Topological Methods in Nonlinear Analysis

Infinitely many solutions of superlinear fourth order boundary value problems

Bryan P. Rynne

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Abstract

We consider the boundary value problem \begin{gather*} u^{(4)}(x) =g(u(x)) + p(x,u^{(0)}(x),\dots,u^{(3)}(x)) , \quad x \in (0,1), \\ u(0) =u(1) =u^{(b)}(0) =u^{(b)}(1) =0, \end{gather*} where:

(i) $g \colon \mathbb R \to \mathbb R$ is continuous and satisfies $\lim_{|\xi| \to \infty} g(\xi)/\xi =\infty$ ($g$ is superlinear as $|\xi| \to \infty$),

(ii) $p \colon [0,1] \times \mathbb R^4 \to \mathbb R$ is continuous and satisfies $$ |p(x,\xi_0,\xi_1,\xi_2,\xi_3)| \le C + \frac{1}{4} |\xi_0| , \quad x \in [0,1],\ (\xi_0,\xi_1,\xi_2,\xi_3) \in \mathbb R^4, $$ for some $C> 0$,

(iii) either $b=1$ or $b=2$.

We obtain solutions having specified nodal properties. In particular, the problem has infinitely many solutions.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 19, Number 2 (2002), 303-312.

Dates
First available in Project Euclid: 2 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1470138766

Mathematical Reviews number (MathSciNet)
MR1921051

Zentralblatt MATH identifier
1017.34015

Citation

Rynne, Bryan P. Infinitely many solutions of superlinear fourth order boundary value problems. Topol. Methods Nonlinear Anal. 19 (2002), no. 2, 303--312. https://projecteuclid.org/euclid.tmna/1470138766


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References

  • M. Conti, S. Terracini and G. Verzini, Infinitely many solutions to fourth order superlinear periodic problems , preprint \ref\no2
  • A. Cappieto, M. Henrard, J. Mawhin and F. Zanolin, A continuation approach to some forced superlinear Sturm-Liouville boundary value problems , Topol. Methods Nonlinear Anal., 3 (1994), 81–100 \ref\no3
  • A. Cappieto, J. Mawhin and F. Zanolin, A continuation approach to superlinear periodic boundary value problems , J. Differential Equations, 88 (1990), 347–395 \ref\no4
  • A. Cappieto, J. Mawhin and F. Zanolin, On the existence of two solutions with a prescribed number of zeros for a superlinear two point boundary value problem , Topol. Methods Nonlinear Anal., 6 (1995), 175–188 \ref\no5
  • C. De Coster and M. Gaudenzi, On the existence of infinitely many solutions for superlinear $n$-th order boundary value problems , Nonlinear World, 4 (1997), 505–524 \ref\no6
  • E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems , Indiana Univ. Math. J., 23 (1974), 1069–1076 \ref\no7
  • U. Elias, Eigenvalue problems for the equation $Ly + \la p(x) y =0$ , J. Differential Equations, 29 (1978), 28–57 \ref\no8
  • J. Mawhin and F. Zanolin, A continuation approach to fourth order superlinear periodic boundary value problems , Topol. Methods Nonlinear Anal., 2 (1993), 55–74 \ref\no9
  • P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems , J. Funct. Anal., 7 (1971), 487–513 \ref\no10
  • B. P. Rynne, Global bifurcation for $2m$'th order boundary value problems and infinitely many solutions of superlinear problems , submitted