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 consider the following generalized nonlinear k-Hessian system G(S_(k)^(1/k)(λ(D^(2)z1)))S_(k)^(1/k)(λ(D^(2)z1))=φ(|x|,z1,z2),x∈R^(N),G(S_(k)^(1/k)(λ(D^(2)z2)))S_(k)^(1/k)(λ(D^(2)z2))=ψ(|x|,z1,z...In this paper,we consider the following generalized nonlinear k-Hessian system G(S_(k)^(1/k)(λ(D^(2)z1)))S_(k)^(1/k)(λ(D^(2)z1))=φ(|x|,z1,z2),x∈R^(N),G(S_(k)^(1/k)(λ(D^(2)z2)))S_(k)^(1/k)(λ(D^(2)z2))=ψ(|x|,z1,z2),x∈R^(N),where G is a nonlinear operator and Sk(λ(D^(2)z))stands for the k-Hessian operator.We first are interested in the classification of positive entire k-convex radial solutions for the k-Hessian system ifφ(|x|,z1,z2)=b(|x|)φ(z1,z2)andψ(|x|,z1,z2)=h(|x|)ψ(z1).Moreover,with the help of the monotone iterative method,some new existence results on the positive entire k-convex radial solutions of the k-Hessian system with the special non-linearitiesψ,φare given,which improve and extend many previous works.展开更多
The existence of positive radial solutions to the systems of m(m≥1) semilinear elliptic equations Δu+p(r)f(u)=0,0<A<r<B in annuli with Dirichlet(Dirichlet/Neumann)boundary conditions,is studied,whe...The existence of positive radial solutions to the systems of m(m≥1) semilinear elliptic equations Δu+p(r)f(u)=0,0<A<r<B in annuli with Dirichlet(Dirichlet/Neumann)boundary conditions,is studied,where r=x 2 1+...+x 2 n,n≥1.u=(u 1,...,u m),p(r)f(u)=(p 1(r)f 1(u),...,p m(r)f m(u)), and p(r) may be singular at r=A or r=B,f may be singular at u=0.展开更多
In this paper, we study the positive radial solutions for elliptic systems to the nonlinear BVP:<br /> <p> <img src="Edit_4da56369-d8f9-42d0-9650-c15af375d30c.bmp" alt="" />, whe...In this paper, we study the positive radial solutions for elliptic systems to the nonlinear BVP:<br /> <p> <img src="Edit_4da56369-d8f9-42d0-9650-c15af375d30c.bmp" alt="" />, where Δ<em>u</em> = <em>div</em> (<span style="white-space:nowrap;">∇</span><em>u</em>) and Δ<em>v</em> = <em>div</em> (<span style="white-space:nowrap;">∇</span><em>v</em>) are the Laplacian of <em>u</em>, <span style="white-space:nowrap;"><em>λ</em> </span>is a positive parameter, Ω = {<em>x</em> ∈ R<sup><em>n</em></sup> : <em>N</em> > 2, |<em>x</em>| > <em>r</em><sub>0</sub>, <em>r</em><sub>0</sub> > 0}, let <em>i</em> = [1,2] then <em>K<sub>i</sub></em> :[<em>r</em><sub>0</sub>,∞] → (0,∞) is a continuous function such that lim<sub><em>r</em>→∞</sub> <em>k<sub>i</sub></em>(<em>r</em>) = 0 and <img src="Edit_19f045da-988f-43e9-b1bc-6517f5734f9c.bmp" alt="" /> is The external natural derivative, and <img src="Edit_3b36ed6b-e780-46de-925e-e3cf7c6a125f.bmp" alt="" />: [0, ∞) → (0, ∞) is a continuous function. We discuss existence and multiplicity results for classes of <em>f </em>with a) <em>f<sub>i </sub></em>> 0, b) <em>f<sub>i </sub></em>< 0, and c) <em>f<sub>i </sub></em>= 0. We base our presence and multiple outcomes via the Sub-solutions method. We also discuss some unique findings. </p>展开更多
By the Schauder-Tychonoff fixed-point theorem, we inyestigate the existence and asymptotic behavior of positive radial solutions of fully nonlinear elliptic equations in Re. We give some sufficient conditions to guara...By the Schauder-Tychonoff fixed-point theorem, we inyestigate the existence and asymptotic behavior of positive radial solutions of fully nonlinear elliptic equations in Re. We give some sufficient conditions to guarantee the existence of bounded and unbounded radial solutions and consider the nonexistence of positive solution in Rn.展开更多
In this paper,we focus on a general n-dimension system of k-Hessian equations.By introducing some new suitable growth conditions,the existence results of radial k-admissible solutions of the k-Hessian system are obtai...In this paper,we focus on a general n-dimension system of k-Hessian equations.By introducing some new suitable growth conditions,the existence results of radial k-admissible solutions of the k-Hessian system are obtained.Our approach is largely based on the well-known fixed-point theorem.展开更多
In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE anal- ysis, we extend previous results to cases ...In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE anal- ysis, we extend previous results to cases where nonlinear terms may have sublinear growth. As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified SchrSdinger equations.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
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
