On the existence of stationary solutions for higher order p-Kirchhoff problems

In this paper we establish the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains. The p-polyharmonic operators \Delta^L_p were recently introduced by Colasuonno and Pucci in 2011 for all orders L and independently in the same volume of the journal by Lubyshev only for L even. In Section 4 the results are then extended to non-degenerate p(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given by Colasuonno, Pucci and Varga in 2012. Several useful properties of the underlying functional solution space [W^{L,p}_0(\Omega)]^d, endowed with the natural norm arising from the variational structure of the problem, are also proved both in the homogeneous case p=Const. and in the non-homogeneous case p=p(x). In the latter some sufficient conditions on the variable exponent p are given to prove the positivity of the the first eigenvalue of the p(x)-polyharmonic operator \Delta^L_{p(x)}.