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March  2015, 14(2): 361-371. doi: 10.3934/cpaa.2015.14.361

Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type

Nemat Nyamoradi 1, and  Kaimin Teng 2,

1. 

Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah, 67149, Iran
2. 

Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, 030024, China

Received  July 2013 Revised  July 2014 Published  December 2014

In this paper by using the minimal principle and Morse theory, we prove the existence of solutions to the following Kirchhoff nonlocal fractional equation: \begin{eqnarray} & M \left (\int_{\mathbb{R}^n\times \mathbb{R}^n} |u (x) - u (y)|^2 K (x - y) d x d y \right) (- \Delta)^s u = f (x, u (x)),\quad \textrm{in}\;\; \Omega,\\ & u = 0, \quad \textrm{in}\;\; \mathbb{R}^n \setminus \Omega, \end{eqnarray} where $(- \Delta)^s$ is the fractional Laplace operator, $s \in (0, 1)$ is a fix, $\Omega$ an open bounded subset of $\mathbb{R}^n$, $n > 2 s$, with Lipschitz boundary, $f: \Omega \times \mathbb{R} \to \mathbb{R}$ Carathéodory function and $M : \mathbb{R}^+ \to \mathbb{R}^+$ is a function that satisfy some suitable conditions.
Keywords: variational methods, Palais-Smale condition., Kirchhoff-type-nonlocal operators.
Mathematics Subject Classification: Primary: 49J52, 35A15; Secondary: 34A0.
Citation: Nemat Nyamoradi, Kaimin Teng. Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Communications on Pure & Applied Analysis, 2015, 14 (2) : 361-371. doi: 10.3934/cpaa.2015.14.361
References:
[1]

B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator,, \emph{J. Differential Equations}, 252 (2012), 6133. doi: 10.1016/j.jde.2012.02.023. Google Scholar

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A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator,, \emph{Nonlinear Anal.}, 94 (2014), 156. doi: 10.1016/j.na.2013.08.011. Google Scholar

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J. Liu, The Morse index of a saddle point,, \emph{Syst. Sci. Math. Sci.}, 2 (1989), 32. Google Scholar

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J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems,, \emph{J. Math. Anal. Appl.}, 258 (2001), 209. doi: 10.1006/jmaa.2000.7374. Google Scholar

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R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, \emph{Discrete Continuous Dynam. Systems}, 33 (2013), 2105. Google Scholar

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R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,, \emph{Rev. Mat. Iberoam.}, 29 (2013). doi: 10.4171/RMI/750. Google Scholar

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R. Servadei and E. Valdinoci, The Brézis-Nirenberg result for the fractional Laplacian,, \emph{Trans. AMS}, (). Google Scholar

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R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent,, preprint, (): 12. Google Scholar

[16]

G. Sun and K. Teng, Existence and multiplicity of solutions for a class of fractional Kirchhoff-type problem,, \emph{Math. Commu.}, 19 (2014), 183. Google Scholar

[17]

J. Tan, The Brézis-Nirenberg type problem involving the square root of the Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 36 (2011), 21. doi: 10.1007/s00526-010-0378-3. Google Scholar

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K. Teng, Multiplicity results for hemivariational inequalities driven by nonlocal elliptic operators,, \emph{J. Math. Anal. Appl.}, 396 (2012), 386. doi: 10.1016/j.jmaa.2012.06.041. Google Scholar

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K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators,, \emph{Nonlinear Anal. RWA}, 14 (2013), 867. doi: 10.1016/j.nonrwa.2012.08.008. Google Scholar

show all references

References:
[1]

B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator,, \emph{J. Differential Equations}, 252 (2012), 6133. doi: 10.1016/j.jde.2012.02.023. Google Scholar

[2]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[3]

A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annularshaped domains,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 1645. doi: 10.3934/cpaa.2011.10.1645. Google Scholar

[4]

A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators,, preprint, (): 12. Google Scholar

[5]

A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by the fractional Laplacian operator,, preprint, (): 12. Google Scholar

[6]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator,, \emph{Nonlinear Anal.}, 94 (2014), 156. doi: 10.1016/j.na.2013.08.011. Google Scholar

[7]

J. Liu, The Morse index of a saddle point,, \emph{Syst. Sci. Math. Sci.}, 2 (1989), 32. Google Scholar

[8]

J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems,, \emph{J. Math. Anal. Appl.}, 258 (2001), 209. doi: 10.1006/jmaa.2000.7374. Google Scholar

[9]

R. Servadei, A critical fractional Laplace equation in the resonant case,, \emph{Topol. Methods Nonlinear Anal.}, (). Google Scholar

[10]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity,, \emph{Contemp. Math.}, (). doi: 10.1090/conm/595/11809. Google Scholar

[11]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators,, \emph{J. Math. Anal. Appl.}, 389 (2012), 887. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar

[12]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, \emph{Discrete Continuous Dynam. Systems}, 33 (2013), 2105. Google Scholar

[13]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,, \emph{Rev. Mat. Iberoam.}, 29 (2013). doi: 10.4171/RMI/750. Google Scholar

[14]

R. Servadei and E. Valdinoci, The Brézis-Nirenberg result for the fractional Laplacian,, \emph{Trans. AMS}, (). Google Scholar

[15]

R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent,, preprint, (): 12. Google Scholar

[16]

G. Sun and K. Teng, Existence and multiplicity of solutions for a class of fractional Kirchhoff-type problem,, \emph{Math. Commu.}, 19 (2014), 183. Google Scholar

[17]

J. Tan, The Brézis-Nirenberg type problem involving the square root of the Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 36 (2011), 21. doi: 10.1007/s00526-010-0378-3. Google Scholar

[18]

K. Teng, Multiplicity results for hemivariational inequalities driven by nonlocal elliptic operators,, \emph{J. Math. Anal. Appl.}, 396 (2012), 386. doi: 10.1016/j.jmaa.2012.06.041. Google Scholar

[19]

K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators,, \emph{Nonlinear Anal. RWA}, 14 (2013), 867. doi: 10.1016/j.nonrwa.2012.08.008. Google Scholar

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