Nehari‐type ground state solutions for a Choquard equation with lower critical exponent and local nonlinear perturbation

Tang, Xianhua; Wei, Jiuyang; Chen, Sitong

doi:10.1002/mma.6404

Cited by 17 publications

(7 citation statements)

References 36 publications

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“…By a careful analysis as in Tang et al, 16 we obtain the upper bound of c λ , which enables us to prove the following result.…”

Section: Introductionmentioning

confidence: 72%

“…Based on the celebrated Hardy-type inequality and some new inequalities motivated by Tang et al 16,21 and by a non-Nehari approach introduced in Tang 22 and an accurate estimate of the upper bound of the least energy, we are able to establish the existence of a ground state to Equation (1.1), that is, a solution u ∈  with E 𝜆 (u) = c 𝜆 .…”

Section: (A1)mentioning

confidence: 99%

“…see previous studies [7][8][9][10][11][12][13][14][15][16][17] for example. We now turn to the Choquard-type equation with potential well.…”

Section: Introductionmentioning

confidence: 95%

“…They also showed that () admits no nontrivial solutions if q ≤ ( N + α )/ N or q ≥( N + α )/( N − 2), where the constants ( N + α )/ N and ( N + α )/( N − 2) are called the lower and upper critical exponents, respectively, by the celebrated Hardy type inequality 6 (see also Proposition 2.1). Later, great attention has been paid to the general Choquard‐type equation

({, \begin{matrix} left center left \\ array - Δ u + V (x) u = k (I_{α} * F (u)) f (u) + g (u), x \in ℝ^{N}, \\ array u \in H^{1} (ℝ^{N}); \end{matrix})

see previous studies 7–17 for example.…”

Section: Introductionmentioning

confidence: 99%

“…Define the functional

E_{λ} : X \mapsto ℝ

corresponding to Equation () by

E_{λ} (u) : = \frac{1}{2} \int_{ℝ^{N}} (| \nabla u |^{2} + (1 + λ b (x)) u^{2}) d x - \frac{1}{2} \int_{ℝ^{N}} (I_{α} * F (u)) F (u) d x - \frac{1}{2^{*}} \int_{ℝ^{N}} u^{2^{*}} d x .

From ( F 1) − ( F 3) and the Hardy‐type inequality (see Lieb and Loss, 6 Proposition 2.1), we know that E λ ( u ) is well‐defined and

E_{λ} \in C^{1} false(X, ℝ false)

. Denote the Nehari manifold by

N : = {u \in X \ {0} : ⟨ E_{λ}^{'} (u), u ⟩ = 0},

and set

c_{λ} = \inf_{u \in N} E_{λ} (u) .

Based on the celebrated Hardy‐type inequality and some new inequalities motivated by Tang et al 16,21 and by a non‐Nehari approach introduced in Tang 22 and an accurate estimate of the upper bound of the least energy, we are able to establish the existence of a ground state to Equation...…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Existence and concentration of ground states to a critical Choquard‐type equation involving steep potential well

2021

Math Methods in App Sciences

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“…By a careful analysis as in Tang et al, 16 we obtain the upper bound of c λ , which enables us to prove the following result.…”

Section: Introductionmentioning

confidence: 72%

Section: (A1)mentioning

confidence: 99%

“…see previous studies [7][8][9][10][11][12][13][14][15][16][17] for example. We now turn to the Choquard-type equation with potential well.…”

Section: Introductionmentioning

confidence: 95%

({, \begin{matrix} left center left \\ array - Δ u + V (x) u = k (I_{α} * F (u)) f (u) + g (u), x \in ℝ^{N}, \\ array u \in H^{1} (ℝ^{N}); \end{matrix})

see previous studies 7–17 for example.…”

Section: Introductionmentioning

confidence: 99%

“…Define the functional

E_{λ} : X \mapsto ℝ

corresponding to Equation () by

E_{λ} (u) : = \frac{1}{2} \int_{ℝ^{N}} (| \nabla u |^{2} + (1 + λ b (x)) u^{2}) d x - \frac{1}{2} \int_{ℝ^{N}} (I_{α} * F (u)) F (u) d x - \frac{1}{2^{*}} \int_{ℝ^{N}} u^{2^{*}} d x .

