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Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian
1. | Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla: V-110, Avda. España 1680, Valparaíso, Chile |
2. | Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
$ \begin{eqnarray*}(-Δ)^α u=λ a(x)u-b(x)u^p&\ \ \ {\rm in}\,\,\mathbb{R}^N, \end{eqnarray*}$ |
$a\left( x \right) \to {a^\infty } > 0\;\;\;\;{\rm{and}}\;\;\;b\left( x \right) \to {b^\infty } > 0\;\;\;\;{\rm{as}}\;\;\;{\rm{|}}\mathit{x}{\rm{|}} \to \infty $ |
References:
[1] |
N. Abatangelo, On a fractional keller osserman condition, arXiv: 1412.6298 [math. AP]. |
[2] |
N. Abatangelo,
Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 5555-5607.
doi: 10.3934/dcds.2015.35.5555. |
[3] |
G. Barles, E. Chasseigne and C. Imbert,
On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana Univ. Math. J., 57 (2008), 213-246.
doi: 10.1512/iumj.2008.57.3315. |
[4] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[5] |
L. Caffarelli and L. Silvestre,
Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.
doi: 10.1007/s00205-010-0336-4. |
[6] |
H. Chen, P. Felmer and A. Quaas,
Large solutions to elliptic equations involving fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1199-1228.
doi: 10.1016/j.anihpc.2014.08.001. |
[7] |
H. Chen, H. Hajaiej and Y. Wang,
Boundary blow-up solutions to fractional elliptic equations in a measure framework, Discrete Contin. Dyn. Syst., 36 (2016), 1881-1903.
|
[8] |
M. G. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Soc., 27 (1992), 1-67.
|
[9] |
Y. Du and Z. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math. , 89 (2003), 277-302; Corrigendum. J. Anal. Math. , 107 (2009), 391-393. |
[10] |
Y. Du and L. Ma,
Logistic type equations on $\mathbb{R}^N$ by a squeezing method involving boundary blow up solutions, J. London Math. Soc.(2), 64 (2001), 107-124.
doi: 10.1017/S0024610701002289. |
[11] |
P. Felmer and A. Quaas,
Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144.
|
[12] |
A. Fiscella, R. Servadei and E. Valdinoci,
Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.
doi: 10.5186/aasfm.2015.4009. |
[13] |
L. del Pezzo and A. Quaas, Global bifurcation for fractional $p$-Laplacian and application, To appear in Zeitschrift Fur Analysis und ihre Anwendungen. arXiv: 1412.4722 [math. AP] |
[14] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[15] |
R. Servadei and E. Valdinoci,
Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.
doi: 10.5565/PUBLMAT_58114_06. |
[16] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990. xiv+244 pp. |
show all references
References:
[1] |
N. Abatangelo, On a fractional keller osserman condition, arXiv: 1412.6298 [math. AP]. |
[2] |
N. Abatangelo,
Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 5555-5607.
doi: 10.3934/dcds.2015.35.5555. |
[3] |
G. Barles, E. Chasseigne and C. Imbert,
On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana Univ. Math. J., 57 (2008), 213-246.
doi: 10.1512/iumj.2008.57.3315. |
[4] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[5] |
L. Caffarelli and L. Silvestre,
Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.
doi: 10.1007/s00205-010-0336-4. |
[6] |
H. Chen, P. Felmer and A. Quaas,
Large solutions to elliptic equations involving fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1199-1228.
doi: 10.1016/j.anihpc.2014.08.001. |
[7] |
H. Chen, H. Hajaiej and Y. Wang,
Boundary blow-up solutions to fractional elliptic equations in a measure framework, Discrete Contin. Dyn. Syst., 36 (2016), 1881-1903.
|
[8] |
M. G. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Soc., 27 (1992), 1-67.
|
[9] |
Y. Du and Z. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math. , 89 (2003), 277-302; Corrigendum. J. Anal. Math. , 107 (2009), 391-393. |
[10] |
Y. Du and L. Ma,
Logistic type equations on $\mathbb{R}^N$ by a squeezing method involving boundary blow up solutions, J. London Math. Soc.(2), 64 (2001), 107-124.
doi: 10.1017/S0024610701002289. |
[11] |
P. Felmer and A. Quaas,
Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144.
|
[12] |
A. Fiscella, R. Servadei and E. Valdinoci,
Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.
doi: 10.5186/aasfm.2015.4009. |
[13] |
L. del Pezzo and A. Quaas, Global bifurcation for fractional $p$-Laplacian and application, To appear in Zeitschrift Fur Analysis und ihre Anwendungen. arXiv: 1412.4722 [math. AP] |
[14] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[15] |
R. Servadei and E. Valdinoci,
Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.
doi: 10.5565/PUBLMAT_58114_06. |
[16] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990. xiv+244 pp. |
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