This paper considers the following fractional $(p,q)$-Laplacian equation:

$$(-\mathrm{\Delta }{)}_p^{s}u+(-\mathrm{\Delta }{)}_q^{s}u+V(x)\left(|u{|}^{p-2}u+|u{|}^{q-2}u\right)=\lambda f(u)+|u{|}^{q_s^{\ast }-2}u\text{in}{\mathbb{R}}^N,$$

where $s\in (0,1),\lambda >0,2<p<q<\frac{N}{s}$, $(-\mathrm{\Delta }{)}_t^s$ with $t\in \{p,q\}$ is the fractional $t$-Laplacian operator, and potential $V$ is a continuous function. Using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove that the above problem has a least energy sign-changing solution $u_{\lambda }$ under suitable conditions on $f$, $V$ and $\lambda$. Moreover, we show that the energy of $u_{\lambda }$ is strictly larger than two times the ground state energy.

In this paper, we consider the following fractional $p$-Laplacian equation involving Trudinger-Moser nonlinearity:

$$(-\mathrm{\Delta }{)}_{N/s}^{s}u+V(x)|u{|}^{\frac{N}{s}-2}u=f(u)\mathrm{i}\mathrm{n}{\mathbb{R}}^N,$$

where $s\in (0,1),2<\frac{N}{s}=p$. The nonlinear function $f$ has exponential critical growth, and potential $V$ is a continuous function. By using the constrained variational methods, quantitative Deformation Lemma and Brouwer degree theory, we prove the existence of least energy sign-changing solutions.

This paper is dedicated to studying the following Kirchhoff-Schrödinger-Poisson system:

$$\{\begin{array}{ll}-\left(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^{2}dx\right)\mathrm{\Delta }u+V(|x|)u+\lambda \varphi u=K(|x|)f(u),& x\in {\mathbb{R}}^3,\\ -\mathrm{\Delta }\varphi =u^2,& x\in {\mathbb{R}}^3,\end{array}$$

where $V,K$ are radial and bounded away from below by positive numbers. Under some weaker assumptions on the nonlinearity $f$, we develop a direct approach to establish the existence of infinitely many nodal solutions $\{u_k^{b,\lambda }\}$ with a prescribed number of nodes $k$, by using the Gersgorin disc's theorem, Miranda theorem and Brouwer degree theory. Moreover, we prove that the energy of $\{u_k^{b,\lambda }\}$ is strictly increasing in $k$, and give a convergence property of $\{u_k^{b,\lambda }\}$ as $b\to 0$ and $\lambda \to 0$.