On the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity

We consider the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity $\varepsilon^{2s}M([u]_{s,A_\varepsilon}^2)(-\Delta)_{A_\varepsilon}^su + V(x)u =$ $|u|^{2_s^\ast-2}u + h(x,|u|^2)u,$ $\ \ x\in \mathbb{R}^N,$ where $u(x) \rightarrow 0$ as $|x| \rightarrow \infty,$ and $(-\Delta)_{A_\varepsilon}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^\ast = 2N/(N-2s),$ $M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+}$ is a continuous nondecreasing function, $V:\mathbb{R}^N \rightarrow \mathbb{R}^+_0,$ and $A: \mathbb{R}^N \rightarrow \mathbb{R}^N$ are the electric and the magnetic potential, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that $\varepsilon < \mathcal {E}$; and (ii) for any $m^\ast \in \mathbb{N}$, has $m^\ast$ pairs of solutions if $\varepsilon < \mathcal {E}_{m^\ast}$, where $\mathcal {E}$ and $\mathcal {E}_{m^\ast}$ are sufficiently small positive numbers. Moreover, these solutions $u_\varepsilon \rightarrow 0$ as $\varepsilon \rightarrow 0$

On a Kirchhoff type problems with potential well and indefinite potential

In this paper, we study the following Kirchhoff type problem:% \left\{\aligned&-\bigg(\alpha\int_{\bbr^3}|\nabla u|^2dx+1\bigg)\Delta u+(\lambda a(x)+a_0)u=|u|^{p-2}u&\text{ in }\bbr^3,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{\alpha,\lambda})}% where $4<p<6$, $\alpha$ and $\lambda$ are two positive parameters, a_0\in\bbr is a (possibly negative) constant and $a(x)\geq0$ is the potential well. By the variational method, we investigate the existence of nontrivial solutions to $(\mathcal{P}_{\alpha,\lambda})$. To our best knowledge, it is the first time that the nontrivial solution of the Kirchhoff type problem is found in the indefinite case. We also obtain the concentration behaviors of the solutions as $\lambda\to+\infty$.Comment: 1

Multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth

We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon$ is a small positive parameter, $a, b>0$ are fixed constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential verifying the global condition due to Rabinowitz \cite{Rab}, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinearity. Due to the presence of the magnetic field and the critical growth of the nonlinearity, several difficulties arise in the study of our problem and a careful analysis will be needed. The main results presented here are established by using minimax methods, concentration compactness principle of Lions \cite{Lions}, a fractional Kato's type inequality and the Ljusternik-Schnirelmann theory of critical points.Comment: arXiv admin note: text overlap with arXiv:1808.0929