Unilateral global bifurcation phenomena and nodal solutions for p-Laplacian

In this paper, we establish a Dancer-type unilateral global bifurcation result for one-dimensional p-Laplacian problem { -(phi(p)(u'))' = mu m(t)phi(p)(u) + g(t, u; mu), t is an element of (0, 1), u(0) = u(1) = 0. Under some natural hypotheses on the perturbation function g : (0,1) x R x R -> R, we show that mu(k)(p) is a bifurcation point of the above problem and there are two distinct unbounded continua, C(k)(+) and C(k)(-), consisting of the bifurcation branch C(k) from (mu(k)(p), 0), where mu(k)(p) is the k-th eigenvalue of the linear problem corresponding to the above problem. As the applications of the above result, we study the existence of nodal solutions for the following problem { -(phi(p)(u'))' + f(t, u) = 0, t is an element of (0, 1), u(0) = u(1) = 0. Moreover, based on the bifurcation result of Girg and Takac (2008) [13], we prove that there exist at least a positive solution and a negative one for the following problem { -div(phi(p)(del u)) = f(x, u), in Omega, u = 0, on partial derivative Omega. (C) 2011 Elsevier Inc. All rights reserved.