In this paper, by adapting the perturbation method, we study normalized standing wave solutions for the following nonlinear Schrödinger-Bopp-Podolsky system: { – ∆ u + ϕ u = G u + f ( u ) in Ω , – ∆ ϕ – ε 4 ∆ 4 ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω , where Ω ⊂ R 3 is a smooth bounded domain, a>0, G∈R is the Lagrange multiplier associated with the L 2 -mass constraint ∫ Ω u 2 d x = µ , and f:R→R is a continuous function satisfying some technical conditions. In particular, we prove the existence of normalized solutions for small µ. Moreover, when f is odd, we obtain multiplicity of normalized solutions.

In this paper, we study the existence and concentration of positive solutions for the following fractional Schr\”odinger logarithmic equation: \begin{equation*} \left\{ \begin{aligned} & \varepsilon^{2s} (-\Delta)^{s} u+V(x)u =u\log u^2,\ x\in \mathbb{R}^N,\\ &u\in H^s(\mathbb{R}^N), \end{aligned} \right. \end{equation*} where $\varepsilon > 0$ is a small parameter, $N>2s,$ $s \in ( 0 ,1), (-\Delta)^{s}$ is the fractional Laplacian, the potential $V$ is a continuous function having a global minimum. Using variational method to modify the nonlinearity with the sum of a $C^1 $ functional and a convex lower semicontinuous functional, we prove the existence of positive solutions and concentration around of a minimum point of $V$ when $\varepsilon$ tends to zero.

In this paper, we study the existence result for a class of fractional Schrödinger-Possion system. Using variational method and Nehari method, we obtianed a uniqueness result for positive solutions.

In the present paper, we deal with the following Kirchhoff-Schr\”{o}dinger-Poisson system with logarithmic and critical nonlinearity: \begin{equation*} \begin{array}{ll} \left \{ \begin{array}{ll} \ds\B(a+b\int_\Omega|\nabla u|^2\mathrm{d}x \B)\Delta u+V(x)u-\frac{1}{2}u\Delta (u^2)+\phi u=\lambda |u|^{q-2}u\ln|u|^2+|u|^4u, &x\in \Omega, \\ -\Delta \phi=u^2,& x\in \Omega, \\ u=0,& x\in \R^3\setminus\Omega, \end{array} \right . \end{array} \end{equation*} where $\lambda,b>0,a>\frac{1}{4},4

\begin{abstract} {In this paper, we study a class of critical fractional Kirchhoff-type equations involving logarithmic nonlinearity and steep potential well in $\R^N$ as following: \begin{align*} \renewcommand{\arraystretch}{1.25} \begin{array}{ll} \ds \left \{ \begin{array}{ll} \ds \left(a+b\int_{\R^{N}}|(-\Delta)^\frac{s}{2}u|^2\, dx\right)(-\Delta)^s u+\mu V(x)u=\lambda a(x)u\ln|u|+|u|^{2_{s}^{*}-2}u~~~\text{in}~\mathbb{R}^N, \\ u\in H^s(\R^N), \\ \end{array} \right . \end{array} \end{align*} where $a>0$ is a constant, $b$ is a positive parameter, $s\in(0,1)$ and $N>4s,$ $\mu>0$ is a parameter and $V(x)$ satisfies some assumptions that will be specified later. By applying the Nehari manifold method, we obtain that such equation with sign-changing weight potentials admits at least one positive ground state solution and the associated energy is negative. Moreover, we also explore the asymptotic behavior as $b\to 0$ and $\mu\to\infty,$ respectively.}

{We study the existence and asymptotic behavior of positive solutions for the following fractional magnetic Kirchhoff equation with steep potential well \begin{align*} \renewcommand{\arraystretch}{1.25} \begin{array}{ll} \ds \left \{ \begin{array}{ll} \ds \left(a+b\int_{\R^{3}}|(-\Delta)_A^\frac{s}{2}u|^2\, dx\right)(-\Delta)_A^s u+V_{\lambda}u=|u|^{p-2}u~~~in~\mathbb{R}^3, \\ u\in H^s(\R^3), \\ \end{array} \right . \end{array} \end{align*} where $a,~b>0$ are constants, $2

\begin{abstract} {In this paper we consider the following fractional Kirchhoff equation with steep potential well \begin{align*} \renewcommand{\arraystretch}{1.25} \begin{array}{ll} \ds \left \{ \begin{array}{ll} \ds \left(a+b\int_{\R^{3}}|(-\Delta)^\frac{s}{2}u|^2\, dx\right)(-\Delta)^s u+\lambda V(x)u=|u|^{p-2}u,\,\,x\in\mathbb{R}^3, \\ u\in H^s(\R^3), \\ \end{array} \right . \end{array} \end{align*} where $a>0$ is a constant, $b$ and $\lambda$ are positive parameters. $2

In this paper we consider the following fractional $(p,q)$-Laplacian equation $$ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right)=\lambda f(u)+|u|^{q^*_s-2}u \quad \text { in } \mathbb{R}^{N}, $$ where $s \in(0,1), \lambda>0, 2

This paper is devoted to the following fractional Schrödinger-Poisson systems: \begin{equation*} \left\{\aligned &(-\Delta)^{s} u+V(x)u+\phi(x)u= f(x,u) \,\,\,&\text{in } \mathbb{R}^3, \\ & (-\Delta)^{t} \phi(x)=u^2 \,\,\,&\text{in } \mathbb{R}^3, \endaligned \right. \end{equation*} where $(-\Delta)^{s}$ is the fractional Lapalcian, $s, t \in (0, 1),$ $V : \R^3 \to \R$ is continuous. In contrast to most studies, we consider that the potentials $V$ is indefinite. With the help of Morse theory, the existence of nontrivial solutions for the above problem is obtained.