Unilateral Global Bifurcation for Fourth-Order Problems and Its Applications

Abstract EN

We will establish unilateral global bifurcation result for a class of fourth-order problems.

Under some natural hypotheses on perturbation function, we show that ( λ k , 0 ) is a bifurcation point of the above problems and there are two distinct unbounded continua, C k + and C k - , consisting of the bifurcation branch C k from ( μ k , 0 ) , where μ k is the k th eigenvalue of the linear problem corresponding to the above problems.

As the applications of the above result, we study the existence of nodal solutions for the following problems: x ′ ′ ′ ′ + k x ′ ′ + l x = r h ( t ) f ( x ) , 0 < t < 1 , x ( 0 ) = x ( 1 ) = x ′ ( 0 ) = x ′ ( 1 ) = 0 , where r ∈ R is a parameter and k , l are given constants; h ( t ) ∈ C ( [ 0,1 ] , [ 0 , ∞ ) ) with h ( t ) ≢ 0 on any subinterval of [ 0,1 ] ; and f : R → R is continuous with s f ( s ) > 0 for s ≠ 0 .

We give the intervals for the parameter r ≠ 0 which ensure the existence of nodal solutions for the above fourth-order Dirichlet problems if f 0 ∈ [ 0 , ∞ ] or f ∞ ∈ [ 0 , ∞ ] , where f 0 = l i m | s | → 0 f ( s ) / s and f ∞ = l i m | s | → + ∞ f ( s ) / s .

We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.