Previous |  Up |  Next


multiplicity results; eigenvalues; bifurcation methods; nodal zeros; multi-point boundary value problems
We study the existence of nodal solutions of the $m$-point boundary value problem \[ u^{\prime \prime }+ f(u)=0, \quad 0<t<1, u^{\prime }(0)=0, \quad u(1)=\sum ^{m-2}_{i=1} \alpha _i u(\eta _i) \] where $\eta _i\in \mathbb{Q}$ $(i=1, 2, \cdots , m-2)$ with $0<\eta _1<\eta _2<\cdots <\eta _{m-2}<1$, and $\alpha _i\in \mathbb{R}$ $(i=1, 2, \cdots , m-2)$ with $\alpha _i>0$ and $0<\sum \nolimits ^{m-2}_{i=1} \alpha _i < 1$. We give conditions on the ratio $f(s)/s$ at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.
[1] A.  Ambrosetti, P.  Hess: Positive solutions of asymptotically linear elliptic eigenvalue problems. J.  Math. Anal. Appl. 73 (1980), 411–422. DOI 10.1016/0022-247X(80)90287-5 | MR 0563992
[2] A.  Castro, P.  Drábek, J. M.  Neuberger: A sign-changing solution for a superlinear Dirichlet problem.  II. Proceedings of the Fifth Mississippi State Conference on Differential Equations and Computational Simulations (Mississippi State, MS, 2001), pp. 101–107. MR 1976635
[3] L. H. Erbe, H.  Wang: On the existence of positive solutions of ordinary differential equations. Proc. Amer. Math. Soc. 120 (1994), 743–748. DOI 10.1090/S0002-9939-1994-1204373-9 | MR 1204373
[4] R.  Ma: Existence of positive solutions for superlinear semipositone $m$-point boundary-value problems. Proc. Edin. Math. Soc. 46 (2003), 279–292. DOI 10.1017/S0013091502000391 | MR 1998561 | Zbl 1069.34036
[5] R.  Ma, B.  Thompson: Nodal solutions for nonlinear eigenvalue problems. Nonlinear Analysis, Theory Methods Appl. 59 (2004), 717–718. MR 2096325
[6] Y.  Naito, S.  Tanaka: On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations. Nonlinear Analysis TMA 56 (2004), 919–935. DOI 10.1016/ | MR 2036055
[7] P. H. Rabinowitz: Some global results for nonlinear eigenvalue problems. J.  Funct. Anal. 7 (1971), 487–513. DOI 10.1016/0022-1236(71)90030-9 | MR 0301587 | Zbl 0212.16504
[8] I.  Rachůnková: On four-point boundary value problem without growth conditions. Czechoslovak Math.  J. 49 (1999), 241–248. DOI 10.1023/A:1022491900369 | MR 1692485
[9] B. Ruf, P. N.  Srikanth: Multiplicity results for ODEs with nonlinearities crossing all but a finite number of eigenvalues. Nonlinear Analysis TMA 10 (1986), 157–163. DOI 10.1016/0362-546X(86)90043-X | MR 0825214
[10] B. P.  Rynne: Global bifurcation for $2m$th-order boundary value problems and infinity many solutions of superlinear problems. J.  Differential Equations 188 (2003), 461–472. DOI 10.1016/S0022-0396(02)00146-8 | MR 1954290
[11] J. R.  Webb: Positive solutions of some three-point boundary value problems via fixed point index theory. Nonlinear Analysis 47 (2001), 4319–4332. DOI 10.1016/S0362-546X(01)00547-8 | MR 1975828 | Zbl 1042.34527
[12] X. Xu: Multiple sign-changing solutions for some $m$-point boundary value problems. Electronic Journal of Differential Equations 89 (2004), 1–14. MR 2075428 | Zbl 1058.34013
Partner of
EuDML logo