In this paper, we consider a class of semilinear parabolic differential equation where nonlinear term has local bounded coefficients. Under some assumptions, we get the existence and uniqueness of solution. Terminate ...In this paper, we consider a class of semilinear parabolic differential equation where nonlinear term has local bounded coefficients. Under some assumptions, we get the existence and uniqueness of solution. Terminate to the time variable, we obtain the so called generalized Emden-Fowler equation and the asymptotic behavior of positive radial solutions have been given in all dimensions. At the end of this paper, we give its application to critical branching Brownian motion (also called measure-valued branching processes).展开更多
Existence and multiplicity results of positive radially symmetric solution ofthe eigenvalue problemsare obtained for fi being a N-ball or an annulus via Leray-Schauder degreetheory and variational method.
In this paper, the nonexistence of positive entire solutions for div(|Du| p-2 Du)q(x)f(u),x∈R N, is established, where p】1,Du=(D 1u,...,D Nu),q∶R N→(0,∞) and f∶(0,∞)→(0,∞) are continuous fun...In this paper, the nonexistence of positive entire solutions for div(|Du| p-2 Du)q(x)f(u),x∈R N, is established, where p】1,Du=(D 1u,...,D Nu),q∶R N→(0,∞) and f∶(0,∞)→(0,∞) are continuous functions.展开更多
By the fixed point theorem on a cone and monotone iterativc technique, the existence and multiplicity of the positive radial solutions to a class of quasilinear elliptic equations are considered. Also, using the monot...By the fixed point theorem on a cone and monotone iterativc technique, the existence and multiplicity of the positive radial solutions to a class of quasilinear elliptic equations are considered. Also, using the monotone iteration method the authors deal with the boundary value problem as the nonlinear term f(t, u) increases in u.展开更多
In this paper, we establish the existence of positive radially symmetric solutions of div(|Du|p-2Du) + λf(r,u(r) ) = 0 in domain R1 < r < R0 or 0 < r < ∞ with a variety of Dirichlet boundary conditions. ...In this paper, we establish the existence of positive radially symmetric solutions of div(|Du|p-2Du) + λf(r,u(r) ) = 0 in domain R1 < r < R0 or 0 < r < ∞ with a variety of Dirichlet boundary conditions. The function f is allowed to be singular when u = 0.展开更多
In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation me...In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.展开更多
Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical ...Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:{-div(|△u|p-2△u)=|x|s|u|p*(s)-2u+λ|x|t|u|p-2u, x∈B1, u|σB1 =0, where t, s〉-p, 2≤p〈N, p*(s)= (N+s)pN-p andλ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N 〉p(p-1)t+p(p2-p+1) andλ∈(0,λ1,t), whereλ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ (ps+p) min{1, p+t/p+s}+p2p-(p-1) min{1, p+tp+s} andλ〉0 is small.展开更多
We prove the two existence results of the radially symmetric strong solutions to the Navier- Stokes-Poisson equations for isentropic compressible fluids. The important point in this paper is that the initial density i...We prove the two existence results of the radially symmetric strong solutions to the Navier- Stokes-Poisson equations for isentropic compressible fluids. The important point in this paper is that the initial density is vacuum. It is different from weak solutions. Now we need some compatibility condition.展开更多
Here we generalize the "BBH"-asymptotic analysis to a simplified mathematical model for tho planar ferromaguets and antiferromagnets. To develop such a static theory is a necessary step for a rigorous mathem...Here we generalize the "BBH"-asymptotic analysis to a simplified mathematical model for tho planar ferromaguets and antiferromagnets. To develop such a static theory is a necessary step for a rigorous mathematical justification of dynamical laws for the magnetic vortices formally derived in [1] and [2].展开更多
Existence and uniqueness results are obtained for positive radial solutions of a class of quasilinear elliptic equations in a N-ball or an annulus without monotone assumptions on the nonlinear term f.It is also proved...Existence and uniqueness results are obtained for positive radial solutions of a class of quasilinear elliptic equations in a N-ball or an annulus without monotone assumptions on the nonlinear term f.It is also proved that there is no non-radial positive solution.展开更多
In this paper,we study superlinear elliptic equations with mixed boundary value conditions in annular domains.It is assumed that the nonlinearities depend on the derivative terms.Some results about existence of soluti...In this paper,we study superlinear elliptic equations with mixed boundary value conditions in annular domains.It is assumed that the nonlinearities depend on the derivative terms.Some results about existence of solutions are established by using the Nehari manifold technique,as well as iterative technique.展开更多
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.
