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{−(φp(u′))′=μm(t)φp(u)+g(t,u;μ),t∈(0,1),u(0)=u(1)=0. Under some natural hypotheses on the perturbation function g:(0,1)×R×R→Rg:(0,1)×R×R→R, we show that μk(p)μk(p) is a bifurcation point of the above problem and there are two distinct unbounded continua, Ck+ and Ck−, consisting of the bifurcation branch CkCk from (μk(p),0μk(p),0), where μk(p)μk(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{(φp(u′))′+f(t,u)=0,t∈(0,1),u(0)=u(1)=0. Moreover, based on the bifurcation result of Girg and Takáč (2008) [13], we prove that there exist at least a positive solution and a negative one for the following problem{−div(φp(∇u))=f(x,u),in Ω,u=0,on ∂Ω.