Solvability of a Kind of Sturm-Liouville Boundary Value Problems with Impulses via Variational Methods

The purpose of this paper is to use a three critical point theorem due to Ricceri to obtain the existence of at least three solutions for the following Sturm-Liouville boundary value problem with impulses {(phi p(x'(t)))' = (a(t)phi p(x) + lambda f(t, x) + mu h(x))g(x't)), a.e. t is an element of [0, 1], Delta G(x'(t(i))) = I(i)(x(t(i))), i = 1, 2, ..., k, alpha(1)x(0) - alpha(2)x'(0) = 0, beta(1)x(1) + beta(2)x'(1) = 0, where p > 1, phi(p)(x) = vertical bar x vertical bar(p-2)x, lambda, mu are positive parameters, G(x) = integral(x)(0) (p-1)vertical bar s vertical bar(p-2)/g(s) ds. The interest is that the nonlinear term includes x'. We exhibit the existence of at least three solutions and h(x) can be an arbitrary C(1) functional with compact derivative. As an application, an example is given to illustrate the results.