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 [Formula: see text] were recently introduced in [F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal.74 (2011) 5962–5974] for all orders L and independently, in the same volume of the journal, in [V. F. Lubyshev, Multiple solutions of an even-order nonlinear problem with convex-concave nonlinearity, Nonlinear Anal.74 (2011) 1345–1354] only for L even. In Sec. 3, the results are then extended to non-degeneratep(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given in [F. Colasuonno, P. Pucci and Cs. Varga, Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Anal.75 (2012) 4496–4512]. Several useful properties of the underlying functional solution space [Formula: see text], 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 infimum λ 1 of the Rayleigh quotient for the p(x)-polyharmonic operator [Formula: see text].