Existence and multiplicity results for a new p(x)-Kirchhoff problem

In this work, we study the existence and multiplicity results for the following nonlocal p(x)-Kirchhoff problem: { - (a - b integral(Omega) 1/p(x) vertical bar del u vertical bar(p(x)) dx) div (vertical bar del u vertical bar(p(x)-2)del u) = lambda vertical bar u vertical bar(p(x)-2) u + g(x, u) in Omega, (0.1) u = 0, on partial derivative Omega, where a >= b > 0 are constants, Omega subset of R-N is a bounded smooth domain, p is an element of C ((Omega) over bar) with N > p(x) > 1, lambda real parameter and g is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties. (C) 2019 Elsevier Ltd. All rights reserved.