基金Supported by the National Natural Science Foundation of China(11501342,12001344).
文摘In this paper,we consider the following generalized nonlinear k-Hessian system G(S_(k)^(1/k)(λ(D^(2)z1)))S_(k)^(1/k)(λ(D^(2)z1))=φ(|x|,z1,z2),x∈R^(N),G(S_(k)^(1/k)(λ(D^(2)z2)))S_(k)^(1/k)(λ(D^(2)z2))=ψ(|x|,z1,z2),x∈R^(N),where G is a nonlinear operator and Sk(λ(D^(2)z))stands for the k-Hessian operator.We first are interested in the classification of positive entire k-convex radial solutions for the k-Hessian system ifφ(|x|,z1,z2)=b(|x|)φ(z1,z2)andψ(|x|,z1,z2)=h(|x|)ψ(z1).Moreover,with the help of the monotone iterative method,some new existence results on the positive entire k-convex radial solutions of the k-Hessian system with the special non-linearitiesψ,φare given,which improve and extend many previous works.
基金The work was supported by NNSF(1 9771 0 0 7) of China
文摘The existence of positive radial solutions to the systems of m(m≥1) semilinear elliptic equations Δu+p(r)f(u)=0,0<A<r<B in annuli with Dirichlet(Dirichlet/Neumann)boundary conditions,is studied,where r=x 2 1+...+x 2 n,n≥1.u=(u 1,...,u m),p(r)f(u)=(p 1(r)f 1(u),...,p m(r)f m(u)), and p(r) may be singular at r=A or r=B,f may be singular at u=0.
文摘In this paper, we study the positive radial solutions for elliptic systems to the nonlinear BVP:<br /> <p> <img src="Edit_4da56369-d8f9-42d0-9650-c15af375d30c.bmp" alt="" />, where Δ<em>u</em> = <em>div</em> (<span style="white-space:nowrap;">∇</span><em>u</em>) and Δ<em>v</em> = <em>div</em> (<span style="white-space:nowrap;">∇</span><em>v</em>) are the Laplacian of <em>u</em>, <span style="white-space:nowrap;"><em>λ</em> </span>is a positive parameter, Ω = {<em>x</em> ∈ R<sup><em>n</em></sup> : <em>N</em> > 2, |<em>x</em>| > <em>r</em><sub>0</sub>, <em>r</em><sub>0</sub> > 0}, let <em>i</em> = [1,2] then <em>K<sub>i</sub></em> :[<em>r</em><sub>0</sub>,∞] → (0,∞) is a continuous function such that lim<sub><em>r</em>→∞</sub> <em>k<sub>i</sub></em>(<em>r</em>) = 0 and <img src="Edit_19f045da-988f-43e9-b1bc-6517f5734f9c.bmp" alt="" /> is The external natural derivative, and <img src="Edit_3b36ed6b-e780-46de-925e-e3cf7c6a125f.bmp" alt="" />: [0, ∞) → (0, ∞) is a continuous function. We discuss existence and multiplicity results for classes of <em>f </em>with a) <em>f<sub>i </sub></em>> 0, b) <em>f<sub>i </sub></em>< 0, and c) <em>f<sub>i </sub></em>= 0. We base our presence and multiple outcomes via the Sub-solutions method. We also discuss some unique findings. </p>
文摘By the Schauder-Tychonoff fixed-point theorem, we inyestigate the existence and asymptotic behavior of positive radial solutions of fully nonlinear elliptic equations in Re. We give some sufficient conditions to guarantee the existence of bounded and unbounded radial solutions and consider the nonexistence of positive solution in Rn.
