Existence of nodal solutions of a nonlinear eigenvalue problem with indefinite weight function

doi:10.1016/j.na.2009.01.046

Nonlinear Analysis: Theory, Methods & Applications

Volume 71, Issues 5–6, 1–15 September 2009, Pages 2119-2125

https://doi.org/10.1016/j.na.2009.01.046 Get rights and content

Abstract

We investigate the existence of nodal solutions of an indefinite weight boundary value problem $u^{″} + r h (t) f (u) = 0, 0 < t < 1, u (0) = u (1) = 0,$ where $h \in C [0, 1]$ changes sign. The proof of our main result is based upon bifurcation techniques.

Section snippets

Introduction and the main result

The nonlinear eigenvalue problem $u^{″} + r a (t) f (u) = 0, 0 < t < 1,$ $u (0) = u (1) = 0$ has been studied by many authors, where $a$ satisfies:

(A) $a : [0, 1] \to [0, \infty)$ is continuous and does not vanish identically on any subinterval of $[0, 1]$ .

In [1], Henderson and Wang were concerned with determining values of $r$ for which there exist positive solutions of (1.1) under condition (A) and some suitable conditions on $f$ . Recently in [2], Ma and Thompson were concerned with determining values of $r$ for which there exist nodal

Proof of the main result

Let $Y = C [0, 1]$ with the norm ${‖ u ‖}_{\infty} = max_{t \in [0, 1]} | u (t) | .$ Let $E = {u \in C^{1} [0, 1] : u (0) = u (1) = 0}$ with the norm $‖ u ‖ = max_{t \in [0, 1]} | u (t) | + max_{t \in [0, 1]} | u^{'} (t) | .$ Define $L : D (L) \to Y$ by setting $L u ≔ - u^{″}, u \in D (L),$ where $D (L) = {u \in C^{2} [0, 1] : u (0) = u (1) = 0} .$ Then $L^{- 1} : Y \to E$ is compact.

Letting $ζ, ξ \in C (R)$ be such that

$f (u) = f_{0} u + ζ (u), f (u) = f_{\infty} u + ξ (u)$ then clearly $lim_{| u | \to 0} \frac{ζ (u)}{u} = 0, lim_{| u | \to \infty} \frac{ξ (u)}{u} = 0 .$ Letting $\bar{ξ} (u) = max_{0 \leq | s | \leq u} | ξ (s) |,$ then $\bar{ξ}$ is nondecreasing and $lim_{u \to \infty} \frac{\bar{ξ} (u)}{u} = 0 .$

Proof of Theorem 1.1

We divide the proof into two cases: $r > 0$ and $r < 0$ .

Case 1: $r > 0$ .

Let us consider $L u - λ h (t) r f$

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions. The first author was supported by NSFC (No. 10671158), the NSF of Gansu Province (No. 3ZS051-A25-016), NWNU-KJCXGC-03-17, the Spring-sun program (No. Z2004-1-62033), SRFDP (No. 20060736001), and the SRF for ROCS, SEM (2006[311]).

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Existence of nodal solutions of a nonlinear eigenvalue problem with indefinite weight function

Abstract

Section snippets

Introduction and the main result

Proof of the main result

Acknowledgements

J. Math. Anal. Appl.

Nonlinear Anal.

J. Funct. Anal.

Linear Algebra Appl.