Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

Chen, Sitong; Tang, Xianhua; Wei, Jiuyang

doi:10.1515/anona-2020-0118

Cited by 16 publications

(10 citation statements)

References 39 publications

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“…When the spectrum

σ false(- normalΔ + V false) > 0

, that is, the positive definite situation which includes the case

V false(x false) > 0

and the case that the potential

V false(x false)

is nonnegative and vanishes on an open bounded domain in

ℝ^{N}

, Equation () with

N \geq 3

has been extensively studied in recent years via the mountain pass theorem, Ekeland's variational principle, Strauss's lemma and the method of moving plane, and so forth; see previous works for the existence, multiplicity, and uniqueness results, 3,5,7–14 for the sign‐changing solutions, 15–17 and for the semiclassical solutions 18–21 . In recent review article, 22 a guide to the Choquard equation was given by Moroz and Schaftingen.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Gao

Chen

Qin

et al. 2022

Math Methods in App Sciences

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In this paper, we show the existence of nontrivial solutions for the following planar Choquard equation: −normalΔu+Vfalse(xfalse)u=false(Iα∗Ffalse(ufalse)false)ffalse(ufalse),3.0235pt3.0235pt0.30emx∈ℝ2,u∈H1false(ℝ2false), where Iα:ℝ2→ℝ is the Riesz potential. We mainly consider the case that Vfalse(xfalse) is 1‐periodic, f:ℝ→ℝ has critical exponential growth at infinity and 0 lies in a gap of the spectrum of −△+V. There are few works in the literature concerned with such problem due to the compactness issue, appearance of the convolution term Iα∗Ffalse(ufalse), and the difficulties aroused by the strongly indefinite characteristic and the critical exponential growth of f. By employing an approximation scheme and some fine estimates of the mountain pass minimax levels in combination with related variational arguments, we show nontrivial solution for the above strongly indefinite problem.

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“…When the spectrum

σ false(- normalΔ + V false) > 0

, that is, the positive definite situation which includes the case

V false(x false) > 0

and the case that the potential

V false(x false)

is nonnegative and vanishes on an open bounded domain in

ℝ^{N}

, Equation () with

N \geq 3

Section: Introduction and Main Resultsmentioning

confidence: 99%

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Gao

Chen

Qin

et al. 2022

Math Methods in App Sciences

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“…Finally, we highlight the recent paper [2], where Chen, Tang and Wei established the existence of solutions for with and having exponential critical growth.…”

Section: Introductionmentioning

confidence: 97%

“…2 ( vn ||vn|| )|v n | 2 dx + b 1 |v n | p e (4π−ω0δ)p ( vn ||vn|| ) = p/(p − 1). Choosing p > 0 sufficiently large in (6.3) in such way that p (4π − ω 0 δ) 4π follows from Trudinger-Moser inequality (2.1) thatR 2 g(v n )v n dx ε|v n | 2 2 + C|v n | p , (6.4)for some C > 0.Using the compact embeddingH 1 (R 2 ) → L 2 loc (R 2) once more and recalling that we are supposing that (u n , v n ) (0, 0) in E, we obtain lim…”

mentioning

confidence: 99%

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ²

Maia¹,

Miyagaki²

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem \[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \] when $(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if $N\geq 3$ ) and $(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if $N=2$ ), where $I_{\alpha }$ and $I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem \[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \] where $\alpha ,\,\beta \in (0,\,2)$ and $f,\,g$ have exponential critical growth in the Trudinger–Moser sense.

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“…$$ when

V\equiv 1

V(x)

is coercive and

f(u)

has exponential growth of Trudinger–Moser type. See, for example, Alves and Souto, 28 Alves et al, 29 Cao, 27 Chen‐Tang, 30 and references therein 31–34 …”

Section: Introductionmentioning

confidence: 99%

“…See, for example, Alves and Souto, 28 Alves et al, 29 Cao, 27 Chen-Tang, 30 and references therein. [31][32][33][34] In the recent papers, [35][36][37] the authors studied the existence of nontrivial solutions and ground state solutions to the following nonlocal Kirchhoff problem of the type:…”

Section: Introductionmentioning

confidence: 99%

Ground state solutions for planar periodic Kirchhoff type equation with critical exponential growth

Wei

Tang

Zhang

2022

Math Methods in App Sciences

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In this paper, we prove the following nonlocal Kirchhoff problem of the type {left leftarray−a+b∫ℝ2|∇u|2dxΔu+V(x)u=f(x,u),arrayx∈ℝ2;arrayu∈H1(ℝ2)array$$ \left\{\begin{array}{ll}-\left(a+b\underset{{\mathbb{R}}^2}{\int }{\left|\nabla u\right|}^2\mathrm{d}x\right)\Delta u+V(x)u=f\left(x,u\right),& x\in {\mathbb{R}}^2;\\ {}u\in {H}^1\left({\mathbb{R}}^2\right)& \end{array}\right. $$ has a Nehari‐type ground state solution when Vfalse(xfalse)$$ V(x) $$ and ffalse(x,ufalse)$$ f\left(x,u\right) $$ are periodic on x$$ x $$ and ffalse(x,ufalse)$$ f\left(x,u\right) $$ has critical exponential growth in the sense of Trudinger–Moser inequality on u$$ u $$. We develop some new approaches to estimate precisely the minimax level of the energy functional and to recover the compactness of Cerami sequences of the associated Euler–Lagrange functional. With this in hand, we can overcome difficulties arising from the appearance of the nonlocal term ∫ℝ2false|∇ufalse|2normaldx$$ {\int}_{{\mathbb{R}}^2}{\left|\nabla u\right|}^2\mathrm{d}x $$ and the nonlinearity which is of critical growth of Trudinger–Moser type in whole Euclidean space ℝ2$$ {\mathbb{R}}^2 $$.

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Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

Cited by 16 publications

References 39 publications

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ²

Ground state solutions for planar periodic Kirchhoff type equation with critical exponential growth

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Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

Cited by 16 publications

References 39 publications

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ2

Ground state solutions for planar periodic Kirchhoff type equation with critical exponential growth

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Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ²