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January  2018, 17(1): 21-37. doi: 10.3934/cpaa.2018002

## Unilateral global interval bifurcation for Kirchhoff type problems and its applications

 Department of Basic Courses, Lanzhou Institute of Technology, Lanzhou, 730050, China

Received  April 2016 Revised  June 2017 Published  September 2017

Fund Project: The author is supported by NNSF of China (No. 11561038) and the National Science Foundation of Gansu (No.145RJZA087).

In this paper, we establish a unilateral global bifurcation result from interval for a class of Kirchhoff type problems with nondifferentiable nonlinearity. By applying the above result, we shall prove the existence of one-sign solutions for the following Kirchhoff type problems.
 $\left\{ {\begin{array}{*{20}{l}} { - M(\int_\Omega {|\nabla u{|^2}dx} )\Delta u = \alpha (x){u^ + } + \beta (x){u^ - } + ra(x)f(u),}&{{\text{in}}{\mkern 1mu} \;\Omega ,} \\ {u = 0,}&{{\text{on}}{\mkern 1mu} \;\partial \Omega ,} \end{array}} \right.$
where Ω is a bounded domain in
 $\mathbb{R}^{N}$
with a smooth boundary
 $\partial$
Ω,
 $M$
is a continuous function, r is a parameter,
 $a(x) \in C(\overline \Omega )$
is positive,
 $u^{+} = \max\{u, 0\}, u^{-}= -\min\{u, 0\}$
,
 $\alpha ,\beta \in C\left( {\overline \Omega } \right)$
;
 $f \in C\left( {\mathbb{R},\mathbb{R}} \right)$
,
 $sf(s)>0$
for
 $s \in {\mathbb{R}^ + },$
and
 ${f_0} \in \left( {0,\infty } \right)$
and
 ${f_\infty } \in \left( {0,\infty } \right]$
or
 ${f_0} \in \infty$
and f∈[0, ∞], where
 ${f_0} = {\lim _{\left| s \right| \to 0}}f\left( s \right)/s,{f_\infty } = {\lim _{\left| s \right| \to + \infty }}f\left( s \right)/s$
. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.
Citation: Wenguo Shen. Unilateral global interval bifurcation for Kirchhoff type problems and its applications. Communications on Pure and Applied Analysis, 2018, 17 (1) : 21-37. doi: 10.3934/cpaa.2018002
##### References:
 [1] W. Allegretto and Y. X. Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.  doi: 10.1016/S0362-546X(97)00530-0. [2] C. O. Alves and F. J. S. A. Corrêa, On the existence of solutions for a class of problem involving a nonlinear operator, Math. Methods Appl. Sci., 8 (2001), 43-56. [3] A. Ambrosetti, R. M. Calahorrano and F. R. Dobarro, Global branching for discontinuous problems, Nonlinear Anal. RWA, 31 (1990), 213-222. [4] G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math. , 16 (2014). doi: 10.1142/S0219199714500023. [5] H. Berestycki, On some nonlinear Sturm-Liouville problems, J. Differ. Equ., 26 (1977), 375-390.  doi: 10.1016/0022-0396(77)90086-9. [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differ. Equ., 6 (2001), 701-730. [7] C. Chen, Y. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differ. Equ., 250 (2011), 1876-1908.  doi: 10.1016/j.jde.2010.11.017. [8] S. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on $R^{N}$, Nonlinear Anal. RWA, 14 (2013), 1477-1486.  doi: 10.1016/j.nonrwa.2012.10.010. [9] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Math. Methods Appl. Sci. , 30 (1997), 4619-4627. doi: 10.1016/S0362-546X(97)00169-7. [10] 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.  doi: 10.1016/S0096-3003(02)00740-3. [11] G. Dai and R. Ma, Global bifurcation, Berestycki's conjecture and one-sign solutions for p-Laplacian, Nonlinear Anal., 91 (2013), 51-59.  doi: 10.1016/j.na.2013.06.003. [12] G. Dai, H. Wang and B. Yang, Global bifurcation and positive solution for a class of fully nonlinear problems, Comput. Math. Appl., 69 (2015), 771-776. [13] E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1974), 1069-1076.  doi: 10.1512/iumj.1974.23.23087. [14] P. D'Ancona and Y. Shibata, On global solvability of nonlinear viscoelastic equations in the analytic category, Math. Methods Appl. Sci., 17 (1994), 477-486.  doi: 10.1002/mma.1670170605. [15] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605. [16] D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Lecture in Math., 957 (1982), 34-87. [17] G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401 (2013), 706-713.  doi: 10.1016/j.jmaa.2012.12.053. [18] G. M. Figueiredo, C. Morales-Rodrigo, J. R. S. Júnior and A. Suárez, Study of a nonlinear Kirchhoff equation with non-homogeneous material, J. Math. Anal. Appl., 416 (2014), 597-608.  doi: 10.1016/j.jmaa.2014.02.067. [19] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [20] Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, Math. Methods Appl. Sci., 253 (2012), 2285-2294.  doi: 10.1016/j.jde.2012.05.017. [21] S. Liang and S. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in $R^{N}$, Math. Methods Appl. Sci., 81 (2013), 31-41.  doi: 10.1016/j.na.2012.12.003. [22] Z. Liang, F. Li and J. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincar Anal. Non Linéaire, 31 (2014), 155-167.  doi: 10.1016/j.anihpc.2013.01.006. [23] 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, 1977), in: North-Holland Math. Stud., 30 (1978), 284-346. [24] R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, Nonlinear Anal., 71 (2009), 4364-4376.  doi: 10.1016/j.na.2009.02.113. [25] R. Ma and G. Dai, Global bifurcation and nodal solutions for a Sturm-Liouville problem with a nonsmooth nonlinearity, J. Funct. Anal., 265 (2013), 1443-1459.  doi: 10.1016/j.jfa.2013.06.017. [26] T. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Math. Methods Appl. Sci., 16 (2003), 243-248.  doi: 10.1016/S0893-9659(03)80038-1. [27] A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287.  doi: 10.1016/j.na.2008.02.011. [28] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ., 221 (2006), 246-255.  doi: 10.1016/j.jde.2005.03.006. [29] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. [30] P. H. Rabinowitz, On bifurcation from infinity, J. Differ. Equ., 14 (1973), 462-475.  doi: 10.1016/0022-0396(73)90061-2. [31] G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958.

