On the planar Choquard equation with indefinite potential and critical exponential growth

Qin, Dongdong; Tang, Xianhua

doi:10.1016/j.jde.2021.03.011

Cited by 44 publications

(14 citation statements)

References 34 publications

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“…The following assumptions on

f

was introduced by Qin and Tang, 32 taking advantage of which existence of nontrivial solution for () was proved there.

false| f false(t false) false| = o false(false| t {false|}^{α false/ 2} false)

false| t false| \to 0

; There exist

M_{0} > 0

and

t_{0} > 0

such that

F (t) \leq M_{0} | f (t) |, \forall | t | \geq t_{0};

{lim inf}_{t \to \infty} \frac{f false(t false)}{e^{β_{0} t^{2}}} = κ > \frac{\sqrt{α false(1 + α false)} false(2 + α false)}{\sqrt{2 π A_{α}} ρ^{1 + α false/ 2}} e^{4 false(2 + α false) π {false(1 + ρ false)}^{2} {scriptB}_{0}^{2} false/ false(2 + ρ false)}

, where

ρ \in false(0,4 false/ false‖ V {false‖}_{\infty} false)

satisfies

2 false(2 + α false) π ρ^{2} {scriptB}_{0}^{2} < 1

. …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The following assumptions on 𝑓 was introduced by Qin and Tang, 32 taking advantage of which existence of nontrivial solution for (1.1) was proved there.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Remark In the marginal case, that is

s = 1

in (), the following inequality does not hold, but it can be recovered by replacing the Lebesgue spaces by the weak Lebesgue spaces,

\int_{ℝ^{2}} false(I_{false(2 + α false) false/ 2} * g false) h normald x \leq {scriptC}_{1} false‖ g {false‖}_{1} false‖ h {false‖}_{4 false/ false(2 + α false)},

where

{scriptC}_{1} : = scriptC false(false(2 - α false) false/ 2,1, 4 false/ false(2 + α false) false)

. Though () is assumed in Lemma 2.1 and used in the proof (3.81), 32 we can use the following inequality to show (3.81) instead of (),

\int_{ℝ^{2}} false(I_{2 false(1 + α false) false/ 3} * g false) h normald x \leq {scriptC}_{2} false‖ g {false‖}_{6 false/ false(4 + α false)} false‖ h {false‖}_{6 false/ false(4 + α false)},

where

{scriptC}_{2} : = scriptC false(2 false(2 - α false) false/ 3,6 false/ false(4 + α false), 6 false/ false(4 + α false) false)

. Indeed, for any

φ \in C_{0}^{\infty} false(ℝ^{2} false)

, one has

\begin{matrix} alignleft \\ align-1 & align-2 \int_{B_{R_{1}}} |F (ū (y))| [ \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…Though () is assumed in Lemma 2.1 and used in the proof (3.81), 32 we can use the following inequality to show (3.81) instead of (),

\int_{ℝ^{2}} false(I_{2 false(1 + α false) false/ 3} * g false) h normald x \leq {scriptC}_{2} false‖ g {false‖}_{6 false/ false(4 + α false)} false‖ h {false‖}_{6 false/ false(4 + α false)},

where

{scriptC}_{2} : = scriptC false(2 false(2 - α false) false/ 3,6 false/ false(4 + α false), 6 false/ false(4 + α false) false)

. Indeed, for any

φ \in C_{0}^{\infty} false(ℝ^{2} false)

, one has

\begin{matrix} alignleft \\ align-1 & align-2 \int_{B_{R_{1}}} |F (ū (y))| [\int_{ℝ^{2} \ B_{R_{1} + ε^{- 1}}} \frac{A_{α}}{| x - y |^{2 - α}} | f (v_{n} (x)) | | φ (x) | d x \\ align-1 \leq & align-2 \frac{ε^{(2 - α) / 3} A_{α}}{A_{2 (1 + α) / 3}} \int_{B} \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…) is assumed in Lemma 2.1 and used in the proof (3.81), 32 we can use the following inequality to show (3.81) instead of (2.7),…”

Section: Preliminariesmentioning

confidence: 99%

See 4 more Smart Citations

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.

show abstract

“…The following assumptions on

f

was introduced by Qin and Tang, 32 taking advantage of which existence of nontrivial solution for () was proved there.

false| f false(t false) false| = o false(false| t {false|}^{α false/ 2} false)

false| t false| \to 0

; There exist

M_{0} > 0

and

t_{0} > 0

such that

F (t) \leq M_{0} | f (t) |, \forall | t | \geq t_{0};

