Ground state solutions for a kind of superlinear elliptic equations with variable exponent

In this paper, we focus on the existence of ground state solutions for the $p(x)$ $p(x)$ -Laplacian equation $\{\begin{array}{c}-{\mathrm{\Delta }}_{p(x)}u+\lambda |u{|}^{p(x)-2}u=f(x,u)+h(x)\text{in }\mathrm{\Omega },\hfill \\ u=0,\text{on }\partial \mathrm{\Omega }.\hfill \end{array}$ \( \textstyle\begin{cases} -\Delta _{p(x)}u+\lambda \vert u \vert ^{p(x)-2}u=f(x,u)+h(x) \quad \text{in } \Omega , \\ u=0,\quad \text{on }\partial \Omega . \end{cases} \) Using the constraint variational method, quantitative deformation lemma, and strong maximum principle, we proved that the above problem admits three ground state solutions, especially speaking, one solution is sign-changing, one is positive, and one is negative. Our results improve on those existing in the literature.