This note is a continuation of the work[17].We study the following quasilinear elliptic equations- △pu-μ/｜x｜p ｜u｜p-2 u=Q（x）｜u｜Np/N-p -2u,x∈R N,where 1 〈 p 〈 N,0 ≤ μ 〈（（N-p）/p）p and Q ∈ L∞（RN）.O...This note is a continuation of the work[17].We study the following quasilinear elliptic equations- △pu-μ/｜x｜p ｜u｜p-2 u=Q（x）｜u｜Np/N-p -2u,x∈R N,where 1 〈 p 〈 N,0 ≤ μ 〈（（N-p）/p）p and Q ∈ L∞（RN）.Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.展开更多
The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational...The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.展开更多
We investigate the following elliptic equations:⎧⎩⎨−M(∫R Nϕ(|∇u|2)dx)div(ϕ′(|∇u|2)∇u)+|u|α−2 u=λh(x,u),u(x)→0,as|x|→∞,in R N,where N≥2,1<p<q<N,α<q,1<α≤p∗q′/p′with p∗=NpN−p,ϕ(t)behaves like ...We investigate the following elliptic equations:⎧⎩⎨−M(∫R Nϕ(|∇u|2)dx)div(ϕ′(|∇u|2)∇u)+|u|α−2 u=λh(x,u),u(x)→0,as|x|→∞,in R N,where N≥2,1<p<q<N,α<q,1<α≤p∗q′/p′with p∗=NpN−p,ϕ(t)behaves like t q/2 for small t and t p/2 for large t,and p′and q′are the conjugate exponents of p and q,respectively.We study the existence of nontrivial radially symmetric solutions for the problem above by applying the mountain pass theorem and the fountain theorem.Moreover,taking into account the dual fountain theorem,we show that the problem admits a sequence of small-energy,radially symmetric solutions.展开更多
Sufficient conditions are obtained for oscillation of certain quasilinear elliptic equations div(｜Du｜m-2A（x）Du)+p（x）｜u｜m-2u=0, x∈ΩRn, where Ω is an exterior domain, m>1, and p（x） is an alternating func...Sufficient conditions are obtained for oscillation of certain quasilinear elliptic equations div(｜Du｜m-2A（x）Du)+p（x）｜u｜m-2u=0, x∈ΩRn, where Ω is an exterior domain, m>1, and p（x） is an alternating function. The integral averaging technique is employed to establish our results.展开更多
Since [1] established the Pohozaev identity in bounded domains, this identity has been the principal tool to deal with the non-existence of the equation
The existence ofpositive radialsolutions ofthe equation - div(｜Du｜p- 2Du)= f(u) is studied in annular dom ains in Rn,n≥2. Itisproved thatiff(0)≥0, f is somewhere negativein (0,∞), lim u→0+ f′(u)= 0and lim u→...The existence ofpositive radialsolutions ofthe equation - div(｜Du｜p- 2Du)= f(u) is studied in annular dom ains in Rn,n≥2. Itisproved thatiff(0)≥0, f is somewhere negativein (0,∞), lim u→0+ f′(u)= 0and lim u→∞(f(u)/up- 1)= ∞, then thereisa largepositiveradialsolution on allannuli.Iff(0)< 0 and satisfiescertain condi- tions, then the equation has no radialsolution ifthe annuliare too wide.展开更多
In this paper we give a priori estimates for the maximum modulus of generalizedsolulions of the quasilinear elliplic equations irith anisotropic growth condition.
