In this paper,we consider a class of quasilinear equations involving a nonlinearity term having critical exponential growth.By using Mountain Pass Theorem,Ekeland's variational principle and inequalities of the ty...In this paper,we consider a class of quasilinear equations involving a nonlinearity term having critical exponential growth.By using Mountain Pass Theorem,Ekeland's variational principle and inequalities of the type Trudinger-Moser,we obtain the existence of at least two positive weak solutions.展开更多
In this paper,(ⅰ)we present unified approaches to studying the existence of ground state solutions and mountain-pass type solutions for the following quasilinear equation:-Δ_(N)^(u)+V(x)|u|^(N-2)u=f(u)in R~N,N≥2 in...In this paper,(ⅰ)we present unified approaches to studying the existence of ground state solutions and mountain-pass type solutions for the following quasilinear equation:-Δ_(N)^(u)+V(x)|u|^(N-2)u=f(u)in R~N,N≥2 in three different cases allowing the potential V∈C(R^(N),R)to be periodic,radially symmetric,or asymptotically constant,whereΔ_(Nu):=div(|?u|~(N-2)?u)and f has critical exponential growth;(ⅱ)two new compactness lemmas in W^(1,N)(R^(N))for general nonlinear functionals are established,which generalize the ones obtained in the radially symmetric space W_(rad)^(1,N)(R^(N));(ⅲ)based on some key observations,we construct a special path allowing us to control the mountain-pass minimax level by a fine threshold under which the compactness can be restored for the critical case.In particular,some delicate analyses are developed to overcome non-standard difficulties due to both the quasilinear characteristic of the equation and the lack of compactness aroused by the critical exponential growth of f.Our results extend and improve the ones of Alves et al.(2012),Ibrahim et al.(2015)(N=2),and Masmoudi and Sani(2015)(N≥3)for the constant potential case,Alves and Figueiredo(2009)for the periodic potential case,Lam and Lu(2012)and Yang(2012)for the coercive potential case,and Chen et al.(Sci China Math,2021)for the degenerate potential case,which are totally new even for the simpler semilinear case of N=2.We believe that our approaches and strategies may be adapted and modified to attack more variational problems with critical exponential growth.展开更多
By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε...By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε^(N)(-△)^(s)N/sμ+V(x)|μ|^(N/s-2μ)=Q(x)h(μ)in R^(N),where ε>0 is a parameter,s ∈(0,1),2≤p<+oo and N=ps.The nonlinear term h is a diferentiable function with exponential critical growth,the absorption potential V and the reaction potential Q are continuous functions.展开更多
In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 rep...In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 represents a trapping potential,and f has an exponential critical growth.Under the appropriate assumptions of f,we have obtained the existence of normalized solutions to the above Schr?dinger equation by introducing a variational method.And these solutions are also high energy solutions with positive energy.展开更多
This paper is concerned with the following Chern-Simons-Schrodinger equation -Δu+V(|x|)u+(∫_(|x|)^(∞)h(s)/su^(2)(s)ds+h^(2)(|x|)/|x|^(2))u=a(|x|)f(u)in R^(2),where h(s)=∫_(0)^(s)l/2u^(2)(l)dl,V,a:R^(+)→R are radi...This paper is concerned with the following Chern-Simons-Schrodinger equation -Δu+V(|x|)u+(∫_(|x|)^(∞)h(s)/su^(2)(s)ds+h^(2)(|x|)/|x|^(2))u=a(|x|)f(u)in R^(2),where h(s)=∫_(0)^(s)l/2u^(2)(l)dl,V,a:R^(+)→R are radially symmetric potentials and the nonlinearity f:R→R is of subcritical or critical exponential growth in the sense of Trudinger-Moser.We give some new sufficient conditions on f to obtain the existence of nontrivial solutions or ground state solutions.In particular,some new estimates and techniques are used to overcome the difficulty arising from the critical growth of Trudinger-Moser type.展开更多
This paper is devoted to studying the existence and multiplicity of nontrivial solutions for the following boundary value problem{-d i v(ω(x)|∇u(x)|^(N-2)∇u(x))=f(x,u)+εh(x),in B;u=0,on ∂B,where B is the unit ball i...This paper is devoted to studying the existence and multiplicity of nontrivial solutions for the following boundary value problem{-d i v(ω(x)|∇u(x)|^(N-2)∇u(x))=f(x,u)+εh(x),in B;u=0,on ∂B,where B is the unit ball in R^(N),the radial positive weight ω(x)is of logarithmic type function,the functional f(x,u)is continuous in B×R and has double exponential critical growth,which behaves like exp{e^(α|u|^(N/N-1))}as|u|→∞ for some α>0.Moreover,ϵ>0,and the radial function h belongs to the dual space of W_(0,rad)^(1,N)(B)h≠0.展开更多
