### The sign-changing solutions for a class of nonlocal elliptic problem in an annulus

DOI: http://dx.doi.org/10.12775/TMNA.2019.081

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R.P. Agarwal and D. O’Regan, A survey of recent results for initial and boundary value problems singular in the dependent variable, Original Research Article Handbook of Differential Equations: Ordinary Differential Equations 1 (2000), 1–68.

C.O. Alves and F.J.S.A. Corrêa, On existence of solutions for a class of problems involving a nonlinear operator, Comm. Appl. Nonlinear Anal. 8 (2001), no. 2, 43–56.

C.O. Alves, F.J.S.A. Corrêa and G.M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl., 2 (2010), 409–417.

C.O. Alves and D.P. Covei, Existence of solution for a class of nonlocal elliptic problem via sub-supersolution method, Nonlinear Anal. 23 (2015), 1–8.

J.J. Aly, Thermodynamics of a two-dimensional self-gravitating system, Phys. Rev. E 49 (1994), 3771–3783.

F. Bavaud, Equilibrium properties of the Vlasov functional: the generalized Poisson–Boltzmann–Emden equation, Rev. Mod. Phys. 63 (1991), 129–148.

P. Biler, A. Krzywicki and T. Nadzieja, Self-interaction of Brownian particles coupled with thermodynamic processes, Reports Math. Phys. 42 (1998), 359–372.

P. Biler and T. Nadzieja, A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq. Math. 66 (1993), 131–145.

P. Biler and T. Nadzieja, Nonlocal parabolic problems in statistical mechanics, Nonlinear Anal. 30 (1997), 5343–5350.

T. Bortsch and Z.Q. Wang, On the existence of sign-changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal. 7 (1996), 115–131.

A. Castra, J. Cossion and J.M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mount. J. Math. 27 (1997), 1041–1053.

H. Cheng and R. Yuan, Existence and stability of traveling waves for Leslie–Gower predator-prey system with nonlocal diffusion, Discrete Contin. Dyn. Syst. Ser. A 37 (2017), no. 10, 5433–5454.

M. Chipot and F.J.S.A. Corrêa, Boundary layer solutions to functional elliptic equations, Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 3, 381–393.

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), 4619–4627.

M. Chipot and P. Roy, Existence results for some functional elliptic equations, Differential Integral Equations 27 (2014), 289–300.

F.J.S.A. Corrêa, On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Anal. 59 (2004), 1147–1155.

F.J.S.A. Corrêa, M. Delgado and A. Suárez, A variational approach to a nonlocal elliptic problem with a sign-changing nonlinearity, Adv. Nonlinear Stud. 11 (2011), 361–375.

F.J.S.A. Corrêa, M. Delgado and A. Suárez, Some non-local problems with nonlinear diffusion, Math. Comput. Model. Dyn. Syst. 54 (2011), 2293–2305.

F.J.S.A. Corrêa and G.M. Figueiredo, On an elliptic equation of p-Kirchhoff type via variational methods, Bull. Aust. Math. Soc. 74 (2006), 263–277.

F.J.S.A. Corrêa, S.D.B. Menezes and J. Ferreira, On a class of problems involving a nonlocal operator, Appl. Math. Comput. 147 (2004), 475–489.

F.J.S.A. Corrêa and R.G. Nascimento, On a nonlocal elliptic system of p-Kirchhofftype under Neumann boundary condition, Math. Comput. Model. 49 (2009), 598–604.

E.N. Dancer and Y. Du, Existence of sign-changing solutions for some semilinear problems with jumping nonlinearities at zero, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 1165–1176.

K. Di and B. Yan, The existence of positive solution for singular Kirchhoff equation with two parameters, Bound. Value Probl. 2019 (2019), no. 40, 1–13.

J.M. do Ó, S. Lorca, J. Sánchez and P. Ubilla, Positive solutions for some nonlocal and nonvariational elliptic systems, Complex Var. Elliptic Equ. 61 (2016), no. 3, 1–18.

G.M. Figueiredo, Existence of positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (1974), no. 2, 324–353.

A. Granas and J. Dugundji, Fixed Point Theory, Springer Science + Business Media, New York, 2003.

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, 1988.

X. Li and S. Song, Stabilization of Delay Systems: Delay-dependent Impulsive Control, IEEE Trans. Automat. Control 62 (2017), no. 1, 406–411.

X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica 64 (2017), 63–69.

A.M. Mao and Z.T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the PS condition, Nonlinear Anal. 70 (2009), 1275–1287.

P.H. Rabinowitz, Nonlinear Sturm–Lionville problems for second order ordinary differential equations, Comm. Pure Appl. Math. 23 (1970), no. 6, 939–961.

P. Roy, Existence results for some nonlocal problems, Differ. Equ. Appl. 6 (2014), no. 3, 361–381.

W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations 259 (2015), 1256–1274.

D. Wang and B. Yan, Existence and multiplicity of positive solutions for p-Kirchhoff type problem with singularity, Bound. Value Probl. 2017 (2017), no. 38, 1–17.

D. Wang and B. Yan, A uniqueness result for some Kirchhoff-type equations with negative exponents, Appl. Math. Lett. 92 (2019), 93–98.

G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rational Mech. Anal. 119 (1992), 355–391.

B. Yan and T. Ma, The existence and multiplicity of positive solutions for a class of nonlocal elliptic problems, Bound. Value Probl. 165 (2016), 1–35.

B. Yan, D. O’Regan and R.P. Agarwal, The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method, An. Univ. Ovidius 26 (2018), no. 1, 5–41.

B. Yan, D. O’Regan and R.P. Agarwal, On spectral asymptotics and bifurcation for some elliptic equations of Kirchhoff-type with odd superlinear term, J. Appl. Anal. Comput. 8 (2018), no. 2, 509–523.

B. Yan and D. Wang, The multiplicity of positive solutions for a class of nonlocal elliptic problem, J. Math. Anal. Appl. 442 (2016), no. 1, 72–102.

X. Xu and J. Sun, On sign-changing solution for some three-point boundary value problems, Nonlinear Anal. 59 (2004), 491–505.

Z.T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006), 456–463.

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