We consider a class of quasilinear elliptic boundary problems,including the following Modified Nonlinear Schrödinger Equation as a special case:{Δu+1/2 uΔ(u^(2))−V(x)u+|u|^(q−2)u=0 in Ω,u=0 on∂Ω,whereΩis the...We consider a class of quasilinear elliptic boundary problems,including the following Modified Nonlinear Schrödinger Equation as a special case:{Δu+1/2 uΔ(u^(2))−V(x)u+|u|^(q−2)u=0 in Ω,u=0 on∂Ω,whereΩis the entire space R^(N) orΩ⊂R^(N) is a bounded domain with smooth boundary,q∈(2,22^(∗)]with 2^(∗)=2 N/(N−2)being the critical Sobolev exponent and 22^(∗)=4 N/(N−2).We review the general methods developed in the last twenty years or so for the studies of existence,multiplicity,nodal property of the solutions within this range of nonlinearity up to the new critical exponent 4 N/(N−2),which is a unique feature for this class of problems.We also discuss some related and more general problems.展开更多
In this paper, we establish the existence of three weak solutions for quasilinear elliptic equations in an Orlicz-Sobolev space via an abstract result recently obtained by Ricceri in [13].
In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger ...In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.展开更多
In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogene...In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.展开更多
In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an un...In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.展开更多
Sufficient conditions for the existence of positive solutions to a class of quasilinear elliptic equations in two-dimensional exterior domains are given.
This paper investigates 2-dimensional singular,quasilinear elliptic equations and gives some suffcient conditions ensuring the equations have infinitely many positive entire solutions. The super-subsolution method is ...This paper investigates 2-dimensional singular,quasilinear elliptic equations and gives some suffcient conditions ensuring the equations have infinitely many positive entire solutions. The super-subsolution method is used to prove the existence of such solutions.展开更多
In this paper, we introduce quasilinear elliptic equations in divergent form. It is permitted that the growth orders of A(x, u, ζ) and B(x, u, ζ) with respect to u arrive at the critical exponents and the growth ord...In this paper, we introduce quasilinear elliptic equations in divergent form. It is permitted that the growth orders of A(x, u, ζ) and B(x, u, ζ) with respect to u arrive at the critical exponents and the growth order of B(x, u, ζ) with respect to ζ is supercritical. Many aspects can be solved satisfactorily.展开更多
In this paper, the existence and nonexistence of solutions to a class of quasilinear elliptic equations with nonsmooth functionals are discussed, and the results obtained are applied to quasilinear SchrSdinger equatio...In this paper, the existence and nonexistence of solutions to a class of quasilinear elliptic equations with nonsmooth functionals are discussed, and the results obtained are applied to quasilinear SchrSdinger equations with negative parameter which arose from the study of self-channeling of high-power ultrashort laser in matter.展开更多
In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable function which is bounded away from 0 in Ω,...In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable function which is bounded away from 0 in Ω, and the domain Ω is a smooth bounded open set in R^N（N≥2）. Especially, under the condition that g（x, s） = 1/｜s｜^α （α〉0） is singular at s = 0, we obtain that α 〈 p is necessary and sufficient for the existence of solutions in W0^1,p（Ω） to problem （0.1） when f is sufficiently regular.展开更多
The present paper investigates the asymptotic behavior of solutions for a class of second order inhomogeneous quasilinear equations on a three dimensional semiinfinite cylinder. A Phragmen-Lindelof type alternative is...The present paper investigates the asymptotic behavior of solutions for a class of second order inhomogeneous quasilinear equations on a three dimensional semiinfinite cylinder. A Phragmen-Lindelof type alternative is obtained, i.e., it is shown that in appropriate norms solutions of the equations either grow or decay as some spatial variable tends to infinity.展开更多
In this paper, we are concerned with a weighted quasilinear elliptic equation involving critical Hardy-Sobolev exponent in a bounded G-symmetric domain. By using the symmetric criticality principle of Palais and varia...In this paper, we are concerned with a weighted quasilinear elliptic equation involving critical Hardy-Sobolev exponent in a bounded G-symmetric domain. By using the symmetric criticality principle of Palais and variational method, we establish several existence and multiplicity results of positive G-symmetric solutions under certain appropriate hypotheses on the potential and the nonlinearity.展开更多
This paper considers the quasilinear elliptic equation where , and 0 < m < p-1 < q < +∞, Ω is a bounded domain in RN(N 3).λ is a positive number. Object is to estimate exactly the magnitute of λ* su...This paper considers the quasilinear elliptic equation where , and 0 < m < p-1 < q < +∞, Ω is a bounded domain in RN(N 3).λ is a positive number. Object is to estimate exactly the magnitute of λ* such that (1)λ has at least one positive solution if λ ∈ (0, λ*) and no positive solutions if λ > λ*. Furthermore, (1)λ has at least one positive solution when λ = λ*, and at least two positive solutions when λ ∈ (0, λ*) and . Finally, the authors obtain a multiplicity result with positive energy of (1)λ when 0 < m < p - 1 < q = (Np)/(N-p) - 1.展开更多
基金financially supported by the Academy of Finland,project 259224
文摘This note is a continuation of the work[17].We study the following quasilinear elliptic equations- △pu-μ/｜x｜p ｜u｜p-2 u=Q（x）｜u｜Np/N-p -2u,x∈R N,where 1 〈 p 〈 N,0 ≤ μ 〈（（N-p）/p）p and Q ∈ L∞（RN）.Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.
