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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 & 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. Google Scholar [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. Google Scholar [3] A. Ambrosetti, R. M. Calahorrano and F. R. Dobarro, Global branching for discontinuous problems, Nonlinear Anal. RWA, 31 (1990), 213-222. Google Scholar [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. Google Scholar [5] H. Berestycki, On some nonlinear Sturm-Liouville problems, J. Differ. Equ., 26 (1977), 375-390. doi: 10.1016/0022-0396(77)90086-9. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Lecture in Math., 957 (1982), 34-87. Google Scholar [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. Google Scholar [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. Google Scholar [19] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [29] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. Google Scholar [30] P. H. Rabinowitz, On bifurcation from infinity, J. Differ. Equ., 14 (1973), 462-475. doi: 10.1016/0022-0396(73)90061-2. Google Scholar [31] G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958. Google Scholar

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. Google Scholar [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. Google Scholar [3] A. Ambrosetti, R. M. Calahorrano and F. R. Dobarro, Global branching for discontinuous problems, Nonlinear Anal. RWA, 31 (1990), 213-222. Google Scholar [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. Google Scholar [5] H. Berestycki, On some nonlinear Sturm-Liouville problems, J. Differ. Equ., 26 (1977), 375-390. doi: 10.1016/0022-0396(77)90086-9. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Lecture in Math., 957 (1982), 34-87. Google Scholar [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. Google Scholar [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. Google Scholar [19] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [29] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. Google Scholar [30] P. H. Rabinowitz, On bifurcation from infinity, J. Differ. Equ., 14 (1973), 462-475. doi: 10.1016/0022-0396(73)90061-2. Google Scholar [31] G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958. Google Scholar
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