On the superlinear Kirchhoff problem involving the double phase operator with variable exponents

The paper deals with the following Kirchhoff-double phase problem \(\begin{aligned} \left\{ \begin{array}{ll} m \left( L(u) \right) D(u) = \vert u \vert ^{p(x)-2} u + b(x) \vert u \vert ^{q(x)-2} u &{}\quad \text {in } \Omega ,\\ m \left( L(u) \right) \left( \vert \nabla u \vert ^{p(x)-2} u + b(x) \vert \nabla u \vert ^{q(x)-2} u \right) \cdot \nu = \lambda g(x,u) &{}\quad \text {on } \partial \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}m\left(L,(,u,)\right)D(u)=|u|_{}^{p(x)-2}u+b(x)|u|_{}^{q(x)-2}u\hfill & \text{in}\mathrm{\Omega },\hfill \\ m\left(L,(,u,)\right)\left(|\mathrm{\nabla }u|_{}^{p(x)-2},u,+,b,(x),|\mathrm{\nabla }u|_{}^{q(x)-2},u\right)·\nu =\lambda g(x,u)\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(L(u)=\int _{\Omega } ( \frac{1}{p(x)} \vert \nabla u \vert ^{p(x)}+ \frac{b(x)}{q(x)} \vert \nabla u \vert ^{q(x)}) dx\) $L(u)={\int }_{\mathrm{\Omega }}(\frac{1}{p(x)}|\mathrm{\nabla }u|_{}^{p(x)}+\frac{b(x)}{q(x)}|\mathrm{\nabla }u|_{}^{q(x)})dx$ and D is the double phase operator with variable exponents. The goal is to determine the precise positive interval of \(\lambda \) $\lambda$ for which the above problem admits at least two nontrivial weak solutions without assuming the Ambrosetti–Rabinowitz condition. Next, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the Fountain Theorem with Cerami condition.