We are concerned with the following Schrödinger type equation with variable exponents

$$(-{\mathrm{\Delta }}_{p(x)}{)}^{s}u+V(x)|u{|}^{p(x)-2}u=f(x,u)\text{in}{\mathbb{R}}^N,$$

where $(-{\mathrm{\Delta }}_{p(x)}{)}^s$ is the fractional $p(x)$-Laplace operator, $s\in (0,1)$, $V:{\mathbb{R}}^N\to (0,+\mathrm{\infty })$ is a continuous potential function, and $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ satisfies the Carathéodory condition. We study the nonlinearity of this equation which is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. By using variational techniques and the fountain theorem, we obtain the existence and multiplicity of nontrivial solutions. Furthermore, we show that the problem has a sequence of solutions with high energies.

This paper was focused on the solvability of a class of doubly critical sub-Laplacian problems on the Carnot group $\mathbb{G}$:

$$-{\mathrm{\Delta }}_{\mathbb{G}}u-\mu \frac{{\psi }^2(\xi )u}{\text{d}(\xi {)}^2}=|u{|}^{p-2}u+{\psi }^{\alpha }(\xi )\frac{|u{|}^{2^{\ast }(\alpha )-2}u}{\text{d}(\xi {)}^{\alpha }},u\in S^{1,2}(\mathbb{G}).$$

Here, $p\in (1,2^{\ast }]$, $\alpha \in (0,2)$, $\mu \in [0,{\mu }_{\mathbb{G}})$, $2^{\ast }=\frac{2Q}{Q-2}$, and $2^{\ast }(\alpha )=\frac{2(Q-\alpha )}{Q-2}$. By means of variational techniques, we extended the arguments developed in ^[1]. In addition, we also established the existence result for the subelliptic system which involved sub-Laplacian and critical homogeneous terms.

We are devoted to the study of the following sub-Laplacian system with Hardy-type potentials and critical nonlinearities

$$\{\begin{array}{r}-{\mathrm{\Delta }}_{\mathbb{G}}u-{\mu }_1\frac{{\psi }^{2}u}{\text{d}(z{)}^2}={\lambda }_1\frac{{\psi }^{\alpha }|u{|}^{2^{\ast }(\alpha )-2}u}{\text{d}(z{)}^{\alpha }}+\beta p_{1}f(z)\frac{{\psi }^{\gamma }|u{|}^{p_1-2}u|v{|}^{p_2}}{\text{d}(z{)}^{\gamma }}\text{in}\mathbb{G},\\ -{\mathrm{\Delta }}_{\mathbb{G}}v-{\mu }_2\frac{{\psi }^{2}v}{\text{d}(z{)}^2}={\lambda }_2\frac{{\psi }^{\alpha }|v{|}^{2^{\ast }(\alpha )-2}v}{\text{d}(z{)}^{\alpha }}+\beta p_{2}f(z)\frac{{\psi }^{\gamma }|u{|}^{p_1}|v{|}^{p_2-2}v}{\text{d}(z{)}^{\gamma }}\text{in}\mathbb{G},\end{array}$$

where $-{\mathrm{\Delta }}_{\mathbb{G}}$ is the sub-Laplacian on Carnot group $\mathbb{G}$, ${\mu }_1$, ${\mu }_2\in [0,{\mu }_{\mathbb{G}})$, $\alpha ,\gamma \in (0,2)$, ${\lambda }_1$, ${\lambda }_2$, $\beta$, $p_1$, $p_2>0$ with $1<p_1+p_2<2$, $\text{d}(z)$ is the ${\mathrm{\Delta }}_{\mathbb{G}}$-gauge, $\psi =|{\mathrm{\nabla }}_{\mathbb{G}}\text{d}(z)|$, $2^{\ast }(\alpha ):=\frac{2(Q-\alpha )}{Q-2}$ is the critical Sobolev-Hardy exponents, and ${\mu }_{\mathbb{G}}=(\frac{Q-2}{2}{)}^2$ is the best Hardy constant on $\mathbb{G}$. By combining a variant of the symmetric mountain pass theorem with the genus theory, we prove the existence of infinitely many weak solutions whose energy tends to zero when $\beta$ or ${\lambda }_1$, ${\lambda }_2$ belong to a suitable range.