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

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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

Abstract
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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 and 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, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.
[2]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.
[3]	A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annularshaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662. doi: 10.3934/cpaa.2011.10.1645.
[4]	A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators, preprint, available at http://www.ma.utexas.edu/mparc-bin/mpa?yn=12-61.
[5]	A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by the fractional Laplacian operator, preprint, available at http://www.ma.utexas.edu/mp arc-bin/mpa?yn=12-128.
[6]	A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011.
[7]	J. Liu, The Morse index of a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32-39.
[8]	J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222. doi: 10.1006/jmaa.2000.7374.
[9]	R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal., to appear.
[10]	R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., to appear. doi: 10.1090/conm/595/11809.
[11]	R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.
[12]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Continuous Dynam. Systems, 33 (2013), 2105-2137.
[13]	R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013). doi: 10.4171/RMI/750.
[14]	R. Servadei and E. Valdinoci, The Brézis-Nirenberg result for the fractional Laplacian, Trans. AMS, to appear.
[15]	R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, preprint, available at http://www.math.utexas.edu/mp arc-bin/mpa?yn=12-58.
[16]	G. Sun and K. Teng, Existence and multiplicity of solutions for a class of fractional Kirchhoff-type problem, Math. Commu., 19 (2014), 183-194.
[17]	J. Tan, The Brézis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.
[18]	K. Teng, Multiplicity results for hemivariational inequalities driven by nonlocal elliptic operators, J. Math. Anal. Appl., 396 (2012), 386-395. doi: 10.1016/j.jmaa.2012.06.041.
[19]	K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. RWA, 14 (2013), 867-874. doi: 10.1016/j.nonrwa.2012.08.008.

show all references

References:

[1]	B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.
[2]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.
[3]	A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annularshaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662. doi: 10.3934/cpaa.2011.10.1645.
[4]	A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators, preprint, available at http://www.ma.utexas.edu/mparc-bin/mpa?yn=12-61.
[5]	A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by the fractional Laplacian operator, preprint, available at http://www.ma.utexas.edu/mp arc-bin/mpa?yn=12-128.
[6]	A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011.
[7]	J. Liu, The Morse index of a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32-39.
[8]	J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222. doi: 10.1006/jmaa.2000.7374.
[9]	R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal., to appear.
[10]	R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., to appear. doi: 10.1090/conm/595/11809.
[11]	R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.
[12]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Continuous Dynam. Systems, 33 (2013), 2105-2137.
[13]	R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013). doi: 10.4171/RMI/750.
[14]	R. Servadei and E. Valdinoci, The Brézis-Nirenberg result for the fractional Laplacian, Trans. AMS, to appear.
[15]	R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, preprint, available at http://www.math.utexas.edu/mp arc-bin/mpa?yn=12-58.
[16]	G. Sun and K. Teng, Existence and multiplicity of solutions for a class of fractional Kirchhoff-type problem, Math. Commu., 19 (2014), 183-194.
[17]	J. Tan, The Brézis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.
[18]	K. Teng, Multiplicity results for hemivariational inequalities driven by nonlocal elliptic operators, J. Math. Anal. Appl., 396 (2012), 386-395. doi: 10.1016/j.jmaa.2012.06.041.
[19]	K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. RWA, 14 (2013), 867-874. doi: 10.1016/j.nonrwa.2012.08.008.

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