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GROUND STATE SOLUTIONS OF NEHARI-POHOZAEV TYPE FOR A FR ACTIONAL SCHRODINGER-POISSON SYSTEM WITH critical growth

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Acta Mathematica Scientia 2020年第4期40卷 1064-1080页

作者：黄文涛王莉 School of Basic Science East China Jiaotong UniversityNanchang 330013China

We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 03 and 2s=6/3-2s is the critical Sobolev exponent in 1R3.Under some more general assumptions on f,we prove that(0.1)admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.

关键词： fractional Schrodinger-Poisson system Nehari-Pohozaev manifold ground state solutions critical growth

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF CONCAVE-CONVEX ELLIPTIC EQUATIONS WITH critical growth

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Acta Mathematica Scientia 2018年第2期38卷 497-518页

作者：廖家锋蒲洋唐春雷 School of Mathematics and Statistics Southwest University Chongqing 400715 China School of Mathematics and Information China West Normal University Nanchong 637002 China

In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g（x）｜u｜2＊-2u＋λf（x）｜u｜q-2u,x∈Ω u=0,x∈δΩ where Ω RN（N ≥ 3） is an open bounded domain with smooth boundary, 1 〈 q 〈 2, λ 〉 0. 2＊= 2N/N-2 is the critical Sobolev exponent, f ∈L2＊/2N/N-2 is nonzero and nonnegative, and g E （Ω） is a positive function with k local maximum points. By the Nehari method and variational method, k ＋ 1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75（2012） 2660-26711.

关键词： Semilinear elliptic equations critical growth positive solutions Nehari method variational method

EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING critical growth

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Acta Mathematica Scientia 2017年第5期37卷 1348-1360页

作者：章国庆张卫国刘三阳 College of Sciences University of Shanghai for Science and TechnologyShanghai 200093China College of Mathematics and Statistics Xidian UniversityXi'an 710071China

In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|^(N-2) u + f(x, u), x ∈ ?,u ∈ W_0^(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R^N(N > 2), f(x, u) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ > λ_1, λ = λ_?(? = 2, 3, · · ·), and λ_? is the eigenvalues of the operator(-?_N, W_0^(1,N)(?)),which is defined by the Z_2-cohomological index.

关键词： nonstandard linking theorem N-Laplacian equation critical growth

Positive Solutions for a Class of Fractional p-Laplacian Equation with critical Sobolev Exponent and Decaying Potentials

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Acta Mathematicae Applicatae Sinica 2022年第2期38卷 463-483页

作者： Na LI Xiao-ming HE College of Science Minzu University of ChinaBeijing 100081China

In this paper,we study the existence of positive solution for the p-Laplacian equations with frac-tional critical nonlinearity{-Δ)_(p)^(s)u+V(x)|u|^(p-2)u=K(x)f(u)+P(x)|u|p_(s)^(*)-^(2)u,x∈R^(N),u∈Ds,p(RN),where s∈(0,1),p_(s)^(*)=Np/N-sp,N>sp,p>1 and V(x),K(x)are positive continuous functions which vanish at infinity,f is a function with a subcritical growth,and P(x)is bounded,nonnegative continuous function.By using variational method in the weighted spaces,we prove the above problem has at least one positive solution.

关键词： Fractional p-Laplacian Variational methods Mountain Pass Theorem critical growth Vanishing potential

ESTIMATES FOR EXTREMAL VALUES FOR A critical FRACTIONAL EQUATION WITH CONCAVE-CONVEX NONLINEARITIES

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Acta Mathematica Scientia 2022年第3期42卷 903-918页

作者：郝江浩张亚静 School of Mathematical Sciences Shanxi UniversityTaiyuan 030006China

In this paper we study the critical fractional equation with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the crit... 详细信息

关键词： Multiple positive solutions fractional Laplace problems critical growth

A Positive Solution for Some critical p-Laplacian Systems

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Acta Mathematica Sinica,English Series 2015年第3期31卷 479-500页

作者： Xiao Zhi WANG College of Science University of Shanghai for Science and Technology

This paper deals with the existence of a positive solution for two classes of critical quasilinear system. We prove these results by a variant of mountain pass lemma, combining two convergence theorems and two estimate results. Here we avoid the usual compactness arguments（e.g., Palais-Smale condition or Cerami condition） and reveal the potential of some energy level estimates for the existence of nontrivial solutions.

关键词： p-Laplacian system critical growth convergence theorem estimate result

Multiplicity of Solutions for Quasilinear Elliptic Systems with Singularity

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Acta Mathematicae Applicatae Sinica 2015年第1期31卷 277-286页

作者： Juan LI Yu-xia TONG Department of Mathematics Ningbo University College of Science Heibei United University

In this paper, we study the existence of multiple solutions for the following quasilinear ellipticsystem：-△pu-μ1︳u︳p-1u/︳x︳=α1up（t）-2/︳x︳t＋β1︳v︳β2︳u︳β1-2u,x∈Ω,-△qu-μ2︳v︳q-2v/︳x︳q=α2 vq（s）-2/︳x︳sv＋β2︳u︳β1︳v︳β2-2v,x∈Ω u（x）=v（x）=0 x∈Ω Multiplicity of solutions for the quasilinear problem is obtained via variational method.

关键词： quasilinear elliptic system singularity critical growth

Existence of Single-Peaked Solution of a Semilinear Elliptic Problem

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Wuhan University Journal of Natural Sciences 2000年第4期5卷 386-390页

作者： Zhong Xin-hua College of Mathematics and Computer Science,Wuhan University,Wuhan 430072,China College of Mathematics and Computer Science Wuhan UniversityWuhan 430072China

We discussed the Dirichlet problem of semilinear elliptic equation (P β,ε):β 2Δu=u p +εu,u>0, in Ω;u=0, on ?Ω, where Ω?R N (N≥4) is smooth and bounded domain, ,β,ε>0. We have proved that there exist positiveε 0 andε 1, such that when 0?ε?ε 0,β>√ε 1, (P β,ε) has a single-peaked solutionu β,ε; furthermore, |?u β0|2?0 in the sense of measure as ε→0 and β→0.

关键词： semilinear critical growth single-peaked solution

A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in R^n

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Journal of Partial Differential Equations 2012年第1期25卷 90-102页

作者： ZHAO Liang School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China and Key Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijing Normal UniversityBeijing 100875China

We establish sufficient conditions under which the quasilinear equation [GRAPHICS] has at least two nontrivial weak solutions in W"(Rn) when e> 0 is small enough, 0 0 and h 0 belongs to the dual space of W"(Rn).

关键词： Moser-Trudinger inequality exponential growth COMPACT RIEMANNIAN MANIFOLD critical growth EQUATIONS