Bifurcation from Infinity and Nodal Solutions of Quasilinear Elliptic Differential Equations

In this article, we establish a unilateral global bifurcation theorem from infinity for a class of N -dimensional p-Laplacian problems. As an application, we study the global behavior of the components of nodal solutions of the problem div(φp(∇u)) + λa(x)f(u) = 0, x ∈ B, u = 0, x ∈ ∂B, where 1 < p < ∞, φp(s) = |s|p−2s, B = {x ∈ RN : |x| < 1}, and a ∈ C(B̄, [0,∞)) is radially symmetric with a 6≡ 0 on any subset of B̄, f ∈ C(R,R) and there exist two constants s2 < 0 < s1, such that f(s2) = f(s1) = 0, and f(s)s > 0 for s ∈ R \ {s2, 0, s1}. Moreover, we give intervals for the parameter λ, where the problem has multiple nodal solutions if lims→0 f(s)/φp(s) = f0 > 0 and lims→∞ f(s)/φp(s) = f∞ > 0. We use topological methods and nonlinear analysis techniques to prove our main results.