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The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains

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Science China Mathematics 2014年第9期57卷 1927-1952页

作者： HE Yi li gongbao School of Mathematics and Statistics Central China Normal University

In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u>0 in Ω,N>p>2,where Ω is a domain in RN,possibly unbounded,with empty or smooth boundary,εis a small positive parameter,f∈C1(R+,R)is of subcritical and V:RN→R is a locally Hlder continuous function which is bounded from below,away from zero,such that infΛV0 such that for anyε∈(0,ε0],the above mentioned problem possesses a weak solution uεwith exponential ***,uεconcentrates around a minimum point of the potential V inΛ.Our result generalizes a similar result by del Pino and Felmer(1996)for semilinear elliptic equations to the p-Laplacian type problem.

关键词：拉普拉斯型无界域弱解浓度半线性椭圆方程椭圆型问题光滑边界连续函数

MULTIPLE CRITICAL POINTS OF PERTURBED SYMMETRIC FUNCTIONALS IN W01,p

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Acta Mathematica Scientia 1987年第4期7卷 431-445页

作者：李工宝 Department of Mathematics Wuhan University China

This paper shudies the existence of multiple critical points of perturbed symmetric functionals in W01,p（Ω）（p≥2）. For p>2, this kind of problems have never been studied befor and the main result of this paper g... 详细信息

关键词： MULTIPLE CRITICAL POINTS OF PERTURBED SYMMETRIC FUNCTIONALS IN W0 刁刀

SOME RESULTS ON CALCULUS OF VARIATIONS IN CERTAIN ORliCZSOBOLEV SPACES

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数学物理学报(英文版) 1986年第1期6卷 45-65页

作者：李工宝 Department of Mathematics Wuhan University Wuhan China

§0. Introduction This paper is devoted to discuss two classes of variational problems in certain Orlicz-Sobolev spaces. In the first part of the paper, we discuss a higher order problem in calculus of variations wit... 详细信息

关键词： devoted variational calculus nonlinearity regularity generalize convergent bedding thank valid

NONTRIVIAL SOLUTIONS OF QUASIliNEAR ELliPTIC EQUATIONS IN W01,p INVOLVING liMITING NONliNEARITIES

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Acta Mathematica Scientia 1987年第3期7卷 329-339页

作者：李工宝 Dept. of Math. Wuhan University China

In this paper, We get the existence of non-trivial solutions of the following elliptic equations where n>p≥2 and p~x=np/n-p.

关键词： lim INVOLVING liMITING NONliNEARITIES NONTRIVIAL SOLUTIONS OF QUASIliNEAR ELliPTIC EQUATIONS IN W0

THE EXISTENCE OF INFINITELY MANY SO UTIONS OF QUASIliNEAR PARTIAL DIFFERENTIAL EQUATIONS IN UNBOUNDED DOMAINS

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Acta Mathematica Scientia 1989年第2期9卷 175-188页

作者：李工宝 Inst. of Math. Sci. Academia Sinica Wuhan China

In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem need not b... 详细信息

关键词： THE EXISTENCE OF INFINITELY MANY SO UTIONS OF QUASIliNEAR PARTIAL DIFFERENTIAL EQUATIONS IN UNBOUNDED DOMAINS

MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT

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Acta Mathematica Scientia 2009年第4期29卷 903-918页

作者：李工宝张国 College of Mathematics and Statistics Central China Normal University

In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p＆q-Laplacian type involving the critical Sobolev exponent：{-△pu-△qu=│u│^p＊-2u＋μ│u│^r-2u in Ω u│δΩ=0,where Ω belong to R^N is a bounded domain,N〉p,p^＊=Np/N-p is the critical Sobolev exponent and μ 〉0. We prove that if 1 〈 r 〈 q 〈 p 〈 N, then there is a μ0 〉 0, such that for any μ∈ （0, μ0）, the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.

关键词： p&q-Laplacian multiplicity of solutions critical exponent

THE　EXISTENCE　OF　A　WEAK　SOLUTION　OF　QUASIliNEAR　ELliPTIC　EQUATION　WITH　CRITICAL　SOBOLV　EXPONENT　ON　UNBOUNDED　DOMAIN

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Acta Mathematica Scientia 1994年第1期14卷 64-74页

作者：李工宝 Inst. of Math. Sci. The Chinese Academy of Sciences Wuhan 430071 China

In this paper, the existence of a weak solution to the following elliptic equation with critical Sobolew exponent on unbounded domain Ω is provedwhere N ≥ p ≥ 2 and Ω is a smooth domain in RN, f(x,u) ～ |u|p-z at... 详细信息

关键词：临界Sobolev指数拟线性椭圆型方程弱解

The Existence of a Weak Solution of Inhomogeneous Quasilinear Elliptic Equation with Critical Growth Conditions

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Acta Mathematica Sinica,English Series 1995年第2期11卷 146-155页

作者： li gongbao Zhou Huansong Wuhan Institute of Mathematical Sciences Academia Sinica P. O. Box 71007 Wuhan, 430071 China Wuhan Institute of Mathematical Sciences Academia Sinica Wuhan People's Republic of China

In this paper, we get the existence of a weak solution of the following inhomogeneous quasilinear elliptic equation with critical growth conditions: where N≥2, f（x,u）～|u|m-1eb|u|γat +∞, with γ=N/N-1, m≥1, b>0.

关键词： lim The Existence of a Weak Solution of Inhomogeneous Quasilinear Elliptic Equation with Critical Growth Conditions Math

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMIliNEAR ELliPTIC SYSTEM ON R N WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

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Acta Mathematica Scientia 2010年第6期30卷 1917-1936页

作者：李工宝王春花 School of Mathematics and Statistics Huazhong Normal University

In this paper, we prove the existence of at least one positive solution pair （u, v）∈ H1（RN） × H1（RN） to the following semilinear elliptic system {-△u＋u=f（x,v）,x∈RN,-△u＋u=g（x,v）,x∈RN （0.1）,by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈C0（RN× R1） are that, f（x, t） and g（x, t） are superlinear at t = 0 as well as at t =＋∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245（2008）, 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem {-△u＋u=f（x,u）,x∈Ω,u∈H0^1（Ω） where Ω ∩→RN is bounded and a result of li and Yang [G. li and J. Yang： Communications in P.D.E. Vol. 29（2004） Nos.5＆ ***.925-954, 2004] concerning （0.1） when f and g are asymptotically linear.

关键词： existence nontrivial solution semilinear elliptic system without the (AR) condition