Eigenvalues, global bifurcation and positive solutions for a class of nonlocal elliptic equations

Guowei Dai


In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem: \begin{equation*} \begin{cases} \displaystyle -\bigg(a+b\int_\Omega \vert \nabla u\vert^2dx\bigg)\Delta u=\lambda u+h(x,u,\lambda) &\text{in } \Omega,\\ u=0 &\text{on }\Omega. \end{cases} \end{equation*} Under some natural hypotheses on $h$, we show that $(a\lambda_1,0)$ is a bifurcation point of the above problem. As an application of the above result, we shall determine the interval of $\lambda$, in which there exist positive solutions for the above problem with $h(x,u;\lambda)=\lambda f(x,u)-\lambda u$, where $f$ is asymptotically linear at zero and asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we provide a positive answer to an open problem involving the case $a=0$.


Bifurcation; eigenvalue; Kirchhoff type equation; positive solutions

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W. Allegretto and Y.X. Huang, A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal. 32 (1998), 819–830.

A. Arosio and S. Pannizi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), 305–330.

G. Autuori, P. Pucci and M.C. Salvatori, Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl. 352 (2009), 149–165.

M.M. Cavalcante, V.N. Cavalcante and J.A. Soriano, Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation, Adv. Differential Equations 6 (2001), 701–730.

C. Chen, Y. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (4) (2011), 1876–1908.

B. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal. 71 (10) (2009), 4883–4892.

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

G. Dai and R. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009), 275–284.

G. Dai and D. Liu, Infinitely many positive solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009), 704–710.

G. Dai and R. Ma, Solutions for a p(x)-Kirchhoff type equation with Neumann boundary data, Nonlinear Anal. Real World Appl. 12 (2011), 2666–2680.

G. Dai and J. Wei, Infinitely many non-negative solutions for a p(x)-Kirchhoff-type problem with Dirichlet boundary condition, Nonlinear Anal. 73 (2010), 3420–3430.

L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 631–652.

E.N. Dancer, The effect of domain shape on the number of positive solu nonlinear equations, J. Differential Equations 74 (1988), 120–156.

P. D’Ancona and Y. Shibata, On global solvability of nonlinear viscoelastic equations in the analytic category, Math. Methods Appl. Sci. 17 (6) (1994), 477–486.

P. D’Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), 247–262.

M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the pLapiacian, J. Differential Equations 92 (1991), 226–251.

M. Dreher, The Kirchhoff equation for the p-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino 64 (2006), 217–238.

M. Dreher, The ware equation for the p-Laplacian, Hokkaido Math. J. 36 (2007), 21–52.

L.C. Evans, Partial differential equations, AMS, Rhode Island, 1998.

X.L. Fan, On nonlocal p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 72 (2010), 3314–3323.

X.L. Fan, Global C 1,α regularity for variable exponent elliptic equations in divergence form, J. Differential Equations 235 (2007), 397–417.

X.L. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal. 36 (1996), 295–318.

X.L. Fan, Y.Z. Zhao and Q.H. Zhang, A strong maximum principle for p(x)-Laplace equations, Chinese J. Contemp. Math. 24 (3) (2003), 277–282.

B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.

X. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009), 1407–1414.

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

Z. Liang, F. Li and J. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poncaré Anal, Nonlinéaire 31 (2014), 155–167.

Z. Liang, F. Li and J. Shi, Positive solutions of Kirchhoff type nonlocal elliptic equation: a bifurcation approach, in press.

C.S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in R2, Manuscripta Math. 84 (1994), 13–19.

J.L. Lions, On some equations in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346.

T.F. Ma and J.E.Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2) (2003), 243–248.

A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal. 70 (3) (2009), 1275–1287.

I.P. Natanson, Theory of Functions of a Real Variable, Nauka, Moscow, 1950.

K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (1) (2006), 246–255.

P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (2) (1973), 161–202.

P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.

J. Sun and C. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal. 74 (4) (2011), 1212–1222.

G.T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958.

E. Zeidler, Nonlinear functional analysis and its applications, Vol. II/B. Berlin–Heidelberg–New York 1985.

Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2) (2006), 456–463.


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