The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation

Gao, Fashun; Yang, Minbo

doi:10.1007/s11425-016-9067-5

Cited by 233 publications

(166 citation statements)

References 42 publications

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“…It is easy to see that {u n } is uniformly bounded in D 1,2 (R 3 ), by using (5) and the definition of Coulomb-Sobolev space, we obtain u n ∈  ,2 * (R 3 ). Applying Lemma 1, we have…”

Section: Lemmamentioning

confidence: 94%

“…Let u 0 >0 be a ground state solution of (

P_{6}

) (see previous work). Then,

\int_{{double-struckR}^{3}} | \nabla u_{0} {false|}^{2} d x = \int_{{double-struckR}^{3}} \int_{{double-struckR}^{3}} \frac{| u_{0} (y) {false|}^{2_{β}^{*}} | u_{0} (x) {false|}^{2_{β}^{*}}}{| x - y {false|}^{3 - β}} d x d y = S_{h, β}^{\frac{3 + β}{2 + β}},

and

I_{0} (u_{0}) = \frac{2 + β}{2 (3 + β)} S_{h, β}^{\frac{3 + β}{2 + β}} .

Moreover, u 0 is a mountain pass type solution.…”

Section: The Proof Of Theoremmentioning

confidence: 99%

“…In an existing study, 4 Moroz and Van Schaftingen considered Equation (1) with lower critical exponent N+ N if the potential 1 − V(x) should not decay to zero at infinity faster than the inverse of |x| 2 . In another research, 5 Gao and Yang investigated Equation 1 with upper critical exponent N+ N−2 in a bounded domain with Lipschitz boundary, and established the existence, multiplicity, and nonexistence of nontrivial solutions to the critical Choquard equation. For details and recent works, we refer to other literature [6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent

Wang

Han

2019

Math Methods in App Sciences

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In this paper, we consider the following Schrödinger‐Poisson system: −normalΔu+λϕfalse|ufalse|2α∗−2u=()∫R3false|ufalse|2β∗false|x−yfalse|3−βnormaldyfalse|ufalse|2β∗−2u,in5ptR3,false(−normalΔfalse)α2ϕ=Aα−1false|ufalse|2α∗,in5ptR3, where parameters α,β∈(0,3),λ>0, Aα=normalΓfalse(3−α2false)2απ32normalΓfalse(α2false), 2α∗=3+α, and 2β∗=3+β are the Hardy‐Littlewood‐Sobolev critical exponents. For α<β and λ>0, we prove the existence of nonnegative groundstate solution to above system. Moreover, applying Moser iteration scheme and Kelvin transformation, we show the behavior of nonnegative groundstate solution at infinity. For β<α and λ>0 small, we apply a perturbation method to study the existence of nonnegative solution. For β<α and λ is a particular value, we show the existence of infinitely many solutions to above system.

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“…It is easy to see that {u n } is uniformly bounded in D 1,2 (R 3 ), by using (5) and the definition of Coulomb-Sobolev space, we obtain u n ∈  ,2 * (R 3 ). Applying Lemma 1, we have…”

Section: Lemmamentioning

confidence: 94%

“…Let u 0 >0 be a ground state solution of (

P_{6}

) (see previous work). Then,

\int_{{double-struckR}^{3}} | \nabla u_{0} {false|}^{2} d x = \int_{{double-struckR}^{3}} \int_{{double-struckR}^{3}} \frac{| u_{0} (y) {false|}^{2_{β}^{*}} | u_{0} (x) {false|}^{2_{β}^{*}}}{| x - y {false|}^{3 - β}} d x d y = S_{h, β}^{\frac{3 + β}{2 + β}},

and

I_{0} (u_{0}) = \frac{2 + β}{2 (3 + β)} S_{h, β}^{\frac{3 + β}{2 + β}} .

Moreover, u 0 is a mountain pass type solution.…”

Section: The Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent

Wang

Han

2019

Math Methods in App Sciences

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“…In particularly, more and more authors have studied the critical problems [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. In the case that ( ) = ( ) = 1, Gao and Yang [22] considered the existence and multiplicity of (1) with upper critical exponent 2 * = (2 − )/( − 2).…”

Section: Introductionmentioning

confidence: 99%

The Choquard Equation with Weighted Terms and Sobolev-Hardy Exponent

Sang

Luo

Wang

2018

Journal of Function Spaces

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“…et al obtained the existence of a non-trivial solution via penalization method of the problem−∆u + V (x)u = |x| −µ * F (u) f (u), in R n ,where 0 < µ < n, n = 3, V is a continuous real valued function and F is the primitive of f . Gao and Yang[18] studied the following Brezis-Nirenberg type problem−∆u = λu + Ω |u(y)| 2 * µ |x − y| µ dy |u(x)| 2 * µ −2 u(x) in Ω, u = 0 on ∂Ω,…”

mentioning

confidence: 99%

Unbalanced (p,2)-fractional problems with critical growth

Kumar

Sreenadh

2021

Journal of Mathematical Analysis and Applications

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We study the existence, multiplicity and regularity results of non-negative solutions of following doubly nonlocal problem:where Ω ⊂ R n is a bounded domain with C 2 boundary ∂Ω, 0 < s 2 < s 1 < 1, n > 2s 1 , 1 < q < p < 2, 1 < r ≤ 2 * µ with 2 * µ = 2n−µ n−2s1 , λ, β > 0 and a ∈ L d d−q (Ω), for some q < d < 2 * s1 := 2n n−2s1 , is a sign changing function. We prove that each nonnegative weak solution of (P λ ) is bounded. Furthermore, we obtain some existence and multiplicity results using Nehari manifold method.

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The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation

Cited by 233 publications

References 42 publications

Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent

Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent

The Choquard Equation with Weighted Terms and Sobolev-Hardy Exponent

Unbalanced (p,2)-fractional problems with critical growth

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