Multiple Solutions to p-Biharmonic Equations of Kirchhoff Type with Vanishing Potential

In this paper, we study the p-biharmonic equation of Kirchhoff type {delta(2)(p)u - ( a + b integral (N )(R)| & nabla; u | (p)dx ) delta(u )(p)+ V (x) | u |(p-2)u = K (x) f (u) + lambda g (x)|u|(q-2)u, x in R-N; u in W-2,W-p (R-N) & cap; W-0(1,p )(R-N). where N >= 5, 1 < q < p < (2)/(N), a > 0, b >= 0, lambda is a positive parameter, delta(p)u = div( |& nabla; u | (p-2)& nabla;u ) is the p-Laplacian operator and delta(2)(p)u = delta( |delta u|( p-2 )delta u) is the p-biharmonic operator, V, K, g are nonnegative functions, V is vanishing at infinity in the sense that lim (|x|->+infinity) V (x) = 0. When the nonlinear term f(u)f(u) satisfies some suitable conditions, we prove that the above problem has at least two nontrivial solutions using the mountain pass theorem combined with the Ekeland variational principle.