Ground states and high energy solutions of the planar Schrödinger–Poisson system

Du, Miao; Weth, Tobias

doi:10.1088/1361-6544/aa7eac

Cited by 64 publications

(67 citation statements)

References 26 publications

(45 reference statements)

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“…Note that

⟨ {normalΨ}^{'} false(s_{n}, v_{n} false), false(τ, w false) ⟩ = ⟨ I^{'} false(h false(s_{n}, v_{n} false) false), h false(s_{n}, w false) ⟩ + scriptJ false(h false(s_{n}, v_{n} false) false) τ, \forall false(τ, w false) \in \overset{H}{true} .

Let

u_{n} : = h false(s_{n}, v_{n} false)

. As the proof in [, Lemma 3.2], we can deduce that

false{u_{n} false}

satisfies .…”

Section: Preliminaries and Variational Settingmentioning

confidence: 99%

“…Besides, (F6) and (F7) are weaker than the following assumptions which are easier to state and verify: (F8)there exist constants

α_{2} > 0

and

p_{3}, p_{4} \in false[2, 3 false)

such that

- α_{2} (| u |^{p_{3}} + | u |^{p_{4}}) \leq f (u) u - 3 F (u) \leq 0, \forall u \in R .

(F9)the function

\frac{\frac{5}{2} f false(u false) u - 3 F false(u false)}{false| u {false|}^{9 false/ 5} u}

is nondecreasing on both

false(- \infty, 0 false)

and

false(0, \infty false)

. We can verify easily that F(8) implies (F6) and F(9) implies (F7). In, Du and Weth proved the existence of ground states and high energy solutions for the logarithmic Choquard equations in the two‐dimensional case with the nonlinearity

f false(u false) = false| u {false|}^{p - 2} u

, which satisfies (F8) when

2 \leq p < 3

. Cingolani and Weth also obtained the existence of nontrivial solutions for the logarithmic Choquard equations in

{double-struckR}^{2}

with the nonlinearity

f false(u false) = b false| u {false|}^{p - 2} u

, which satisfies (AR) when

p > 4

.…”

Section: Introductionmentioning

confidence: 99%

“…When

f false(u false) = b false| u {false|}^{p - 2} u

, Cingolani and Weth proved the existence of solutions of system for

p \geq 4

. Later, Du and Weth covered the range

2 \leq p \leq 4

by applying a new variational approach. By using strict rearrangement inequalities, Stubbe has proved the existence and uniqueness of the ground state solution for planar logarithmic Choquard equations.…”

Section: Introductionmentioning

confidence: 99%

“…Cingolani and Weth also obtained the existence of nontrivial solutions for the logarithmic Choquard equations in

{double-struckR}^{2}

with the nonlinearity

f false(u false) = b false| u {false|}^{p - 2} u

, which satisfies (AR) when

p > 4

. So our work can be seen as a generation and improvement of in the three‐dimensional case.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Ground state solutions to logarithmic Choquard equations in R3

2020

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“…Note that

⟨ {normalΨ}^{'} false(s_{n}, v_{n} false), false(τ, w false) ⟩ = ⟨ I^{'} false(h false(s_{n}, v_{n} false) false), h false(s_{n}, w false) ⟩ + scriptJ false(h false(s_{n}, v_{n} false) false) τ, \forall false(τ, w false) \in \overset{H}{true} .

Let

u_{n} : = h false(s_{n}, v_{n} false)

. As the proof in [, Lemma 3.2], we can deduce that

false{u_{n} false}

satisfies .…”

Section: Preliminaries and Variational Settingmentioning

confidence: 99%

“…Besides, (F6) and (F7) are weaker than the following assumptions which are easier to state and verify: (F8)there exist constants

α_{2} > 0

and

p_{3}, p_{4} \in false[2, 3 false)

such that

- α_{2} (| u |^{p_{3}} + | u |^{p_{4}}) \leq f (u) u - 3 F (u) \leq 0, \forall u \in R .

(F9)the function

\frac{\frac{5}{2} f false(u false) u - 3 F false(u false)}{false| u {false|}^{9 false/ 5} u}

is nondecreasing on both

false(- \infty, 0 false)

and

false(0, \infty false)

f false(u false) = false| u {false|}^{p - 2} u

, which satisfies (F8) when

2 \leq p < 3

. Cingolani and Weth also obtained the existence of nontrivial solutions for the logarithmic Choquard equations in

{double-struckR}^{2}

with the nonlinearity

f false(u false) = b false| u {false|}^{p - 2} u

, which satisfies (AR) when

p > 4

.…”

Section: Introductionmentioning

confidence: 99%

“…When

f false(u false) = b false| u {false|}^{p - 2} u

, Cingolani and Weth proved the existence of solutions of system for

p \geq 4

. Later, Du and Weth covered the range

2 \leq p \leq 4

Section: Introductionmentioning

confidence: 99%

“…Cingolani and Weth also obtained the existence of nontrivial solutions for the logarithmic Choquard equations in

{double-struckR}^{2}

with the nonlinearity

f false(u false) = b false| u {false|}^{p - 2} u

, which satisfies (AR) when

p > 4

. So our work can be seen as a generation and improvement of in the three‐dimensional case.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Ground state solutions to logarithmic Choquard equations in R3

2020

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“…A key difficulty in order to construct solutions variationally is that the functional I is not well-defined on the natural Sobolev space H 1 (R N ). For the two-dimensional logarithmic Newton kernel, variational methods have been applied successfully in the framework of the Hilbert space of functions u : R N → R such that (1.2)ˆR 2 |∇u| 2 + 1 + V − |u| 2 < +∞ (see [6,8,10,11,26,27]). A delicate point in this approach is that although the Choquard equation (C) and the associated energy functional I are invariant under translations of the Euclidean space R N , the Hilbert space naturally defined by the quadratic form (1.2) is not any more invariant under translation.…”

Section: Introductionmentioning

confidence: 99%

Groundstates of the Choquard equations with a sign-changing self-interaction potential

Battaglia

Schaftingen

2018

Z. Angew. Math. Phys.

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We consider a nonlinear Choquard equationwhen the self-interaction potential V is unbounded from below. Under some assumptions on V and on p, covering p = 2 and V being the one-or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution u ∈ H 1 (R N ) \ {0} by solving a relaxed problem by a constrained minimization and then proving the convergence of the relaxed solutions to a groundstate of the original equation.

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