We study the Schrödinger–Poisson system:

where parameter \(\lambda >0\), \(2<p<3\) and \(a\left( x\right) \) is a positive continuous function in \({{\mathbb {R}}}^{3}\). Assuming that \(a\left( x\right) \ge \lim _{\left| x\right| \rightarrow \infty }a\left( x\right) =a_{\infty }>0\) and other suitable conditions, we explore the energy functional corresponding to the system which is bounded below on \( H^{1}\left( {{\mathbb {R}}}^{3}\right) \) and the existence and multiplicity of positive (ground state) solutions for \(\left[ \frac{A\left( p\right) }{p} a_{\infty }\right] ^{2/\left( p-2\right) }<\lambda \le \left[ \frac{A\left( p\right) }{p}a_{1}\right] ^{2/\left( p-2\right) },\) where \(A\left( p\right) :=2^{\left( 6-p\right) /2}\left( 3-p\right) ^{3-p}\left( p-2\right) ^{\left( p-2\right) }\) and \(a_{\infty }<a_{1}<a_{\max }:=\sup _{x\in {{\mathbb {R}}} ^{3}}a\left( x\right) .\) More importantly, when \(a\left( x\right) =a\left( \left| x\right| \right) \) and \(a\left( 0\right) =a_{\max },\) we establish the existence of non-radial ground state solutions.

我们研究Schrödinger-Poisson系统：

其中参数\（\ lambda> 0 \），\（2 <p <3 \）和\（a \ left（x \ right）\）是\（{{\ mathbb {R}}}中的正连续函数^ {3} \）。假设\（a \ left（x \ right）\ ge \ lim _ {\ left | x \ right | \ rightarrow \ infty} a \ left（x \ right）= a _ {\ infty}> 0 \）等在合适的条件下，我们探索与下面\\ H ^ {1} \ left（{{\ mathbb {R}}} ^ {3} \ right）\）上界定的系统相对应的能量函数，以及存在性和多重性\（\ left [\ frac {A \ left（p \ right）} {p} a _ {\ infty} \ right] ^ {2 / \ left（p-2 \ right）}的正（基态）解的数量<\ lambda \ le \ left [\ frac {A \ left（p \ right）} {p} a_ {1} \ right] ^ {2 / \ left（p-2 \ right）}，\）其中\（A \ left（p \ right）：= 2 ^ {\ left（6-p \ right）/ 2} \ left（3-p \ right）^ {3-p} \ left（p-2 \ right ）^ {\ left（p-2 \ right）} \）和\（a _ {\ infty} <a_ {1} <a _ {\ max}：= \ sup _ {x \ in {{\ mathbb {R} }} ^ {3}} a \ left（x \ right）。\）更重要的是，当\（a \ left（x \ right）= a \ left（\ left | x \ right | \ right）\）和\（a \ left（0 \ right）= a _ {\ max}，\）我们建立了非径向基态解的存在。