MULTIPLICITY OF SOLUTIONS FOR A CLASS OF FOURTH-ORDER ELLIPTIC EQUATIONS OF P(X)-KIRCHHOFF TYPE

This paper deals with a class of fourth order elliptic equations of Kirchhoff type with variable exponent {Delta(2)(p(x))u - M(integral(Omega)1/p(x)vertical bar del u vertical bar(p(x))dx)Delta(p(x))u + vertical bar u vertical bar(p(x)-2) u = lambda f(x,u)+mu g(x,u) in Omega, u=Delta u=0 on partial derivative Omega, where p(-) := inf(x is an element of(Omega) over bar) p(x) > max{1, N/2}, lambda>0 and mu >= 0 are real numbers, Omega subset of R-N (N >= 1) is a smooth bounded domain, Delta(2)(p(x))u = Delta(vertical bar Delta u vertical bar(p(x)-2)Delta u) is the operator of fourth order called the p(x)-biharmonic operator, Delta(p(x))u = div(vertical bar del u vertical bar(p(x)-2)del u) is the (x)-Laplacian, p: Omega -> R is a log-Holder continuous function, M: [0, +infinity) -> R is a continuous function and f, g: Omega x R -> R are two L-1-Caratheodory functions satisfying some certain conditions. Using two kinds of three critical point theorems, we establish the existence of at least three weak solutions for the problem in an appropriate space of functions. (C) 2021 Mathematical Institute Slovak Academy of Sciences