This article studies a class of nonlinear Kirchhoff equations with exponential critical growth,trapping potential,and perturbation.Under appropriate assumptions about f and h,the article obtained the existence of norm...This article studies a class of nonlinear Kirchhoff equations with exponential critical growth,trapping potential,and perturbation.Under appropriate assumptions about f and h,the article obtained the existence of normalized positive solutions for this equation via the Trudinger-Moser inequality and variational methods.Moreover,these solutions are also ground state solutions.Additionally,the article also characterized the asymptotic behavior of solutions.The results of this article expand the research of relevant literature.展开更多
We study a quasilinear Schrodinger equation {-εN△Nu＋V（x）｜u｜N-2= Q（x）f（u） in R^N,0〈u∈W1,N（RN）,u（x）^｜x｜→∞0,where V, Q are two continuous real functions on R^N and c 〉 0 is a real parameter. Assume ...We study a quasilinear Schrodinger equation {-εN△Nu＋V（x）｜u｜N-2= Q（x）f（u） in R^N,0〈u∈W1,N（RN）,u（x）^｜x｜→∞0,where V, Q are two continuous real functions on R^N and c 〉 0 is a real parameter. Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger-Moser inequality, we are able to establish the existence and concentration of the semiclassical solutions by variational methods. Keywords Exponential critical growth, semiclassical solutions, variational methods展开更多
In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger...In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.展开更多
In this paper,the authors consider the following singular Kirchhoff-Schrodinger problem M(∫_(R^(N))|∇u|^(N)+V(x)|u|^(N)dx)(−Δ_(N)u+V(x)|u|^(N-2)u)=f(x,u)/|x|^(η)in R^(N),(P_(η))where 0<η<N,M is a Kirchhoff-...In this paper,the authors consider the following singular Kirchhoff-Schrodinger problem M(∫_(R^(N))|∇u|^(N)+V(x)|u|^(N)dx)(−Δ_(N)u+V(x)|u|^(N-2)u)=f(x,u)/|x|^(η)in R^(N),(P_(η))where 0<η<N,M is a Kirchhoff-type function and V(x)is a continuous function with positive lower bound,f(x,t)has a critical exponential growth behavior at infinity.Combining variational techniques with some estimates,they get the existence of ground state solution for(P_(η)).Moreover,they also get the same result without the A-R condition.展开更多
We study the existence of solutions for the following class of nonlinear Schr?dinger equations-ΔN u+V(x)u=K(x)f(u)in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s)has exponential crit...We study the existence of solutions for the following class of nonlinear Schr?dinger equations-ΔN u+V(x)u=K(x)f(u)in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s)has exponential critical growth.The approaches used here are based on a version of the Trudinger–Moser inequality and a minimax theorem.展开更多
基金Supported by the Natural Science Foundation of Anhui Province(Grant No.1808085QA15)the Scientific Research Project of Anhui University of Finance and Economics(Grant No.ACKYC19050)the Natural Science Foundation of China(Grant No.11571093)。
文摘In this paper,we consider a class of quasilinear equations involving a nonlinearity term having critical exponential growth.By using Mountain Pass Theorem,Ekeland's variational principle and inequalities of the type Trudinger-Moser,we obtain the existence of at least two positive weak solutions.
基金supported by National Natural Science Foundation of China(Grant Nos.11971485,12001542,12171486,and 12371181)the Project for Young Backbone Teachers of Hunan Province(Grant No.10900-150220002)+1 种基金Natural Science Foundation for Excellent Young Scholars of Hunan Province(Grant No.2023JJ20057)supported by the Grant“Nonlinear Differential Systems in Applied Sciences”of the Romanian Ministry of Research,Innovation,and Digitization(Grant No.PNRR-Ⅲ-C9-2022-I8/22)。
文摘In this paper,(ⅰ)we present unified approaches to studying the existence of ground state solutions and mountain-pass type solutions for the following quasilinear equation:-Δ_(N)^(u)+V(x)|u|^(N-2)u=f(u)in R~N,N≥2 in three different cases allowing the potential V∈C(R^(N),R)to be periodic,radially symmetric,or asymptotically constant,whereΔ_(Nu):=div(|?u|~(N-2)?u)and f has critical exponential growth;(ⅱ)two new compactness lemmas in W^(1,N)(R^(N))for general nonlinear functionals are established,which generalize the ones obtained in the radially symmetric space W_(rad)^(1,N)(R^(N));(ⅲ)based on some key observations,we construct a special path allowing us to control the mountain-pass minimax level by a fine threshold under which the compactness can be restored for the critical case.In particular,some delicate analyses are developed to overcome non-standard difficulties due to both the quasilinear characteristic of the equation and the lack of compactness aroused by the critical exponential growth of f.Our results extend and improve the ones of Alves et al.(2012),Ibrahim et al.(2015)(N=2),and Masmoudi and Sani(2015)(N≥3)for the constant potential case,Alves and Figueiredo(2009)for the periodic potential case,Lam and Lu(2012)and Yang(2012)for the coercive potential case,and Chen et al.(Sci China Math,2021)for the degenerate potential case,which are totally new even for the simpler semilinear case of N=2.We believe that our approaches and strategies may be adapted and modified to attack more variational problems with critical exponential growth.