基金Supported by National Natural Science Foundation of China (11071198 11101347)+2 种基金Postdoctor Foundation of China (2012M510363)the Key Project in Science and Technology Research Plan of the Education Department of Hubei Province (D20112605 D20122501)
文摘The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.
基金the National Research Foundation of Korea(NRF)grant funded by the Korea government(MSIT)(NRF-2019R1F1A1057775)Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education(2018R1D1A1B07048620).
文摘We investigate the following elliptic equations:⎧⎩⎨−M(∫R Nϕ(|∇u|2)dx)div(ϕ′(|∇u|2)∇u)+|u|α−2 u=λh(x,u),u(x)→0,as|x|→∞,in R N,where N≥2,1<p<q<N,α<q,1<α≤p∗q′/p′with p∗=NpN−p,ϕ(t)behaves like t q/2 for small t and t p/2 for large t,and p′and q′are the conjugate exponents of p and q,respectively.We study the existence of nontrivial radially symmetric solutions for the problem above by applying the mountain pass theorem and the fountain theorem.Moreover,taking into account the dual fountain theorem,we show that the problem admits a sequence of small-energy,radially symmetric solutions.
文摘Sufficient conditions are obtained for oscillation of certain quasilinear elliptic equations div(｜Du｜m-2A（x）Du)+p（x）｜u｜m-2u=0, x∈ΩRn, where Ω is an exterior domain, m>1, and p（x） is an alternating function. The integral averaging technique is employed to establish our results.
基金This work is supported in port by the Foundation of Zhongshan University Advanced Research Center.
文摘Since [1] established the Pohozaev identity in bounded domains, this identity has been the principal tool to deal with the non-existence of the equation
文摘The existence ofpositive radialsolutions ofthe equation - div(｜Du｜p- 2Du)= f(u) is studied in annular dom ains in Rn,n≥2. Itisproved thatiff(0)≥0, f is somewhere negativein (0,∞), lim u→0+ f′(u)= 0and lim u→∞(f(u)/up- 1)= ∞, then thereisa largepositiveradialsolution on allannuli.Iff(0)< 0 and satisfiescertain condi- tions, then the equation has no radialsolution ifthe annuliare too wide.
文摘In this paper we give a priori estimates for the maximum modulus of generalizedsolulions of the quasilinear elliplic equations irith anisotropic growth condition.
基金The work is supported by NSFC(Grant No.11831009).
文摘We consider a class of quasilinear elliptic boundary problems,including the following Modified Nonlinear Schrödinger Equation as a special case:{Δu+1/2 uΔ(u^(2))−V(x)u+|u|^(q−2)u=0 in Ω,u=0 on∂Ω,whereΩis the entire space R^(N) orΩ⊂R^(N) is a bounded domain with smooth boundary,q∈(2,22^(∗)]with 2^(∗)=2 N/(N−2)being the critical Sobolev exponent and 22^(∗)=4 N/(N−2).We review the general methods developed in the last twenty years or so for the studies of existence,multiplicity,nodal property of the solutions within this range of nonlinearity up to the new critical exponent 4 N/(N−2),which is a unique feature for this class of problems.We also discuss some related and more general problems.
