On a quasilinear Schrödinger-Poisson system

Du, Yihong; Su, Jiabao; Wang, Cong

doi:10.1016/j.jmaa.2021.125446

Cited by 26 publications

(12 citation statements)

References 20 publications

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“…In some sense, these results extend some relevant results in [4] to the quasilinear Schrödinger-Poisson system. Moreover, we extend some results of the subcritical case in [11,12] to the critical case. It should be pointed out that in the critical situation, the quasilinear system (1.1) could be considered only for 3 2 < p < 3.…”

mentioning

confidence: 52%

“…We first collect the following important conclusions related to the Poisson equation in (1.1). In [12] we have proved that for any u ∈ W 1,p (R 3 ) there exists a unique φ u ∈ D 1,2 (R 3 ) solving weakly the equation −∆φ = |u| p . Therefore, the map Φ :…”

Section: Preliminariesmentioning

confidence: 99%

“…with 4 < q < 6. In a recent work [12], we first dealt with via the variational methods the subcritical quasilinear system…”

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confidence: 99%

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On the critical Schrödinger-Poisson system with $ p $-Laplacian

Du¹,

Su²,

Wang³

2022

CPAA

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In this paper we consider the critical quasilinear Schrödinger-Poisson system<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left \{\begin{array}{ll} -\Delta_p u+|u|^{p-2}u+\mu\phi |u|^{p-2}u = \lambda|u|^{q-2}u+|u|^{p^*-2}u,&\mathrm{in} \ \mathbb{R}^3,\\ -\Delta \phi = |u|^p, &\mathrm{in}\ \mathbb{R}^3, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M2">\begin{document}$ \frac{3}{2}<p<3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \Delta_p u = \hbox{div}(|\nabla u|^{p-2}\nabla u) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ p<q<p^*: = \frac{3p}{3-p} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \mu,\lambda>0 $\end{document}</tex-math></inline-formula>. Based upon the variational approach, the ground state solutions and the nontrivial solutions are obtained depending on the parameters <inline-formula><tex-math id="M6">\begin{document}$ q $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>.

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confidence: 52%

Section: Preliminariesmentioning

confidence: 99%

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On the critical Schrödinger-Poisson system with $ p $-Laplacian

Du¹,

Su²,

Wang³

2022

CPAA

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“…This is the main object of our paper. Motivated by the above mentioned works, especially our recent works [14,15], this paper deals with the case q = 2, or to be more precise, 1 < p < 3, q and s satisfy (1.2), see the figure F2. From the figure F2, we see that the case q = 2 is included in the present paper.…”

Section: F1mentioning

confidence: 99%

“…the coordinate point (2, 2, 2) in the figure F1. In [14] the system (1.1) has been considered for the case that 4 3 < p < 12 5 , q = 2 and s = 2, while in [15] the system (1.1) has been considered for the case that 1 < p < 3, q = 2 and s = p, see the blue and the red line segment in the figure F1, respectively.…”

Section: Introductionmentioning

confidence: 99%

The quasilinear Schrödinger--Poisson system

Du¹,

Su²

2022

Preprint

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This paper deals with the (p, q)-Schrödinger-Poisson systemand λ > 0 is a parameter. This quasilinear system is new and has never been considered in the literature. The uniqueness of solutions of the quasilinear Poisson equation is obtained via the Minty-Browder theorem. The variational framework of the quasilinear system is built and the nontrivial solutions of the system are obtained via the mountain pass theorem.

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Concentration of solutions for non-autonomous double-phase problems with lack of compactness

Zhang,

Zuo,

Rădulescu

2024

Z. Angew. Math. Phys.

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The present paper is devoted to the study of the following double-phase equation $$\begin{aligned} -\text {div}(|\nabla u|^{p-2}\nabla u+\mu _{\varepsilon }(x)|\nabla u|^{q-2}\nabla u)+V_{\varepsilon }(x)(|u|^{p-2}u+\mu _{\varepsilon }(x)|u|^{q-2}u)=f(u)\quad \text{ in }\quad \mathbb {R}^{N}, \end{aligned}$$ - div ( | ∇ u | p - 2 ∇ u + μ ε ( x ) | ∇ u | q - 2 ∇ u ) + V ε ( x ) ( | u | p - 2 u + μ ε ( x ) | u | q - 2 u ) = f ( u ) in R N , where $$N\ge 2$$ N ≥ 2 , $$1<p<q<N$$ 1 < p < q < N , $$q<p^{*}$$ q < p ∗ with $$p^{*}=\frac{Np}{N-p}$$ p ∗ = Np N - p , $$\mu :\mathbb {R}^{N}\rightarrow \mathbb {R}$$ μ : R N → R is a continuous non-negative function, $$\mu _{\varepsilon }(x)=\mu (\varepsilon x)$$ μ ε ( x ) = μ ( ε x ) , $$V:\mathbb {R}^{N}\rightarrow \mathbb {R}$$ V : R N → R is a positive potential satisfying a local minimum condition, $$V_{{{\,\mathrm{\varepsilon }\,}}}(x)=V({{\,\mathrm{\varepsilon }\,}}x)$$ V ε ( x ) = V ( ε x ) , and the nonlinearity $$f:\mathbb {R}\rightarrow \mathbb {R}$$ f : R → R is a continuous function with subcritical growth. Under natural assumptions on $$\mu $$ μ , V and f, by using penalization methods and Lusternik–Schnirelmann theory we first establish the multiplicity of solutions, and then, we obtain concentration properties of solutions.

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On a quasilinear Schrödinger-Poisson system

Cited by 26 publications

References 20 publications

On the critical Schrödinger-Poisson system with $ p $-Laplacian

On the critical Schrödinger-Poisson system with $ p $-Laplacian

The quasilinear Schrödinger--Poisson system

Concentration of solutions for non-autonomous double-phase problems with lack of compactness

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