We prove the existence of a positive solution to the problem-Δu=a(x)f(u), x∈Ω, u(x)=0,x∈Ω,where Ω is a bounded domain in R n with smooth boundary, a(x) is allowed to change sign.
The existence and uniqueness of singular solutions decaying like r^-m（see （1.4）） of the equation △u＋k∑i=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe sepa...The existence and uniqueness of singular solutions decaying like r^-m（see （1.4）） of the equation △u＋k∑i=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe separation structure of singular solutions decaying like r^-（n-2） of eq. （0.1） are discussed. moreover, we obtain the explicit critical exponent ps （l） （see （1.9））.展开更多
Applying Krasnosel'skii fixed point theorem of cone expansion-compression type, the existence of positive radial solutions for some second-order nonlinear elliptic equations in annular domains, subject to Dirichle...Applying Krasnosel'skii fixed point theorem of cone expansion-compression type, the existence of positive radial solutions for some second-order nonlinear elliptic equations in annular domains, subject to Dirichlet boundary conditions, is investigated. By considering the properties of nonlinear term on boundary closed intervals, several existence results of positive radial solutions are established. The main results are independent of superlinear growth and sublinear growth of nonlinear term. If nonlinear term has extreme values and satisfies suitable conditions, the main results are very effective.展开更多
In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general in...In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general involving both singular and sublinear terms. Some sufficient conditions are given with the aid of the barrier method and ODE approach, which guarantee the existence of positive entire solutions that tend to any sufficiently large constants arbitrarily prescribed in advance.展开更多
In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong Alle...In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong Allee effect and is inhomogeneous. We use variational methods to prove that the equation has at least two positive solutions for a large parameter if it satisfies some appropriate conditions. We also prove some nonexistence results.展开更多
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 op...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.展开更多
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 an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the a...In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.展开更多
The existence of positive radial solutions to the second order semilinear elliptic BVPΔu(X)+g(|X|)f(u(X))=0, R 1<|X|<R 2, u(X)=0, |X|=R 1 or |X|=R 2is considered. A general existence criterion and se...The existence of positive radial solutions to the second order semilinear elliptic BVPΔu(X)+g(|X|)f(u(X))=0, R 1<|X|<R 2, u(X)=0, |X|=R 1 or |X|=R 2is considered. A general existence criterion and several existence theorems of positive radial solution are established. Here it is not required that lim l→0f(l)/l and lim l→∞f(l)/l exist.展开更多
In this paper,we have investigated the asymptotic behavior of nodal solutions of semilinear elliptic equations in R n. We conclude more precise and extensive results and give the expression of asymptotic behavior near...In this paper,we have investigated the asymptotic behavior of nodal solutions of semilinear elliptic equations in R n. We conclude more precise and extensive results and give the expression of asymptotic behavior near ∞ more detail than that of [3]-[5].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
The paper is concerned with the multiplicity of solutions for some nonlinear elliptic equations involving critical Sobolev exponents and mixed boundary conditions.
This paper is concerned with Neumann problem for semilinear elliptic equations involving Sobolev critical exponents with limit nonlinearity in boundary condition. By critical point theory and dual variational principl...This paper is concerned with Neumann problem for semilinear elliptic equations involving Sobolev critical exponents with limit nonlinearity in boundary condition. By critical point theory and dual variational principle, the author obtains the existence and multiplicity results.展开更多
We study the following elliptic problem：{-div（a（x）Du）=Q（x）｜u｜2-2u＋λu x∈Ω,u=0 onδΩ Under certain assumptions on a and Q, we obtain existence of infinitely many solutions by variational method.
We consider the logarithmic elliptic equation with singular nonlinearity {Δu+ulogu^(2)+λ/u^(γ)=0,in Ω,u>0,in Ω,u=0,on δΩ,where Ω⊂R^(N)(N≥3)is a bounded domain with a smooth boundary,0<γ<1 andλis a ...We consider the logarithmic elliptic equation with singular nonlinearity {Δu+ulogu^(2)+λ/u^(γ)=0,in Ω,u>0,in Ω,u=0,on δΩ,where Ω⊂R^(N)(N≥3)is a bounded domain with a smooth boundary,0<γ<1 andλis a positive constant.By using a variational method and the critical point theory for a nonsmooth functional,we obtain the existence of two positive solutions.This result generalizes and improves upon recent results in the literature.展开更多
This paper is to study semilinear elliptic equation Lu = f(u) in Ω, Ω RN being a smooth bounded domain, with the third boundary value conditions =0, where operator, nonlinear function f(u) increase in the form of up...This paper is to study semilinear elliptic equation Lu = f(u) in Ω, Ω RN being a smooth bounded domain, with the third boundary value conditions =0, where operator, nonlinear function f(u) increase in the form of up, , We get a prior estimate and existence of positive solutions of above equation.展开更多
This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to pro...This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.展开更多
In this paper, we will analyze further to obtain a finer asymptotic behavior of positive solutions of semilinear elliptic equations in R^n by employing the Li＇s method of energy function.
