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May  2017, 37(5): 2653-2668. doi: 10.3934/dcds.2017113

Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian

Alexander Quaas 1, and  Aliang Xia 2,

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

Received  November 2015 Revised  January 2017 Published  February 2017

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation:
$ \begin{eqnarray*}(-Δ)^α u=λ a(x)u-b(x)u^p&\ \ \ {\rm in}\,\,\mathbb{R}^N, \end{eqnarray*}$
where $ α∈(0, 1) $, $ N≥ 2 $, $λ >0$, $a$ and $b$ are positive smooth function in $\mathbb{R}^N$ satisfying
$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 $
Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations involving the fractional Laplacian.
Keywords: Fractional Laplacian, comparison principle, blow-up solution, uniqueness.
Mathematics Subject Classification: 35J60, 47G20.
Citation: 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
References:
[1]

N. Abatangelo, On a fractional keller osserman condition, arXiv: 1412.6298 [math. AP]. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[11]

P. Felmer and A. Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144.   Google Scholar

[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.  Google Scholar

[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] Google Scholar

[14]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[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.  Google Scholar

[16]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990. xiv+244 pp. Google Scholar

show all references

References:
[1]

N. Abatangelo, On a fractional keller osserman condition, arXiv: 1412.6298 [math. AP]. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[11]

P. Felmer and A. Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144.   Google Scholar

[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.  Google Scholar

[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] Google Scholar

[14]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[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.  Google Scholar

[16]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990. xiv+244 pp. Google Scholar

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