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

In this paper, we study a higher order Kirchhoff problem with variable exponent of type M∫Ω|Dru|m(x)m(x)dxΔm(x)ru=f(x,u)inΩ,Dαu=0,on∂Ω,for eachα∈ℝNwith|α|≤r−1,$$ \left\{\begin{array}{ll}M\left({\int}_{\Omega}\frac{{\left&#x0007C;{\mathcal{D}}_ru\right&#x0007C;}&#x0005E;{m(x)}}{m(x)} dx\right){\Delta}_{m(x)}&#x0005E;ru&#x0003D;f\left(x,u\right)& \mathrm{in}\kern0.30em \Omega, \\ {}{D}&#x0005E;{\alpha }u&#x0003D;0,\kern0.30em & \mathrm{on}\kern0.30em \mathrm{\partial \Omega },\kern0.30em \mathrm{for}\ \mathrm{each}\kern0.4em \alpha \in {\mathrm{\mathbb{R}}}&#x0005E;N\kern0.4em \mathrm{with}\kern0.4em \mid \alpha \mid \le r-1,\end{array}\right. $$where Ω⊂ℝN$$ \Omega \subset {\mathrm{\mathbb{R}}}&#x0005E;N $$ is a smooth bounded domain, r∈ℕ∗,m∈C(Ω‾),1<m(x)<Nr$$ r\in {\mathrm{\mathbb{N}}}&#x0005E;{\ast },m\in C\left(\overline{\Omega}\right),1<m(x)<\frac{N}{r} $$ for all x∈Ω‾$$ x\in \overline{\Omega} $$; M$$ M $$ is a Kirchhoff function, and it may be degenerate at zero; f:Ω×ℝN→ℝN$$ f:\Omega \times {\mathrm{\mathbb{R}}}&#x0005E;N\to {\mathrm{\mathbb{R}}}&#x0005E;N $$ is a continuous function; and Dr$$ {D}_r $$ is the main r$$ r $$‐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 mγ$$ m\gamma $$‐superlinear conditions required in [16]. In case of odd nonlinearity f$$ f $$ in u$$ u $$ and without requiring any control on f$$ f $$ 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. [ABSTRACT FROM AUTHOR]