
$ λ>0 $ |
$ 1≤q <2$ |
$ 2_s^* = \frac{2N}{N-2s} $ |
$ 0∈ Ω\subset \mathbb{R} ^N(N>4s) $ |
$ \partialΩ $ |
$ f,\,g $ |
$\bar{Ω} $ |
$ (-Δ)^s $ |
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. S |
[5] |
J. Bertoin, Lévy Processes, Camb. Tracts Math., 121, Cambridge University Press, Cambridge, 1996. |
[6] |
C. Br |
[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. S |
[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. S |
[5] |
J. Bertoin, Lévy Processes, Camb. Tracts Math., 121, Cambridge University Press, Cambridge, 1996. |
[6] |
C. Br |
[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. S |
[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. |
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