Concentration behavior of ground states for a generalized quasilinear Choquard equation

Yang, Xianyong; Tang, Xianhua; Gu, Guangze

doi:10.1002/mma.6138

Cited by 13 publications

(2 citation statements)

References 44 publications

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“…When

g false(x, ψ false) = false(I_{α} * false| ψ {false|}^{p} false) false| ψ {false|}^{p - 2} ψ

, a ground state solution was obtained in Chen et al, 14 and then, existence of positive solution of () was gained in Chen and Wu 15 by a variational argument. When

I_{α} = false| x {false|}^{- μ}

, by using perturbation method, the existence of positive solutions, negative solutions, and high‐energy solutions were obtained in Yang et al 16 Yanget al 17 discussed the concentration behavior of ground states via dual approach. Zhang and Wu 18 considered a quasilinear Choquard equation with critical exponent and researched the existence, multiplicity, and concentration of positive solutions for the problem by a dual approach.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof Let

v \in H_{r}^{1} false(ℝ^{N} false) \ false{0 false}

be fixed. For any

λ \in 𝕀 = ([], \frac{1}{2}, 1)

, one has

I_{λ} (v) \leq I_{\frac{1}{2}} (v) = \frac{1}{2} \int_{ℝ^{N}} | \nabla v |^{2} + \frac{1}{4} \int_{ℝ^{N}} V (x) [v^{2} + f^{2} (v)] - \frac{1}{4} \int_{ℝ^{N}} (I_{α} * G (f (v))) G (f (v)) .

As Chen and Wu 15 and Miyagaki and Soares, 24 we consider

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

such that 0 ≤ φ ≤ 1 and

φ (x) = \{\begin{array}{ll} 1, & if false| x false| \leq 1, \\ 0, & if false| x false| \geq 2 . \end{array}

According to Yang et al, 17 from ( g 3 ) and Lemma …”

Section: Proof Of Theorem 11mentioning

confidence: 99%

See 1 more Smart Citation

Ground state solutions for a generalized quasilinear Choquard equation

Zhang

2021

Math Methods in App Sciences

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This paper is concerned with the following quasilinear Choquard equation: −Δu+V(x)u−uΔ(u2)=(Iα∗G(u))g(u),x∈ℝN, where N ≥ 3, 0 < α < N, V:0.1emℝN→ℝ is radial potential, Gfalse(tfalse)=∫0tgfalse(sfalse)ds, and Iα is a Riesz potential. Using the variational method we establish the existence of ground state solutions under appropriate assumptions, that is, nontrivial solution with least possible energy.

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“…When

g false(x, ψ false) = false(I_{α} * false| ψ {false|}^{p} false) false| ψ {false|}^{p - 2} ψ

, a ground state solution was obtained in Chen et al, 14 and then, existence of positive solution of () was gained in Chen and Wu 15 by a variational argument. When