Existence of nodal solutions of a nonlinear fourth-order two-point boundary value problem
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People's Republic of China
Department of Basic Courses, Lanzhou Polytechnic College, Lanzhou 730050, People's Republic of China
Boundary Value Problems 2012, 2012:31 doi:10.1186/1687-2770-2012-31Published: 20 March 2012
In this article, we give conditions on parameters k, l that the generalized eigenvalue problem x″″ + kx″ + lx = λh(t)x, 0 < t < 1, x(0) = x(1) = x′(0) = x′(1) = 0 possesses an infinite number of simple positive eigenvalues and to each eigenvalue there corresponds an essential unique eigenfunction ψk which has exactly k - 1 simple zeros in (0,1) and is positive near 0. It follows that we consider the fourth-order two-point boundary value problem x″″ + kx″ + lx = f(t,x), 0 < t < 1, x(0) = x(1) = x′(0) = x′(1) = 0, where f(t, x) ∈ C([0,1] × ℝ, ℝ) satisfies f(t, x)x > 0 for all x ≠ 0, t ∈ [0,1] and lim|x|→0 f(t,x)/x = a(t), lim|x|→+∞ f(t,x)/x = b(t) or limx→-∞ f(t,x)/x = 0 and limx→+∞f(t,x)/x = c(t) for some a(t), b(t), c(t) ∈ C([0,1], (0,+∞)) and t ∈ [0,1]. Furthermore, we obtain the existence and multiplicity results of nodal solutions for the above problem. The proofs of our main results are based upon disconjugate operator theory and the global bifurcation techniques.
MSC (2000): 34B15.