In this paper, we consider the following 2-D Schrödinger–Newton equations\[-\Delta u + a(x)u + \frac{\gamma}{2\pi} (\operatorname{log}(\lvert \cdot \rvert) \ast {\lvert u \rvert}^p) {\lvert u \rvert}^{p-2} u = b {\lvert u \rvert}^{q-2} u \quad \textrm{in} \; \mathbb{R}^2 \; \textrm{,}\]where $a \in C(\mathbb{R}^2)$ is a $\mathbb{Z}^2$-periodic function with $\operatorname{inf}_{\mathbb{R}^2} a \gt 0, \gamma \gt 0, b \geq 0, p \geq 2$ and $q \geq 2$. By using ideas from [14, 22, 43], under mild assumptions, we obtain existence of ground state solutions and mountain pass solutions to the above equations for $p \geq 2$ and $q \geq 2p-2$ via variational methods. The auxiliary functional $J_1$ plays a key role in the case $p \geq 3$. We also prove the radial symmetry of positive solutions (up to translations) for $p \geq 2$ and $q \geq 2$. The corresponding results for planar Schrödinger–Poisson systems will also be obtained. Our theorems extend the results in [14] from $p=2$ to general $p \geq 2$ and the results in [22] from $p=2$ and $b=1$ to general $p \geq 2$ and $b \geq 0$.

中文翻译：

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二维Schrödinger–Newton方程解的存在性和对称性

在本文中，我们考虑以下二维Schrödinger–Newton方程\ [-\ Delta u + a（x）u + \ frac {\ gamma} {2 \ pi}（\ operatorname {log}（\ lvert \ cdot \ rvert）\ ast {\ lvert u \ rvert} ^ p）{\ lvert u \ rvert} ^ {p-2} u = b {\ lvert u \ rvert} ^ {q-2} u \ quad \ textrm {在} \; \ mathbb {R} ^ 2 \; \ textrm {，} \]，其中$ a \ in C（\ mathbb {R} ^ 2）$是$ \ mathbb {Z} ^ 2 $周期函数，带有$ \ operatorname {inf} _ {\ mathbb {R } ^ 2} a \ gt 0，\ gamma \ gt 0，b \ geq 0，p \ geq 2 $和$ q \ geq 2 $。通过使用[14、22、43]中的想法在温和的假设下，我们通过变分方法获得了上述方程的$ p \ geq 2 $和$ q \ geq 2p-2 $的基态解和山口解的存在性。辅助功能$ J_1 $在$ p \ geq 3 $情况下起关键作用。我们还证明了$ p \ geq 2 $和$ q \ geq 2 $的正解（直至平移）的径向对称性。还将获得平面Schrödinger-Poisson系统的相应结果。我们的定理将[14]中的结果从$ p = 2 $扩展到一般的$ p \ geq 2 $，将[22]中的结果从$ p = 2 $和$ b = 1 $扩展到一般的$ p \ geq 2 $和$ b \ geq 0 $。