On a m(x)$$ m(x) $$‐polyharmonic Kirchhoff problem without any growth near 0 and Ambrosetti–Rabinowitz conditions

Abstract

In this paper, we study a higher order Kirchhoff problem with variable exponent of type where is a smooth bounded domain, for all ; is a Kirchhoff function, and it may be degenerate at zero; is a continuous function; and is the main ‐order differential operator. The main feature of our paper is the fact that the nonlinearity considered here satisfies some conditions which are much weaker than the classical Ambrosetti–Rabinowitz condition, the standard subcritical polynomial growth, and the strong ‐superlinear conditions required in [16]. In case of odd nonlinearity in and without requiring any control on near 0, we obtain the existence of infinitely many solutions of the above problem via the Symmetric mountain pass theorem. We improve and extend some recent results in the literature.