Existence and concentration behavior of solutions for the logarithmic Schrödinger–Bopp–Podolsky system

Peng, Xueqin; Jia, Gao

doi:10.1007/s00033-021-01633-4

Cited by 8 publications

(4 citation statements)

References 20 publications

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“…Subsequently, Hu-Wu-Tang [26] established the existence of least energy sign-changing solutions of (1.3). For more recent results, we refer to [14,34,36,37,39,47,48].…”

Section: Introductionmentioning

confidence: 99%

Normalized solutions for Sobolev critical Schrodinger-Bopp-Podolsky systems

Li,

Chang,

Feng

2023

ejde

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We study the Sobolev critical Schrodinger-Bopp-Podolsky system $$\displaylines{ -\Delta u+\phi u=\lambda u+\mu|u|^{p-2}u+|u|^4u\quad \text{in }\mathbb{R}^3,\cr -\Delta\phi+\Delta^2\phi=4\pi u^2\quad \text{in } \mathbb{R}^3, }$$ under the mass constraint $\int_{\mathbb{R}^3}u^2\,dx=c $ for some prescribed $c>0$, where $2<p<8/3$, $\mu>0$ is a parameter, and $\lambda\in\mathbb{R}$ is a Lagrange multiplier. By developing a constraint minimizing approach, we show that the above system admits a local minimizer. Furthermore, we establish the existence of normalized ground state solutions. For more inofrmation see https://ejde.math.txstate.edu/Volumes/2023/56/abstr.html

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“…Subsequently, Hu-Wu-Tang [26] established the existence of least energy sign-changing solutions of (1.3). For more recent results, we refer to [14,34,36,37,39,47,48].…”

Section: Introductionmentioning

confidence: 99%

Normalized solutions for Sobolev critical Schrodinger-Bopp-Podolsky systems

Li,

Chang,

Feng

2023

ejde

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“…In fact, the associated energy functional of () would formally be the following form:

\tilde{J} (u) = \frac{1}{2} \int_{ℝ^{N}} (| \nabla u |^{2} + ω (x) | u |^{2}) d x - \int_{ℝ^{N}} F (u) d x,

where

F (t) = \int_{0}^{t} s \log s^{2} d s = - \frac{t^{2}}{2} + \frac{t^{2} \log t^{2}}{2}, for all t \in ℝ .

In the last years, researchers have developed several techniques to solve Problem () or similar one. We cite Cazenave 25 where the author worked in a suitable Banach space endowed with a Luxemburg type norm, in this way, the functional

\overset{J}{true}

is well defined and

C^{1}

smooth; Guerrero et al 26 where the authors penalized the nonlinearity around the origin and tried to obtain a prior estimates to get a nontrivial solution at the limit; d'Avenia et al 27,28 where the authors used the non‐smooth critical point theory introduced in Degiovanni and Zani 29 to obtain the existence and multiplicity of solutions; Tanaka and Zhang 30 where authors tried to construct solutions of () through spatially 2

L

‐periodic solutions; other studies 31–36 where authors decomposed

\overset{J}{true}

into the sum of a

C^{1}

functional and a convex lower semicontinuous functional, and applied the minimax principles for lower semicontinuous functionals to obtain solutions; and previous studies…”

Section: Introductionmentioning

confidence: 99%

“…In the last years, researchers have developed several techniques to solve Problem (1.3) or similar one. We cite Cazenave 25 where the author worked in a suitable Banach space endowed with a Luxemburg type norm, in this way, the functional J is well defined and C 1 smooth; Guerrero et al 26 where the authors penalized the nonlinearity around the origin and tried to obtain a prior estimates to get a nontrivial solution at the limit; d'Avenia et al 27,28 where the authors used the non-smooth critical point theory introduced in Degiovanni and Zani 29 to obtain the existence and multiplicity of solutions; Tanaka and Zhang 30 where authors tried to construct solutions of (1.3) through spatially 2L-periodic solutions; other studies [31][32][33][34][35][36] where authors decomposed J into the sum of a C 1 functional and a convex lower semicontinuous functional, and applied the minimax principles for lower semicontinuous functionals to obtain solutions; and previous studies [37][38][39] where authors considered the sign-changing solutions with logarithmic nonlinearity in a bounded domain. The motivation of this paper is derived from Figueiredo and Siciliano 10 and Zhang, 39 as the fact that the quasilinear Schrödinger-Poisson system with mixed type, that is, exponential critical and logarithmic nonlinearities, has not been investigated before.…”

Section: Introductionmentioning

confidence: 99%

Quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities

Peng

Jia

Huang

2022

Math Methods in App Sciences

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In this paper, we consider the following quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities −normalΔu+ϕu=false|ufalse|p−2ulogfalse|ufalse|2+λffalse(ufalse),0.30em0.1emin0.51emnormalΩ,−normalΔϕ−ε4normalΔ4ϕ=u2,0.30em0.1emin0.51emnormalΩ,u=ϕ=0,0.30em0.1emon0.51em∂normalΩ, where 40 are parameters, normalΔ4ϕ=divfalse(false|∇ϕfalse|2∇ϕfalse),normalΩ⊂ℝ2 is a bounded domain, and f has exponential critical growth. By adopting the reduction argument and a truncation technique, we prove for every ε>0, the above system admits at least one pair of nonnegative solutions false(uε,λ,ϕε,λfalse) for λ>0 large. Furthermore, we research the asymptotical behavior of solutions with respect to the parameters ε and λ. The novelty of this system is the intersection among the quasilinear term, logarithmic term, and exponential critical term. These results are new and improve some existing results in the literature.

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“…In the last few years a wide literature on this topic is developing both in R 3 (see [7,8,11,12,[21][22][23]26,27] and references therein) and on manifolds, due to Hebey (see [16][17][18][19]).…”

Section: Introductionmentioning

confidence: 99%