{lim inf}_{t \to \infty} \frac{f false(t false)}{e^{β_{0} t^{2}}} = κ > \frac{\sqrt{α false(1 + α false)} false(2 + α false)}{\sqrt{2 π A_{α}} ρ^{1 + α false/ 2}} e^{4 false(2 + α false) π {false(1 + ρ false)}^{2} {scriptB}_{0}^{2} false/ false(2 + ρ false)}

, where

ρ \in false(0,4 false/ false‖ V {false‖}_{\infty} false)

satisfies

2 false(2 + α false) π ρ^{2} {scriptB}_{0}^{2} < 1

. …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The following assumptions on 𝑓 was introduced by Qin and Tang, 32 taking advantage of which existence of nontrivial solution for (1.1) was proved there.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Remark In the marginal case, that is

s = 1

in (), the following inequality does not hold, but it can be recovered by replacing the Lebesgue spaces by the weak Lebesgue spaces,

\int_{ℝ^{2}} false(I_{false(2 + α false) false/ 2} * g false) h normald x \leq {scriptC}_{1} false‖ g {false‖}_{1} false‖ h {false‖}_{4 false/ false(2 + α false)},

where

{scriptC}_{1} : = scriptC false(false(2 - α false) false/ 2,1, 4 false/ false(2 + α false) false)

. Though () is assumed in Lemma 2.1 and used in the proof (3.81), 32 we can use the following inequality to show (3.81) instead of (),

\int_{ℝ^{2}} false(I_{2 false(1 + α false) false/ 3} * g false) h normald x \leq {scriptC}_{2} false‖ g {false‖}_{6 false/ false(4 + α false)} false‖ h {false‖}_{6 false/ false(4 + α false)},

where

{scriptC}_{2} : = scriptC false(2 false(2 - α false) false/ 3,6 false/ false(4 + α false), 6 false/ false(4 + α false) false)

. Indeed, for any

φ \in C_{0}^{\infty} false(ℝ^{2} false)

, one has

\begin{matrix} alignleft \\ align-1 & align-2 \int_{B_{R_{1}}} |F (ū (y))| [ \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…Though () is assumed in Lemma 2.1 and used in the proof (3.81), 32 we can use the following inequality to show (3.81) instead of (),

\int_{ℝ^{2}} false(I_{2 false(1 + α false) false/ 3} * g false) h normald x \leq {scriptC}_{2} false‖ g {false‖}_{6 false/ false(4 + α false)} false‖ h {false‖}_{6 false/ false(4 + α false)},

where

{scriptC}_{2} : = scriptC false(2 false(2 - α false) false/ 3,6 false/ false(4 + α false), 6 false/ false(4 + α false) false)

. Indeed, for any

φ \in C_{0}^{\infty} false(ℝ^{2} false)

, one has

\begin{matrix} alignleft \\ align-1 & align-2 \int_{B_{R_{1}}} |F (ū (y))| [\int_{ℝ^{2} \ B_{R_{1} + ε^{- 1}}} \frac{A_{α}}{| x - y |^{2 - α}} | f (v_{n} (x)) | | φ (x) | d x \\ align-1 \leq & align-2 \frac{ε^{(2 - α) / 3} A_{α}}{A_{2 (1 + α) / 3}} \int_{B} \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…) is assumed in Lemma 2.1 and used in the proof (3.81), 32 we can use the following inequality to show (3.81) instead of (2.7),…”

Section: Preliminariesmentioning

confidence: 99%

See 3 more Smart Citations

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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show abstract

Existence of ground state solutions for a biharmonic Choquard equation with critical exponential growth in ℝ⁴

Chen,

Li,

Wang

2024

Math Methods in App Sciences

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In this paper, we study the following singularly perturbed biharmonic Choquard equation: where is a parameter, , ∗ is the convolution product in , and is a continuous real function. is the primitive function of , and has critical exponential growth in the sense of the Adams inequality. By using variational methods, we establish the existence of ground state solutions when small enough.

show abstract

On a planar Choquard equation involving exponential critical growth

Carvalho

Medeiros

Ribeiro

2021

Z. Angew. Math. Phys.

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On the planar Choquard equation with indefinite potential and critical exponential growth

Cited by 44 publications

References 34 publications

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Existence of ground state solutions for a biharmonic Choquard equation with critical exponential growth in ℝ⁴

On a planar Choquard equation involving exponential critical growth

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On the planar Choquard equation with indefinite potential and critical exponential growth

Cited by 44 publications

References 34 publications

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Existence of ground state solutions for a biharmonic Choquard equation with critical exponential growth in ℝ4

On a planar Choquard equation involving exponential critical growth

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Existence of ground state solutions for a biharmonic Choquard equation with critical exponential growth in ℝ⁴