From ( F 1) − ( F 3) and the Hardy‐type inequality (see Lieb and Loss, 6 Proposition 2.1), we know that E λ ( u ) is well‐defined and

E_{λ} \in C^{1} false(X, ℝ false)

. Denote the Nehari manifold by

N : = {u \in X \ {0} : ⟨ E_{λ}^{'} (u), u ⟩ = 0},

and set

c_{λ} = \inf_{u \in N} E_{λ} (u) .

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Existence and concentration of ground states to a critical Choquard‐type equation involving steep potential well

2021

Math Methods in App Sciences

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Ground state solutions of Pohožaev type for Kirchhoff‐type problems with general convolution nonlinearity and variable potential

Zhang

Xie

Jiang

2022

Math Methods in App Sciences

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This paper is devoted to dealing with the following nonlinear Kirchhoff‐type problem with general convolution nonlinearity and variable potential: {left leftarray−(a+b∫ℝ3|∇u|2dx)Δu+V(x)u=(Iα∗F(u))f(u),inℝ3,arrayu∈H1(ℝ3),$$ \left\{\begin{array}{l}-\left(a&amp;amp;amp;amp;amp;#x0002B;b{\int}_{{\mathbb{R}}&amp;amp;amp;amp;amp;#x0005E;3}{\left&amp;amp;amp;amp;amp;#x0007C;\nabla u\right&amp;amp;amp;amp;amp;#x0007C;}&amp;amp;amp;amp;amp;#x0005E;2 dx\right)\Delta u&amp;amp;amp;amp;amp;#x0002B;V(x)u&amp;amp;amp;amp;amp;#x0003D;\left({I}_{\alpha}\ast F(u)\right)f(u),\kern0.30em \mathrm{in}\kern0.50em {\mathbb{R}}&amp;amp;amp;amp;amp;#x0005E;3,\\ {}u\in {H}&amp;amp;amp;amp;amp;#x0005E;1\left({\mathbb{R}}&amp;amp;amp;amp;amp;#x0005E;3\right),\end{array}\right. $$ where a>0$$ a&amp;amp;amp;amp;gt;0 $$, b≥0$$ b\ge 0 $$ are constants; V∈C1false(ℝ3,false[0,+∞false)false)$$ V\in {C}&amp;amp;amp;amp;amp;#x0005E;1\left({\mathbb{R}}&amp;amp;amp;amp;amp;#x0005E;3,\right[0,&amp;amp;amp;amp;amp;#x0002B;\infty \left)\right) $$; f∈Cfalse(ℝ,ℝfalse)$$ f\in C\left(\mathbb{R},\mathbb{R}\right) $$, Ffalse(tfalse)=∫0tffalse(sfalse)ds$$ F(t)&amp;amp;amp;amp;amp;#x0003D;{\int}_0&amp;amp;amp;amp;amp;#x0005E;tf(s) ds $$; and Iα:ℝ3→ℝ$$ {I}_{\alpha }:{\mathbb{R}}&amp;amp;amp;amp;amp;#x0005E;3\to \mathbb{R} $$ is the Riesz potential, α∈false(0,3false)$$ \alpha \in \left(0,3\right) $$. By applying some new analytical tricks introduced by Tang and Chen, the existence results of ground state solutions of Pohožaev type for the above Kirchhoff type problem are obtained under some mild assumptions on V$$ V $$ and the general “Berestycki‐Lions assumptions” on the nonlinearity f$$ f $$. Our results generalize and improve the ones obtained by Chen and Liu and other related results in the literature.

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Semiclassical Ground State Solutions for a Class of Kirchhoff-Type Problem with Convolution Nonlinearity

Tang

Zhang

2022

J Geom Anal

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Nehari‐type ground state solutions for a Choquard equation with lower critical exponent and local nonlinear perturbation

Abstract: This paper deals with the following Choquard equation with a local nonlinear perturbation: −normalΔu+u=()Iα∗false|ufalse|αN+1false|ufalse|αN−1u+λfalse|ufalse|p−2u,x∈double-struckRN;u∈H1false(double-struckRNfalse), where N ≥ 1, α∈(0,N), λ>0, 2 Show more

Cited by 17 publications

References 36 publications

Existence and concentration of ground states to a critical Choquard‐type equation involving steep potential well

Existence and concentration of ground states to a critical Choquard‐type equation involving steep potential well

Ground state solutions of Pohožaev type for Kirchhoff‐type problems with general convolution nonlinearity and variable potential

Semiclassical Ground State Solutions for a Class of Kirchhoff-Type Problem with Convolution Nonlinearity

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