Multiplicity of solutions of the bi-harmonic Schrödinger equation with critical growth

Zhang, Jian; Lou, Zhenluo; Ji, Yanju; Shao, Wei

doi:10.1007/s00033-018-0940-y

Cited by 10 publications

(4 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, the nonlinear biharmonic equation has been extensively studied. We refer readers to existing studies() for the subcritical case and other studies() for the critical case. Now let us briefly comment some known results of them.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Pimenta and Soares in a previous study obtained the existence and concentration behavior of solutions of the biharmonic Equation with γ = 0, considering, respectively, a global and a local condition on the potential V and a subcritical power‐type nonlinearity f . Zhang et al studied multiplicity of solutions of the critical biharmonic equation

ϵ^{4} {normalΔ}^{2} u + V (x) u = h (x) f (u) + g (x) u^{\frac{N + 4}{N - 4}} in {double-struckR}^{N} .

When ϵ > 0 is small, they established the relationship between the number of solutions and the profile of V , h , g . Also, without the restriction on ϵ , they obtained a multiplicity result.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Construction of high‐energy solutions for the supercritical biharmonic Schrödinger equation

Gan

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

We consider the problem Δ 2 u = V(x)u p + in R N with u, Δu → 0 as |x| → + ∞, where p = N+4 N−4 , N ≥ 5, V is a positive continuous potential. Our aim is to construct high-energy solutions for this equation by applying the finite-dimensional reduction method and the penalization method. KEYWORDSbiharmonic equation, reduction method, supercriticalWhen > 0 is small, they established the relationship between the number of solutions and the profile of V, h, g. Also, without the restriction on , they obtained a multiplicity result.Recently, Guo, Tan, and Wang 7 concerned with the existence of least energy solutions for the following biharmonic equations:How to cite this article: Gan L. Construction of high-energy solutions for the supercritical biharmonic Schrödinger equation. Math Meth Appl Sci. 2019;42:883-891. https://doi.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

ϵ^{4} {normalΔ}^{2} u + V (x) u = h (x) f (u) + g (x) u^{\frac{N + 4}{N - 4}} in {double-struckR}^{N} .

When ϵ > 0 is small, they established the relationship between the number of solutions and the profile of V , h , g . Also, without the restriction on ϵ , they obtained a multiplicity result.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Construction of high‐energy solutions for the supercritical biharmonic Schrödinger equation

Gan

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…(6), respectively. For other related work, we refer reader to [17,18,35,36]. In comparison, model (2) was obtained in the case where the vertical motion of the fluid was neglected.…”

Section: Introductionmentioning

confidence: 99%

Solvability and asymptotic properties for an elliptic geophysical fluid flows model in a planar exterior domain

Zhang

Liu

et al. 2021

NAMC

View full text Add to dashboard Cite

In this paper, we study the solvability and asymptotic properties of a recently derived gyre model of nonlinear elliptic Schrödinger equation arising from the geophysical fluid flows. The existence theorems and the asymptotic properties for radial positive solutions are established due to space theory and analytical techniques, some special cases and specific examples are also given to describe the applicability of model in gyres of geophysical fluid flows.

show abstract

“…In [24], Niu, Tang and Wang studied the asymptotic behavior of ground state solutions for a nonlinear biharmonic equation on R N . For more results related to nonlinear biharmonic equations on the Euclidean space, one can refer to [13,26,27,32,33] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and convergence of solutions for nonlinear biharmonic equations on graphs

Han

Shao

Zhao

2020

Journal of Differential Equations

View full text Add to dashboard Cite

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph G = (V, E), which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equationUnder some suitable assumptions, we prove that for any λ > 1 and p > 2, the equation admits a ground state solution u λ . Moreover, we prove that as λ → +∞, the solutions u λ converge to a solution of the equationwhere Ω = {x ∈ V : a(x) = 0} is the potential well and ∂Ω denotes the the boundary of Ω.

show abstract

Multiplicity of solutions of the bi-harmonic Schrödinger equation with critical growth

Cited by 10 publications

References 37 publications

Construction of high‐energy solutions for the supercritical biharmonic Schrödinger equation

Construction of high‐energy solutions for the supercritical biharmonic Schrödinger equation

Solvability and asymptotic properties for an elliptic geophysical fluid flows model in a planar exterior domain

Existence and convergence of solutions for nonlinear biharmonic equations on graphs

Contact Info

Product

Resources

About