In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical nonlinearity: $$ (-\Delta )_{A}^{s}u+V(x)u=f\bigl(x, \vert u \vert ^{2}\bigr)u+\lambda \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, $$ where $(-\Delta )_{A}^{s}$ is the fractional magnetic operator with $0< s<1$, $N>2s$, $\lambda >0$, $2_{s}^{*}=\frac{2N}{N-2s}$, $p\geq 2_{s}^{*}$, f is a subcritical nonlinearity, and $V \in C(\mathbb{R}^{N},\mathbb{R})$ and $A \in C(\mathbb{R}^{N}, \mathbb{R}^{N})$ are the electric and magnetic potentials, respectively. Under some suitable conditions, by variational methods we prove that the equation has a nontrivial solution for small $\lambda >0$. Our main contribution is related to the fact that we are able to deal with the case $p>2_{s}^{*}$.

中文翻译：

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具有电磁场和临界或超临界非线性的分数阶Schrödinger方程非平凡解的存在性

在本文中，我们研究以下具有电磁场和临界或超临界非线性的分数薛定ding方程：$$（-\ Delta）_ {A} ^ {s} u + V（x）u = f \ bigl（x，\ vert u \ vert ^ {2} \ bigr）u + \ lambda \ vert u \ vert ^ {p-2} u，\ quad x \ in \ mathbb {R} ^ {N}，$$其中$（-\ Delta ）_ {A} ^ {s} $是分数磁算子，其中$ 0 <s <1 $，$ N> 2s $，$ \ lambda> 0 $，$ 2_ {s} ^ {*} = \ frac {2N } {N-2s} $，$ p \ geq 2_ {s} ^ {*} $，f是次临界非线性，而$ V \ in C（\ mathbb {R} ^ {N}，\ mathbb {R} C（\ mathbb {R} ^ {N}，\ mathbb {R} ^ {N}）$中的）$和$ A \ $分别是电势和磁势。在一些合适的条件下，通过变分方法，我们证明了方程对于小$ \ lambda> 0 $具有非平凡解。我们的主要贡献与以下事实有关：我们能够处理$ p> 2_ {s} ^ {*} $的情况。