Bifurcation and nodal solutions for -Laplacian problems with non-asymptotic nonlinearity at 0 or

In this work, we study the existence of nodal solutions for 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 ] and f : R → R is continuous with f ( s ) s > 0 for s ≠ 0 . We give the intervals for the parameter λ which ensure the existence of single or multiple nodal solutions for the problem if f 0 ∉ ( 0 , + ∞ ) or f ∞ ∉ ( 0 , + ∞ ) , where f ( s ) / φ p ( s ) approaches f 0 and f ∞ as s approaches 0 and ∞ , respectively. We use bifurcation techniques to prove our main results.