基金supported by National Natural Science Foundation of China(No.12171152)。
文摘By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε^(N)(-△)^(s)N/sμ+V(x)|μ|^(N/s-2μ)=Q(x)h(μ)in R^(N),where ε>0 is a parameter,s ∈(0,1),2≤p<+oo and N=ps.The nonlinear term h is a diferentiable function with exponential critical growth,the absorption potential V and the reaction potential Q are continuous functions.
基金Supported by National Natural Science Foundation of China(Grant Nos.11671403 and 11671236)Henan Provincial General Natural Science Foundation Project(Grant No.232300420113)。
文摘In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 represents a trapping potential,and f has an exponential critical growth.Under the appropriate assumptions of f,we have obtained the existence of normalized solutions to the above Schr?dinger equation by introducing a variational method.And these solutions are also high energy solutions with positive energy.
基金Supported by National Natural Science Foundation of China(Grant Nos.11971485 and 12001542)。
文摘This paper is concerned with the following Chern-Simons-Schrodinger equation -Δu+V(|x|)u+(∫_(|x|)^(∞)h(s)/su^(2)(s)ds+h^(2)(|x|)/|x|^(2))u=a(|x|)f(u)in R^(2),where h(s)=∫_(0)^(s)l/2u^(2)(l)dl,V,a:R^(+)→R are radially symmetric potentials and the nonlinearity f:R→R is of subcritical or critical exponential growth in the sense of Trudinger-Moser.We give some new sufficient conditions on f to obtain the existence of nontrivial solutions or ground state solutions.In particular,some new estimates and techniques are used to overcome the difficulty arising from the critical growth of Trudinger-Moser type.
基金supported by Natural Science Foundation of Chongqing,China(Grant Nos.CSTB2024N and SCQ-LZX0038).
文摘This paper is devoted to studying the existence and multiplicity of nontrivial solutions for the following boundary value problem{-d i v(ω(x)|∇u(x)|^(N-2)∇u(x))=f(x,u)+εh(x),in B;u=0,on ∂B,where B is the unit ball in R^(N),the radial positive weight ω(x)is of logarithmic type function,the functional f(x,u)is continuous in B×R and has double exponential critical growth,which behaves like exp{e^(α|u|^(N/N-1))}as|u|→∞ for some α>0.Moreover,ϵ>0,and the radial function h belongs to the dual space of W_(0,rad)^(1,N)(B)h≠0.
基金Supported by National Natural Science Foundation of China(11671403,11671236)Henan Provincial General Natural Science Foundation Project(232300420113)。
文摘This article studies a class of nonlinear Kirchhoff equations with exponential critical growth,trapping potential,and perturbation.Under appropriate assumptions about f and h,the article obtained the existence of normalized positive solutions for this equation via the Trudinger-Moser inequality and variational methods.Moreover,these solutions are also ground state solutions.Additionally,the article also characterized the asymptotic behavior of solutions.The results of this article expand the research of relevant literature.
基金partially supported by PROCAD/UFG/Un B and FAPDF(Grant No.PRONEX 193.000.580/2009)partially supported by NSFC(Grant Nos.11571317,11101374,11271331)ZJNSF(Grant No.Y15A010026)
文摘We study a quasilinear Schrodinger equation {-εN△Nu＋V（x）｜u｜N-2= Q（x）f（u） in R^N,0〈u∈W1,N（RN）,u（x）^｜x｜→∞0,where V, Q are two continuous real functions on R^N and c 〉 0 is a real parameter. Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger-Moser inequality, we are able to establish the existence and concentration of the semiclassical solutions by variational methods. Keywords Exponential critical growth, semiclassical solutions, variational methods
文摘In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.
基金supported by the National Natural Science Foundation of China(Nos.11790271,12171108,12201089)Guangdong Basic and Applied basic Research Foundation(No.2020A1515011019)Innovation and Development Project of Guangzhou University and Chongqing Normal University Foundation(No.21XLB039)。
文摘In this paper,the authors consider the following singular Kirchhoff-Schrodinger problem M(∫_(R^(N))|∇u|^(N)+V(x)|u|^(N)dx)(−Δ_(N)u+V(x)|u|^(N-2)u)=f(x,u)/|x|^(η)in R^(N),(P_(η))where 0<η<N,M is a Kirchhoff-type function and V(x)is a continuous function with positive lower bound,f(x,t)has a critical exponential growth behavior at infinity.Combining variational techniques with some estimates,they get the existence of ground state solution for(P_(η)).Moreover,they also get the same result without the A-R condition.
基金Natural Science Foundation of China(Grant Nos.11601190 and 11661006)Natural Science Foundation of Jiangsu Province(Grant No.BK20160483)Jiangsu University Foundation Grant(Grant No.16JDG043)。
文摘We study the existence of solutions for the following class of nonlinear Schr?dinger equations-ΔN u+V(x)u=K(x)f(u)in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s)has exponential critical growth.The approaches used here are based on a version of the Trudinger–Moser inequality and a minimax theorem.