文摘We prove the existence of a positive solution to the problem-Δu=a(x)f(u), x∈Ω, u(x)=0,x∈Ω,where Ω is a bounded domain in R n with smooth boundary, a(x) is allowed to change sign.
基金Supported by the Natural Science Foundation of China(10901126)
文摘The existence and uniqueness of singular solutions decaying like r^-m（see （1.4）） of the equation △u＋k∑i=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe separation structure of singular solutions decaying like r^-（n-2） of eq. （0.1） are discussed. moreover, we obtain the explicit critical exponent ps （l） （see （1.9））.
文摘Applying Krasnosel'skii fixed point theorem of cone expansion-compression type, the existence of positive radial solutions for some second-order nonlinear elliptic equations in annular domains, subject to Dirichlet boundary conditions, is investigated. By considering the properties of nonlinear term on boundary closed intervals, several existence results of positive radial solutions are established. The main results are independent of superlinear growth and sublinear growth of nonlinear term. If nonlinear term has extreme values and satisfies suitable conditions, the main results are very effective.
文摘In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general involving both singular and sublinear terms. Some sufficient conditions are given with the aid of the barrier method and ODE approach, which guarantee the existence of positive entire solutions that tend to any sufficiently large constants arbitrarily prescribed in advance.
基金supported by the National Natural Science Foundation of China (No.10671049)the US-NSF grants (No.DMS-0314736)
文摘In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong Allee effect and is inhomogeneous. We use variational methods to prove that the equation has at least two positive solutions for a large parameter if it satisfies some appropriate conditions. We also prove some nonexistence results.
基金Supported by National Natural Science Foundation of China(11471267)the Doctoral Scientific Research Funds of China West Normal University(15D006 and 16E014)+1 种基金Meritocracy Research Funds of China West Normal University(17YC383)Natural Science Foundation of Education of Guizhou Province(KY[2016]046)
文摘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.
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.
基金The project supported by NNSF of China(10071080)
文摘In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.
文摘The existence of positive radial solutions to the second order semilinear elliptic BVPΔu(X)+g(|X|)f(u(X))=0, R 1<|X|<R 2, u(X)=0, |X|=R 1 or |X|=R 2is considered. A general existence criterion and several existence theorems of positive radial solution are established. Here it is not required that lim l→0f(l)/l and lim l→∞f(l)/l exist.
文摘In this paper,we have investigated the asymptotic behavior of nodal solutions of semilinear elliptic equations in R n. We conclude more precise and extensive results and give the expression of asymptotic behavior near ∞ more detail than that of [3]-[5].
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.
文摘The paper is concerned with the multiplicity of solutions for some nonlinear elliptic equations involving critical Sobolev exponents and mixed boundary conditions.
文摘This paper is concerned with Neumann problem for semilinear elliptic equations involving Sobolev critical exponents with limit nonlinearity in boundary condition. By critical point theory and dual variational principle, the author obtains the existence and multiplicity results.
基金supported by Key Project (10631030) of NSFCKnowledge Innovation Funds of CAS in Chinasupported by ARC in Australia
文摘We study the following elliptic problem：{-div（a（x）Du）=Q（x）｜u｜2-2u＋λu x∈Ω,u=0 onδΩ Under certain assumptions on a and Q, we obtain existence of infinitely many solutions by variational method.
基金supported by Natural Science Foundation of Guizhou Minzu University(20185773-YB03)supported by Fundamental Research Funds of China West Normal University(18B015)+2 种基金Innovative Research Team of China West Normal University(CXTD2018-8)supported by National Natural Science Foundation of China(11861021)supported by National Natural Science Foundation of China(11661021)。
文摘We consider the logarithmic elliptic equation with singular nonlinearity {Δu+ulogu^(2)+λ/u^(γ)=0,in Ω,u>0,in Ω,u=0,on δΩ,where Ω⊂R^(N)(N≥3)is a bounded domain with a smooth boundary,0<γ<1 andλis a positive constant.By using a variational method and the critical point theory for a nonsmooth functional,we obtain the existence of two positive solutions.This result generalizes and improves upon recent results in the literature.
基金This research is supported by the National Natural Science Foundation of China.
文摘This paper is to study semilinear elliptic equation Lu = f(u) in Ω, Ω RN being a smooth bounded domain, with the third boundary value conditions =0, where operator, nonlinear function f(u) increase in the form of up, , We get a prior estimate and existence of positive solutions of above equation.
基金the National Natural Science Foundation of China(Grant Nos.10671064,10171029)
文摘This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10901047 and 10971061) Excellent Youth Program of Hunan Normal University (Grant No. 080640)
文摘In this paper, we will analyze further to obtain a finer asymptotic behavior of positive solutions of semilinear elliptic equations in R^n by employing the Li＇s method of energy function.
