an:05323998
Zbl 1166.34012
Yang, Yang; Zhang, Jihui
Existence of solutions for some fourth-order boundary value problems with parameters
EN
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 4, 1364-1375 (2008).
00222450
2008
j
34B15 47N20 58E30
boundary value problem; homological nontrivial critical point; Morse theory; local linking
The authors consider the boundary value problem (BVP)
\[
u^{( 4) }( t) +\eta u^{( 2) }( t) -\xi u( t) =\lambda f( t,u( t)),\quad 0<t<1,
\]
\[
u(0) =u(1) =u^{(2)}(0) =u^{(2)}(1) =0
\]
with continuous nonlinearity \(f:\left[ 0,1\right] \times \mathbb{R}\rightarrow \mathbb{R}\) and fixed \(\eta ,\xi \) such that
\[
\frac{\xi}{\pi^{4}}+\frac{\eta}{\pi^{2}}<1,\;\xi \geq -\frac{\eta ^{2}}{4},\;\eta <2\pi ^{2}
\]
and where \(\lambda \in \mathbb{R}^{+}\) is a parameter. Using Green's function the authors provide a fixed point formulation of \ (BVP) for which they find an action functional and apply variational methods. The investigations of the equivalent variational formulation involve a square root operator of a suitable integral functional.
Depending on the assumptions on the nonlinear term \(f\), they further obtain the values of \(\lambda \) for which (BVP) has at least one and at least two nontrivial solutions.
The proofs are based on variational techniques involving Morse theory and local linking. The paper is interesting since it shows the interplay between the topological and the variational methods.
Marek Galewski (????d??)