基金Supported by the National Natural Science Foundation of China(11961060)Graduate Research Support of Northwest Normal University(2021KYZZ01032)。
文摘In this paper,we focus on a general n-dimension system of k-Hessian equations.By introducing some new suitable growth conditions,the existence results of radial k-admissible solutions of the k-Hessian system are obtained.Our approach is largely based on the well-known fixed-point theorem.
基金supported by JSPS Grant-in-Aid for Scientific Research(C)(15K04970)
文摘In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE anal- ysis, we extend previous results to cases where nonlinear terms may have sublinear growth. As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified SchrSdinger equations.
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.
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.
基金The Project Supported NSF of Guangdong (990444) NSFC (10071014).
文摘In this paper, we consider a class of semilinear parabolic differential equation where nonlinear term has local bounded coefficients. Under some assumptions, we get the existence and uniqueness of solution. Terminate to the time variable, we obtain the so called generalized Emden-Fowler equation and the asymptotic behavior of positive radial solutions have been given in all dimensions. At the end of this paper, we give its application to critical branching Brownian motion (also called measure-valued branching processes).
文摘Existence and multiplicity results of positive radially symmetric solution ofthe eigenvalue problemsare obtained for fi being a N-ball or an annulus via Leray-Schauder degreetheory and variational method.
文摘In this paper, the nonexistence of positive entire solutions for div(|Du| p-2 Du)q(x)f(u),x∈R N, is established, where p】1,Du=(D 1u,...,D Nu),q∶R N→(0,∞) and f∶(0,∞)→(0,∞) are continuous functions.
基金The project was supported by NNSF of China (10371116)
文摘By the fixed point theorem on a cone and monotone iterativc technique, the existence and multiplicity of the positive radial solutions to a class of quasilinear elliptic equations are considered. Also, using the monotone iteration method the authors deal with the boundary value problem as the nonlinear term f(t, u) increases in u.
文摘In this paper, we establish the existence of positive radially symmetric solutions of div(|Du|p-2Du) + λf(r,u(r) ) = 0 in domain R1 < r < R0 or 0 < r < ∞ with a variety of Dirichlet boundary conditions. The function f is allowed to be singular when u = 0.
基金Supported by National Natural Science Foundation of China(11071198)
文摘In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.
基金supported by the National Natural Science Foundation of China(11326139,11326145)Tian Yuan Foundation(KJLD12067)Hubei Provincial Department of Education(Q20122504)
文摘Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:{-div(|△u|p-2△u)=|x|s|u|p*(s)-2u+λ|x|t|u|p-2u, x∈B1, u|σB1 =0, where t, s〉-p, 2≤p〈N, p*(s)= (N+s)pN-p andλ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N 〉p(p-1)t+p(p2-p+1) andλ∈(0,λ1,t), whereλ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ (ps+p) min{1, p+t/p+s}+p2p-(p-1) min{1, p+tp+s} andλ〉0 is small.
基金Supported by NSF of China (No.10531020)the Program of 985 Innovation Engineering on Information in Xiamen University(2004-2007)NCETXMU
文摘We prove the two existence results of the radially symmetric strong solutions to the Navier- Stokes-Poisson equations for isentropic compressible fluids. The important point in this paper is that the initial density is vacuum. It is different from weak solutions. Now we need some compatibility condition.
基金Supported by the Dean's Dissertation FellowshipSupported by a NSF grant
文摘Here we generalize the "BBH"-asymptotic analysis to a simplified mathematical model for tho planar ferromaguets and antiferromagnets. To develop such a static theory is a necessary step for a rigorous mathematical justification of dynamical laws for the magnetic vortices formally derived in [1] and [2].
基金Supported by the Youth Foundations of National Education Commuttee the Committee on Science and Technology of Henan Province
文摘Existence and uniqueness results are obtained for positive radial solutions of a class of quasilinear elliptic equations in a N-ball or an annulus without monotone assumptions on the nonlinear term f.It is also proved that there is no non-radial positive solution.
基金Supported by National Natural Science Foundation of China(Grant No.11871242)Natural Science Foundation of Jilin Province of China(Grant No.20200201248JC)the Fundamental Research Funds for the Central Universities。
文摘In this paper,we study superlinear elliptic equations with mixed boundary value conditions in annular domains.It is assumed that the nonlinearities depend on the derivative terms.Some results about existence of solutions are established by using the Nehari manifold technique,as well as iterative technique.