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

Article Contents

2017, Volume 37, Issue 5: 2653-2668. Doi: 10.3934/dcds.2017113

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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

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: May 2017

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Abstract

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.

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Mathematics Subject Classification: 35J60, 47G20.

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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., 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.