SECONDARY CRITICAL EXPONENT AND LIFE SPAN FOR A DOUBLY SINGULAR PARABOLIC EQUATION WITH A WEIGHTED SOURCE

This paper deals with the Cauchy problem for a doubly singular parabolic equation with a weighted source ut=div｜u｜p-2um）＋｜x｜α uq ,（x,t）∈RN×（0,t）,where N ≥ 1, 1 〈 p 〈 2, m 〉 max（0,3 -p/N} satisfying 2 〈 p＋m 〈 3, q 〉 1, and（α 〉 N（3 - p - m） - p. We give the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and non-existence of global solutions of the Cauchy problem. Moreover, the life span of solutions is also studied.

PERTURBATION METHODS IN SEMILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY-SOBOLEV EXPONENT

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.

Multiple Solutions to the Problem of Kirchhoff Type Involving the Critical Caffareli-Kohn-Niremberg Exponent, Concave Term and Sign-Changing Weights

In this paper, we establish the existence of at least four distinct solutions to an Kirchhoff type problems involving the critical Caffareli-Kohn-Niremberg exponent, concave term and sign-changing weights, by using the Nehari manifold and mountain pass theorem.

Five Nontrivial Solutions of p-Laplacian Problems Involving Critical Exponents and Singular Cylindrical Potential

In this paper, we establish the existence of at least five distinct solutions to a p-Laplacian problems involving critical exponents and singular cylindrical potential, by using the Nehari manifold, concentration-compactness principle and mountain pass theorem

Multiple solutions for weighted nonlinear elliptic system involving critical exponents

In this paper, by using the idea of category, we investigate how the shape of the graph of h（x） affects the number of positive solutions to the following weighted nonlinear elliptic system： = （ N-2-2a 2. where 0 is a smooth bounded domain in ]1N （N 〉 3）, A, cr 〉 0 are parameters, 0 ≤ μ 〈 μa a 2 ＇ h（x）, KI（X） and K2（x） are positive continuous functions in , 1 〈 q 〈 2, a, β 〉 1 and a ＋ β = 2＊（a,b） （2＊ （a, b） 2N = N-2（1＋a-b） is critical Sobolev-Hardy exponent）. We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters （）, r） belongs to a certain subset of N2.

THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN R^N WITH A HARDY TERM

In this paper,we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:(−Δ)s u−γu|x|2s=|u|2∗s(β)−2 u|x|β+[Iμ∗Fα(⋅,u)](x)fα(x,u),u∈H˙s(R n),(0.1)(1)where s∈(0,1),0≤α,β<2s<n,μ∈(0,n),γ<γH,Iμ(x)=|x|−μ,Fα(x,u)=|u(x)|2#μ(α)|x|δμ(α),fα(x,u)=|u(x)|2#μ(α)−2 u(x)|x|δμ(α),2#μ(α)=(1−μ2n)⋅2∗s(α),δμ(α)=(1−μ2n)α,2∗s(α)=2(n−α)n−2s andγH=4 sΓ2(n+2s4)Γ2(n−2s4).We show that problem(0.1)admits at least a weak solution under some conditions.To prove the main result,we develop some useful tools based on a weighted Morrey space.To be precise,we discover the embeddings H˙s(R n)↪L 2∗s(α)(R n,|y|−α)↪L p,n−2s2 p+pr(R n,|y|−pr),(0.2)(2)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α))and r=α2∗s(α).We also establish an improved Sobolev inequality,(∫R n|u(y)|2∗s(α)|y|αdy)12∗s(α)≤C||u||θH˙s(R n)||u||1−θL p,n−2s2 p+pr(R n,|y|−pr),∀u∈H˙s(R n),(0.3)(3)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α)),r=α2∗s(α),0<max{22∗s(α),2∗s−12∗s(α)}<θ<1,2∗s=2nn−2s and C=C(n,s,α)>0 is a constant.Inequality(0.3)is a more general form of Theorem 1 in Palatucci,Pisante[1].By using the mountain pass lemma along with(0.2)and(0.3),we obtain a nontrivial weak solution to problem(0.1)in a direct way.It is worth pointing out that(0.2)and 0.3)could be applied to simplify the proof of the existence results in[2]and[3].

On the Weighted Elliptic Problems Involving Multi-singular Potentials and Multi-critical Exponents

Suppose Ω belong to R^N（N≥3） is a smooth bounded domain,ξi∈Ω,0〈ai〈√μ,μ：=（（N-1）/2）^2,0≤μi〈（√μ-ai）^2,ai〈bi〈ai＋1 and pi：=2N/N-2（1＋ai-bi）are the weighted critical Hardy-Sobolev exponents, i = 1, 2,..., k, k ≥ 2. We deal with the conditions that ensure the existence of positive solutions to the multi-singular and multi-critical elliptic problem ∑i=1^k（-div（｜x-ξi｜^-2ai△↓u）-μiu/｜x-ξi｜^2（1＋ai）-u^pi-1/｜x-ξi｜^bipi）=0with Dirichlet boundary condition, which involves the weighted Hardy inequality and the weighted Hardy-Sobolev inequality. The results depend crucially on the parameters ai, bi and #i, i -- 1, 2,..., k.

