On a planar non-autonomous Schrödinger–Poisson system involving exponential critical growth

Albuquerque, Francisco S. B.; Carvalho, João; Figueiredo, Giovany M.; Medeiros, Everaldo S.

doi:10.1007/s00526-020-01902-6

Cited by 29 publications

(16 citation statements)

References 39 publications

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“…In the sequel, we mean that (V, K) ∈ K if the continuous functions V (x) and K(x) satisfy (V ) 0 and (K) 0 , respectively. Let's mention here that the similar conditions were presented in [3,4]. However, compared with [4] the proof for the nonlocal Chern-Simons case (1.6) is much more complicated and technical.…”

Section: Introduction and Main Resultsmentioning

confidence: 53%

“…In particular, if Ω = R 2 , instead of • L m (R 2 ) , we'll use | • | m for simplicity. Given a constant q ∈ [1, +∞), as in [3,4], we introduce the weighted Lebesgue functions L 2 V (R N ) and L q K (R N ) as follows…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Next, inspired by the Trudinger-Moser inequality in R 2 , see e.g. [5,12,17,26,40,45], we can follow the methods exploited in [3,4] to establish the following two lemmas. Lemma 2.5.…”

Section: Preliminariesmentioning

confidence: 99%

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Critical gauged Schrödinger equations in R^2 with vanishing potentials

Shen¹,

Squassina²,

Yang³

2021

Preprint

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We study a class of gauged nonlinear Schrödinger equations in the planewhere h(s) = ´s 0 r 2 u 2 (r)dr, λ, µ > 0 are constants, V (x) and K(x) are continuous functions vanishing at infinity. Assume that f is of critical exponential growth and g(x) = g(|x|) satisfies some technical assumptions with 1 ≤ q < 2, we obtain the existence of two nontrivial solutions via the Mountain-Pass theorem and Ekeland's variational principle. Moreover, with the help of the genus theory, we prove the existence of infinitely many solutions if f in addition is odd.This system consists of the nonlinear Schrödinger equation augmented by the gauge field A j : R 1+2 → R, where i denotes the imaginary unit,, φ : R 1+2 → C is the complex scalar field and D j = ∂ j iA j is the covariant derivative for j = 0, 1, 2. For each C ∞ 0 (R 1+2 ) function χ, under the following gauge transformation φ → φe iχ , A j → A j − ∂ j χ, system (1.2) is invariant because of the Chern-Simons theory [22].

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Section: Introduction and Main Resultsmentioning

confidence: 53%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Critical gauged Schrödinger equations in R^2 with vanishing potentials

Shen¹,

Squassina²,

Yang³

2021

Preprint

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show abstract

“…The case of dimension N = 2 and α < N has been studied in [4,5]. More recently in [6,10,14] it has been considered the limiting case α = N = 2; see also [3,9] for related results. A major difficulty in the limiting case is to construct a proper function space framework in which to settle the problem.…”

Section: Consider the Following System Of Elliptic Equationsmentioning

confidence: 99%

Quasilinear logarithmic Choquard equations with exponential growth in RN

Bucur

Cassani

Tarsi

2022

Journal of Differential Equations

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Section: Consider the Following System Of Elliptic Equationsmentioning

confidence: 99%

Quasilinear logarithmic Choquard equations with exponential growth in $\mathbb{R}^N$

Bucur¹,

Cassani²,

Tarsi³

2022

Preprint

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We consider the N -Laplacian Schrödinger equation strongly coupled with higher order fractional Poisson's equations. When the order of the Riesz potential α is equal to the Euclidean dimension N , and thus it is a logarithm, the system turns out to be equivalent to a nonlocal Choquard type equation. On the one hand, the natural function space setting in which the Schrödinger energy is well defined is the Sobolev limiting space W 1,N (R N ), where the maximal nonlinear growth is of exponential type. On the other hand, in order to have the nonlocal energy well defined and prove the existence of finite energy solutions, we introduce a suitable log-weighted variant of the Pohozaev-Trudinger inequality which provides a proper functional framework where we use variational methods.

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On a planar non-autonomous Schrödinger–Poisson system involving exponential critical growth

Cited by 29 publications

References 39 publications

Critical gauged Schrödinger equations in R^2 with vanishing potentials

Critical gauged Schrödinger equations in R^2 with vanishing potentials

Quasilinear logarithmic Choquard equations with exponential growth in RN

Quasilinear logarithmic Choquard equations with exponential growth in $\mathbb{R}^N$

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