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

Alexander Quaas; Aliang Xia

doi:10.3934/dcds.2017113

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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 and Continuous Dynamical Systems, 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].
[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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American Institute of Mathematical Sciences

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

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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

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