一类具Hardy-Sobolev临界增长拟线性椭圆方程的径向解

考虑一类具有临界Hardy-Sobolev指数的p-Laplace拟线性椭圆方程及其扰动问题的径向解的存在性,通过Lions引理和非线性泛函理论知识建立Sobolev空间到加权Lebesgue空间的紧嵌入定理,并利用极小化方法,得到了上述问题在全空间和有界区域上的径向解的存在性。

具有加权Hardy-Sobolev临界指数的p-Laplacian方程解的存在性和多重性(英文)

通过变分方法和分析技巧,得到了一类具有加权Hardy-Sobolev临界指数和超线性的非线性项的p-Laplacian方程解的存在性与多重性结果.

具有加权Hardy-Sobolev临界指数的半线性椭圆方程的无穷多个任意小解

考虑了具有加权Hardy-Sobolev临界指数的半线性椭圆方程,运用变分方法和分析技巧得到了1列收剑于零的无穷多个任意小解.

具有临界带权Hardy-Sobolev指数的椭圆方程的多个正解(英文)

通过变分方法和一些分析技巧研究了一类具有临界带权Hardy-Sobolev指数的半线性椭圆方程正解的存在性与多解性问题.

带有加权Hardy-Sobolev临界指数的拟线性椭圆方程正解的存在性和多重性

研究了一类加权拟线性椭圆方程,利用Ekeland变分原理和强极大值原理,证明了该方程正解的存在性和多重性.

一类具临界指数的半线性椭圆方程注

建立一类带第一特征值λ1的具临界指数的半线性椭圆方程-Δu=λ1u+|u|2*-2u零边值问题的非平凡弱解存在的一个必要条件,并在加权情况下得其相应结论.

R^n上一类拟线性椭圆方程正解的存在性(英文)

本文讨论了Rn 上如下一类带临界增长的拟线性椭圆方程正解的存在性 :-div(｜ u｜p- 2 u) -axn｜ u｜p- 2 u xn +｜u｜p- 2u=up - 1 ,xn ≠ 0 ,x∈Rn.这里 ,1<p <n ,0 <a<p-1,其中 p =(n+a) pn +a-p是带权的Sobolev空间W1 ,p(a ,Rn) →Lq(a ,Rn)的临界指数 .通过修改后的集中列紧原理 ,证明了此类问题至少具有一个正解 .

对称性与Sobolev嵌入

建立了一个Sobolev空间上部分对称函数到加权Lp空间的嵌入定理,并给出这一定理对具临界增长非线性椭圆边值问题的应用。过去这类结论主要是关于Holder函数的,笔者将这一结论推广到连续函数。

A Quasilinear Parabolic System with Nonlocal Sources and Weighted Nonlocal Boundary Conditions

In this paper, we investigate the blow-up properties of a quasilinear reaction-diffusion system with nonlocal nonlinear sources and weighted nonlocal Dirichlet boundary conditions. The critical exponent is determined under various situations of the weight functions. It is observed that the boundary weight functions play an important role in determining the blow-up conditions. In addition, the blow-up rate estimate of non-global solutions for a class of weight functions is also obtained, which is found to be independent of nonlinear diffusion parameters m and n.

对称Sobolev空间到加权L^(2^*)空间的紧嵌入

根据LIons引理,建立了Sobolev空间H1(Rn)到一类加权L2*空间紧嵌入定理。将这一理论应用到Rn上的标量场方程,得到具有限能量的正解,推广了文献[2]的部分结果。

对称Sobolev函数到加权函数空间的嵌入

证明Sobolev空间W^(1,p)(R^n)上对称函数到某类加权L^p空间存在紧嵌入定理,进而,作为应用,证明在一定条件下,一类非线性项涉临界Sobolev指标的拟线性椭圆方程具有有限能量解的正解.

Infinitely Many Solutions for a Class of Fourth Order Elliptic Equations in R^N

With the aids of variational method and concentration-compactness principle, infinitely many solutions are obtained for a class of fourth order elliptic equations with singular potentialΔ^2u=μ｜u｜^2＊＊（s）-2u/｜x｜^s＋λk（x）｜u｜^r-2 u, u∈H^2,2（R^N） （P）

On the Existence of Ground State Solutions to a Quasilinear Schr?dinger Equation involving p-Laplacian

We consider the following quasilinear Schrodinger equation involving p-Laplacian-Δpu+V(x)|u|^(p-2)u-Δp(|u|^(2η))|u|^(2η-2)u=λ|u|^(q-2)u/|x|^(μ)+|u|^(2ηp*(v)-2)u/|x|^(v)in R^(N),where N>p>1,η≥p/2(p-1),p<q<2ηp^(*)(μ),p^(*)(s)=(p(N-s))/N-p,andλ,μ,νare parameters withλ>0,μ,ν∈[0,p).Via the Mountain Pass Theorem and the Concentration Compactness Principle,we establish the existence of nontrivial ground state solutions for the above problem.