Positive ground state solutions for Schrödinger–Poisson system with critical nonlocal term in ℝ3

Yin, Rong; Zhang, Jihui; Shang, Xudong

doi:10.1002/mma.6541

Cited by 5 publications

(3 citation statements)

References 19 publications

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“…In view of this, there has been much attention to (1.1), and many interesting works have been devoted to investigating the existence and nonexistence of positive solutions, sign-changing solutions, positive ground states solutions, multiple solutions, and semiclassical states under variant assumptions on V, K, and 𝑓 , via variational methods in recent years. We refer to previous studies 1, [4][5][6][7][8][9] and references therein. We note that Azzollini et al 10 first studied the Schrödinger-Poisson system with critical nonlocal term as follows:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Li and He 15 studied the existence and multiplicity of positive solutions for (1.4) by using the Ljusternik-Schnirelmann theory. Yin et al 8 considered the existence of positive ground state solution to Equation (1.4) by using the variational approach.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In view of this, there has been much attention to (), and many interesting works have been devoted to investigating the existence and nonexistence of positive solutions, sign‐changing solutions, positive ground states solutions, multiple solutions, and semiclassical states under variant assumptions on

V

K

, and

f

, via variational methods in recent years. We refer to previous studies 1,4–9 and references therein. We note that Azzollini et al 10 first studied the Schrödinger‐Poisson system with critical nonlocal term as follows:

$$ \left\{\begin{array}{cc}-\Delta u&#x0003D;\lambda u&#x0002B; q\phi {\left&#x0007C;u\right&#x0007C;}&#x0005E;3u,\kern0.30em &amp; \mathrm{in}\kern0.61em {B}_R,\\ {}-\Delta \phi &#x0003D;{\left&#x0007C;u\right&#x0007C;}&#x0005E;5,\kern0.30em &amp; \mathrm{in}\kern0.61em {B}_R,\\ {}u&#x0003D;\phi &#x0003D;0,\kern0.30em &amp; \mathrm{on}\kern0.61em \partial {B}_R.\end{array}\right.

…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Ground states for critical fractional Schrödinger‐Poisson systems with vanishing potentials

Dou

2022

Math Methods in App Sciences

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This paper deals with a class of fractional Schrödinger‐Poisson system {array(−Δ)su+V(x)u−K(x)ϕ|u|2s∗−3u=a(x)f(u),arrayx∈ℝ3array(−Δ)sϕ=K(x)|u|2s∗−1,arrayx∈ℝ3$$ \left\{\begin{array}{cc}{\left(-\Delta \right)}^su+V(x)u-K(x)\phi {\left|u\right|}^{2_s^{\ast }-3}u=a(x)f(u),\kern0.30em & x\in {\mathbb{R}}^3\\ {}{\left(-\Delta \right)}^s\phi =K(x){\left|u\right|}^{2_s^{\ast }-1},\kern0.30em & x\in {\mathbb{R}}^3\end{array}\right. $$ with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where s∈false(34,1false),0.1em2s∗=63−2s$$ s\in \left(\frac{3}{4},1\right),{2}_s^{\ast }=\frac{6}{3-2s} $$ is the fractional critical exponent. The problem is set on the whole space, and compactness issues have to be tackled. By employing the mountain pass theorem, concentration‐compactness principle, and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on V$$ V $$, K$$ K $$, a$$ a $$, and f$$ f $$.

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