Multiple solutions for the Schrödinger equations with sign-changing potential and Hartree nonlinearity

Che, Guofeng; Chen, Haibo

doi:10.1016/j.aml.2017.12.014

Cited by 14 publications

(10 citation statements)

References 8 publications

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“…Let < 1 − 512 N 2 (N−4) 2 and Ξ ∶= 1 2 − 256 N 2 (N−4) 2 − 2 . According to (17) and Lemma 3, we get…”

Section: Stepmentioning

confidence: 99%

“…is superlinear at origin, K(x)|u| q − 2 u is sublinear at origin, so h (x, ·)is sublinear at origin; (2) since both f(·) and K(x)|u| q − 2 u are sublinear at infinity, so h (x, ·) is sublinear at infinity. For different kinds of nonlinearities, we refer to for example [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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The existence of nontrivial solution for biharmonic equation with sign‐changing potential

Chen

2018

Math Methods in App Sciences

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In this paper, we study the following biharmonic equation: {arraynormalΔ2u+Vλ(x)u=α(x)f(u)+νK(x)|ufalse|q−2uindouble-struckRN,arrayu∈H2(double-struckRN), where N⩾5,ν ∈ (0,ν0],1 < q < 2,Δ2u = Δ(Δu) and Vλ(x) = λa(x) − b(x) with λ > 0. Firstly, we prove the bipolar Rellich inequality. Secondly, by using bipolar Rellich inequality, Gigliardo‐Nirenberg inequality, and Ekeland variational principle, we prove the existence of nontrivial solution for problem Bν.

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“…Let < 1 − 512 N 2 (N−4) 2 and Ξ ∶= 1 2 − 256 N 2 (N−4) 2 − 2 . According to (17) and Lemma 3, we get…”

Section: Stepmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The existence of nontrivial solution for biharmonic equation with sign‐changing potential

Chen

2018

Math Methods in App Sciences

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“…Recently, Sun, Wu, and Feng used the argument of the filtration of Nehari manifold to show the presence of multiple positive solutions for

2 < p < 6

and ground state solutions for

\frac{1 + \sqrt{73}}{3} < p \leq 4

with weight functions

V

and

K

. We refer to previous studies for more related results for .…”

Section: Introductionmentioning

confidence: 99%

Multibump solutions for nonlinear Schrödinger‐Poisson systems

Chen

2020

Math Meth Appl Sci

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In this paper, we study the following Schrödinger‐Poisson equations: −ε2normalΔu+Vfalse(xfalse)u+Kfalse(xfalse)ϕu=false|ufalse|p−2u,x∈double-struckR3,−ε2normalΔϕ=Kfalse(xfalse)u2,x∈double-struckR3, where p∈false(4,6false), ε>0 is a parameter and V and K satisfy the critical frequency conditions. By using variational methods and penalization arguments, we show the existence of multibump solutions for the above system. Furthermore, the heights of these bumps are different order.

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“…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%

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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Multiple solutions for the Schrödinger equations with sign-changing potential and Hartree nonlinearity

Cited by 14 publications

References 8 publications

The existence of nontrivial solution for biharmonic equation with sign‐changing potential

The existence of nontrivial solution for biharmonic equation with sign‐changing potential

Multibump solutions for nonlinear Schrödinger‐Poisson systems

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

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