Structure of positive solutions for quasilinear elliptic systems—degenerate ecological models

We study the degenerate ecological models where are positive numbers. The structure of positive solutions of the models is discussed via bifurcation theory and monotone techniques. Copyright © 2004 John Wiley & Sons, Ltd.     

Existence of solutions for a degenerate auasilinear elliptic system in bounded domain

Using variational methods, we study the existence of weak solutions forthe degenerate quasilinear elliptic system$$\left\{\begin{array}{ll}- \mathrm{div}\Big(h_1(x)|\nabla u|^{p-2}\nabla u\Big) = F_{u}(x,u,v) &\text{ in } \Omega,\\-\mathrm{div}\Big(h_2(x)|\nabla v|^{q-2}\nabla v\Big) = F_{v}(x,u,v) &\text{ in } \Omega,\\u=v=0 & \textrm{ on } \partial\Omega,\end{array}\right.$$where $\Omega\subset \mathbb R^N$ is a smooth bounded domain, $\nabla F= (F_u,F_v)$ stands for the gradient of $C^1$-function $F:\Omega\times\mathbb R^2 \to \mathbb R$, the weights $h_i$, $i=1,2$ are allowed to vanish somewhere,the primitive $F(x,u,v)$ is intimately related to the first eigenvalue of acorresponding quasilinear system.     

Existence and uniqueness of positive solutions for a class of semilinear elliptic systems

The authors prove the uniqueness and existence of positive solutions for the semilinear elliptic system which involves nonlinearities with sublinear growth conditions.     

Existence of non-radially symmetric viscosity solutions to semilinear degenerate elliptic equations with radially symmetric coefficients in the plane, Part I

We prove the existence of non-radially symmetric solutions for semilinear degenerate elliptic equations with radially symmetric coefficients in the plane. We adapt the viscosity solution for the weak solution. The key arguments consist of the analysis of the structure of 2π-periodic solutions for the associated Laplace-Beltrami operator and construction of super- and sub-solutions which have the prescribed asymptotic structures.     

奇异椭圆方程组径向正解的存在性

Abstract. The existence of positive radial solutions to the systems of     

Existence of nonnegative solutions for second order m-point boundary value problems at resonance

In this paper, the minimal and maximal nonnegative solutions for second order m-point boundary value problems at resonance are investigated by using a new fixed point theorem of increasing operators. And the monotone iterative sequences which converge to solutions are given. The method is different essentially from the previous papers used by coincidence degree continuation theorem of Mawhin.     

Existence and uniqueness of nonnegative solutions for a boundary blow-up problem

Assume that Ω is a bounded domain in RN (N?3) with smooth boundary ∂Ω. In this work, we study existence and uniqueness of blow-up solutions for the problem −Δp(u)+c(x)|∇u|^p−1+F(x,u)=0 in Ω, where 2?p. Under some conditions related to the function F, we give a sufficient condition for existence and nonexistence of nonnegative blow-up solutions. We study also the uniqueness of these solutions.     

Existence of solutions for a class of quasilinear elliptic systems with Hardy potential

In this paper, by using the Morse theory, we obtain the existence of nontrivial weak solutions of quasilinear elliptic systems with Hardy potential.     

Integral representations of nonnegative solutions for parabolic equations and elliptic Martin boundaries

We consider nonnegative solutions of a parabolic equation in a cylinder D×(0,T), where D is a noncompact domain of a Riemannian manifold. Under the assumption [IU] (i.e., the associated heat kernel is intrinsically ultracontractive), we establish an integral representation theorem: any nonnegative solution is represented uniquely by an integral on (D×{0})∪(M_∂D×[0,T)), where M_∂D is the Martin boundary of D for the associated elliptic operator. We apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions. We also show that [IU] implies the condition [SP] (i.e., the constant function 1 is a small perturbation of the elliptic operator on D).     

Existence and uniqueness of solutions for quasilinear elliptic systems

We obtain necessary and sufficient conditions for the existence of positive solutions for a class of sublinear Dirichlet quasilinear elliptic systems.

     

Existence and multiplicity of solutions to a class of noncooperative elliptic systems with superlinear nonlinear terms

We study a class of noncooperative elliptic systems. By applying a new superlinear condition, it is shown that there exists a nontrivial weak solution. Moreover, infinitely many solutions are obtained.     

Existence and multiplicity of solutions for asymptotically linear noncooperative elliptic systems

Using the minimax methods in critical point theory and a generalized Landesman-Lazer type condition, we establish two existence results of solutions for asymptotically linear noncooperative elliptic systems at resonance. Besides this, we obtain two solutions in the case of near resonance.     

一类椭圆系统正径向解的存在性

通过考察函数f,g在端点处的性质证明了一类非线性椭圆系统在环域上的正径向解的存在性.     

Existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms

In this paper, the authors establish the existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms by using the theory of Leray-Schauder's degree.     

Large time behavior of solutions to degenerate parabolic equations with weights

We study the degenerate parabolic equation t_∂u=a(δ(x))upΔu−g(u) in Ω×(0,∞), where Ω⊂RN (N?1) is a smooth bounded domain, p?1, δ(x)=dist(x,∂Ω) and a is a continuous nondecreasing function such that a(0)=0. Under some suitable assumptions on a and g we prove the existence and the uniqueness of a classical solution and we study its asymptotic behavior as t→∞.     

Existence of positive radial solutions for some nonvariational superlinear elliptic systems

The system under consideration is

     

Existence and convexity of local solutions to degenerate Hessian equations

In this work, we prove the existence of convex solutions to the following k-Hessian equation

$S_k[u]=K(y)g(y,u,Du)$

in the neighborhood of a point $(y_0,u_0,p_0)\in {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n$, where $g\in C^{\infty },g(y_0,u_0,p_0)>0$, $K\in C^{\infty }$ is nonnegative near $y_0$, $K(y_0)=0$ and $\text{Rank}(D_y^{2}K)(y_0)\ge n?k+1$.     

Existence and regularity of positive solutions of a degenerate elliptic problem

Existence and regularity of positive solutions of a degenerate elliptic Dirichlet problem of the form in Ω, on , where Ω is a bounded smooth domain in , , are obtained via new embeddings of some weighted Sobolev spaces with singular weights and . It is seen that and admit many singular points in Ω. The main embedding results in this paper provide some generalizations of the well‐known Caffarelli–Kohn–Nirenberg inequality.     

Existence and multiplicity of weak solutions for a singular semilinear elliptic equation

In this paper we study existence and multiplicity of weak solutions of the homogenous Dirichlet problem for a singular semilinear elliptic equation with a quadratic gradient term. The proofs for the main results are based on a priori estimates of solutions of approximate problems.