基金Supported by the National Natural Science Foundation of China(Grant No.11626038)
文摘In this paper, we establish the existence of three weak solutions for quasilinear elliptic equations in an Orlicz-Sobolev space via an abstract result recently obtained by Ricceri in [13].
基金the National Natural Science Foundation of China (No.19531060).
文摘In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.
基金supported by Ministry of Education and Training(Vietnam),under grant number B2023-SPS-01。
文摘In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.
基金Supported by the National Natural Science Foundation of China (No. 10671211)
文摘In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.
基金partially supported by the NNsF of China(No.10531040)the NSF of Guangdong Provincethe foundation of Zhongshan University Advanced Research Center
文摘Sufficient conditions for the existence of positive solutions to a class of quasilinear elliptic equations in two-dimensional exterior domains are given.
文摘This paper investigates 2-dimensional singular,quasilinear elliptic equations and gives some suffcient conditions ensuring the equations have infinitely many positive entire solutions. The super-subsolution method is used to prove the existence of such solutions.
基金supported by the National Natural Science Foundation of China under Grant Nos.11401096 and 11326123the Natural Science Foundation of Guangdong Province under Grant Nos.S2013010014485,2014A030313619 and S2013010012463+2 种基金Special fund of the Guangdong College discipline construction under Grant Nos.2013KJCX0189 and 2013B020314020Guangdong provincial major school projects:2014KZDXM063Guangdong Provincial Innovation Project features 2014KTSCX150
文摘In this paper, we introduce quasilinear elliptic equations in divergent form. It is permitted that the growth orders of A(x, u, ζ) and B(x, u, ζ) with respect to u arrive at the critical exponents and the growth order of B(x, u, ζ) with respect to ζ is supercritical. Many aspects can be solved satisfactorily.
基金supported by NSF of China(11201488),supported by NSF of China(11371146)Hunan Provincial Natural Science Foundation of China(14JJ4002)
文摘In this paper, the existence and nonexistence of solutions to a class of quasilinear elliptic equations with nonsmooth functionals are discussed, and the results obtained are applied to quasilinear SchrSdinger equations with negative parameter which arose from the study of self-channeling of high-power ultrashort laser in matter.
基金supported by the Natural Science Foundation of Henan Province(15A110050)
文摘In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable function which is bounded away from 0 in Ω, and the domain Ω is a smooth bounded open set in R^N（N≥2）. Especially, under the condition that g（x, s） = 1/｜s｜^α （α〉0） is singular at s = 0, we obtain that α 〈 p is necessary and sufficient for the existence of solutions in W0^1,p（Ω） to problem （0.1） when f is sufficiently regular.
文摘The present paper investigates the asymptotic behavior of solutions for a class of second order inhomogeneous quasilinear equations on a three dimensional semiinfinite cylinder. A Phragmen-Lindelof type alternative is obtained, i.e., it is shown that in appropriate norms solutions of the equations either grow or decay as some spatial variable tends to infinity.
基金Supported by the Natural Science Foundation of China(1107118011171247)Project supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission(KJ130503)
文摘In this paper, we are concerned with a weighted quasilinear elliptic equation involving critical Hardy-Sobolev exponent in a bounded G-symmetric domain. By using the symmetric criticality principle of Palais and variational method, we establish several existence and multiplicity results of positive G-symmetric solutions under certain appropriate hypotheses on the potential and the nonlinearity.
文摘This paper considers the quasilinear elliptic equation where , and 0 < m < p-1 < q < +∞, Ω is a bounded domain in RN(N 3).λ is a positive number. Object is to estimate exactly the magnitute of λ* such that (1)λ has at least one positive solution if λ ∈ (0, λ*) and no positive solutions if λ > λ*. Furthermore, (1)λ has at least one positive solution when λ = λ*, and at least two positive solutions when λ ∈ (0, λ*) and . Finally, the authors obtain a multiplicity result with positive energy of (1)λ when 0 < m < p - 1 < q = (Np)/(N-p) - 1.