show all references

##### References:
 [1] W. Allegretto and Y. X. Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.  doi: 10.1016/S0362-546X(97)00530-0. [2] C. O. Alves and F. J. S. A. Corrêa, On the existence of solutions for a class of problem involving a nonlinear operator, Math. Methods Appl. Sci., 8 (2001), 43-56. [3] A. Ambrosetti, R. M. Calahorrano and F. R. Dobarro, Global branching for discontinuous problems, Nonlinear Anal. RWA, 31 (1990), 213-222. [4] G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math. , 16 (2014). doi: 10.1142/S0219199714500023. [5] H. Berestycki, On some nonlinear Sturm-Liouville problems, J. Differ. Equ., 26 (1977), 375-390.  doi: 10.1016/0022-0396(77)90086-9. [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differ. Equ., 6 (2001), 701-730. [7] C. Chen, Y. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differ. Equ., 250 (2011), 1876-1908.  doi: 10.1016/j.jde.2010.11.017. [8] S. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on $R^{N}$, Nonlinear Anal. RWA, 14 (2013), 1477-1486.  doi: 10.1016/j.nonrwa.2012.10.010. [9] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Math. Methods Appl. Sci. , 30 (1997), 4619-4627. doi: 10.1016/S0362-546X(97)00169-7. [10] 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.  doi: 10.1016/S0096-3003(02)00740-3. [11] G. Dai and R. Ma, Global bifurcation, Berestycki's conjecture and one-sign solutions for p-Laplacian, Nonlinear Anal., 91 (2013), 51-59.  doi: 10.1016/j.na.2013.06.003. [12] G. Dai, H. Wang and B. Yang, Global bifurcation and positive solution for a class of fully nonlinear problems, Comput. Math. Appl., 69 (2015), 771-776. [13] E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1974), 1069-1076.  doi: 10.1512/iumj.1974.23.23087. [14] P. D'Ancona and Y. Shibata, On global solvability of nonlinear viscoelastic equations in the analytic category, Math. Methods Appl. Sci., 17 (1994), 477-486.  doi: 10.1002/mma.1670170605. [15] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605. [16] D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Lecture in Math., 957 (1982), 34-87. [17] G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401 (2013), 706-713.  doi: 10.1016/j.jmaa.2012.12.053. [18] G. M. Figueiredo, C. Morales-Rodrigo, J. R. S. Júnior and A. Suárez, Study of a nonlinear Kirchhoff equation with non-homogeneous material, J. Math. Anal. Appl., 416 (2014), 597-608.  doi: 10.1016/j.jmaa.2014.02.067. [19] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [20] Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, Math. Methods Appl. Sci., 253 (2012), 2285-2294.  doi: 10.1016/j.jde.2012.05.017. [21] S. Liang and S. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in $R^{N}$, Math. Methods Appl. Sci., 81 (2013), 31-41.  doi: 10.1016/j.na.2012.12.003. [22] Z. Liang, F. Li and J. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincar Anal. Non Linéaire, 31 (2014), 155-167.  doi: 10.1016/j.anihpc.2013.01.006. [23] 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, 1977), in: North-Holland Math. Stud., 30 (1978), 284-346. [24] R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, Nonlinear Anal., 71 (2009), 4364-4376.  doi: 10.1016/j.na.2009.02.113. [25] R. Ma and G. Dai, Global bifurcation and nodal solutions for a Sturm-Liouville problem with a nonsmooth nonlinearity, J. Funct. Anal., 265 (2013), 1443-1459.  doi: 10.1016/j.jfa.2013.06.017. [26] T. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Math. Methods Appl. Sci., 16 (2003), 243-248.  doi: 10.1016/S0893-9659(03)80038-1. [27] A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287.  doi: 10.1016/j.na.2008.02.011. [28] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ., 221 (2006), 246-255.  doi: 10.1016/j.jde.2005.03.006. [29] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. [30] P. H. Rabinowitz, On bifurcation from infinity, J. Differ. Equ., 14 (1973), 462-475.  doi: 10.1016/0022-0396(73)90061-2. [31] G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958.
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