Bifurcation from infinity and nodal solutions of quasilinear problems without the signum condition

In this paper, we shall establish a unilateral global bifurcation theorem from infinity for a class of p -Laplacian problems. As an application of the above result, we shall study the global behavior of the components of nodal solutions of the following problem { ( φ p ( u ′ ) ) ′ + λ a ( t ) f ( u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where φ p ( s ) = | s | p − 2 s , a ∈ C ( [ 0 , 1 ] , [ 0 , + ∞ ) ) with a ≢ 0 on any subinterval of [ 0 , 1 ] ; f : R → R is continuous, and there exist two constants s 2 < 0 < s 1 such that f ( s 2 ) = f ( s 1 ) = f ( 0 ) = 0 , f ( s ) s > 0 for s ∈ R ∖ { s 2 , 0 , s 1 } . Moreover, we give the intervals for the parameter λ which ensure the existence of multiple nodal solutions for the problem if f 0 ∈ ( 0 , + ∞ ) and f ∞ ∈ ( 0 , + ∞ ) , where f ( s ) / φ p ( s ) approaches f 0 and f ∞ as s approaches 0 and ∞ , respectively. We use topological methods and nonlinear analysis techniques to prove our main results.