Existence of sign-changing solutions to equations involving the one-dimensional p-Laplacian

We consider the equations involving the one-dimensional p-laplacian (p): (|u′ (t)|p-2u′(t))′ + λf (u(t)) = 0,0 < t < 1,and u(0) = u(1) = 0,where p > 1,λ > 0,f ∈ c1(r;r),f(s)s > 0,and s ≠ 0. we show the existence of sign-changing solutions under the assumptions f∞= lim|s|→∞(f (s) /sp-1) = +∞ and f0= lim|s|→0(f(s)/sp-1) ∈ [0,∞]. we also show that (p) has exactly one solution having specified nodal properties for λ ∈ (0,λ∗) for some λ∗ ∈ (0,∞). our main results are based on quadrature method. copyright © 2014 r. ma and l. jiang.