论文标题
卷积和关键非线性的分数磁schrödinger-kirchhoff问题
Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities
论文作者
论文摘要
在本文中，我们关注的是具有电磁场和关键非线性的分数Choquard型Schrödinger-Kirchhoff方程的存在和多样性：\ begin {eqnarray*} \ varepsilon^{2S} m（[u] _ {s，a}^2）（ - δ）_ {a}^su + v（x） \ Mathbb {r}^n，\\ u（x）\ rightarrow 0，\ \ \ \ \ \ \ quad \ mbox {as} \ | x | \ rightarrow \ infty，\ end {cases} \ end {eqnarray*}其中$（ - δ）_ {a}^s $是$ 0 <s <s <1 $，$ 2_s^\ ast = 2n/（n-2n/（n-2s）$，$ a {n-2s）$ 0 <s <1 $，$ 0 <s <1 $，$α< \ Mathbb {r}^{+} _ {0} \ rightarrow \ MathBb {r}^{+} _ 0 $是连续的函数，$ a：\ mathbb {r}^n \ rightArrow \ rightArrow \ rightArrow \ Mathbb {r} \ int_0^{| u |} f（t）dt $，$ \ varepsilon> 0 $是一个正参数。 c（\ mathbb {r}^n，\ mathbb {r}^+_ 0）$在$ \ mathbb {r}^n $的某些区域中满足$ v（x）= 0 $的电势$ v \，这是关键的频率情况。我们首先使用浓度紧凑性原理的分数版本证明$（ps）_c $条件。然后，还应用山区通过定理和属理论，我们为上述问题获得了半经典状态的存在和多样性。我们问题的主要特征是Kirchhoff Term $ m $可以在零下消失。
In this paper we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: \begin{eqnarray*} \begin{cases} \varepsilon^{2s}M([u]_{s,A}^2)(-Δ)_{A}^su + V(x)u = (|x|^{-α}*F(|u|^2))f(|u|^2)u + |u|^{2_s^\ast-2}u,\ \ \ x\in \mathbb{R}^N,\\ u(x) \rightarrow 0,\ \ \quad \mbox{as}\ |x| \rightarrow \infty, \end{cases} \end{eqnarray*} where $(-Δ)_{A}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^\ast = 2N/(N-2s)$, $α< \min\{N, 4s\}$, $M : \mathbb{R}^{+}_{0}\rightarrow \mathbb{R}^{+}_0$ is a continuous function, $A: \mathbb{R}^N \rightarrow \mathbb{R}^N$ is the magnetic potential, $F(|u|) = \int_0^{|u|}f(t)dt$, and $\varepsilon > 0$ is a positive parameter. The electric potential $V\in C(\mathbb{R}^N, \mathbb{R}^+_0)$ satisfies $V(x) = 0$ in some region of $\mathbb{R}^N$, which means that this is the critical frequency case. We first prove the $(PS)_c$ condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term $M$ can vanish at zero.
