Multiple positive solutions for a quasilinear system of Schrödinger equations

We consider the quasilinear system

Existence and uniqueness of positive radial solutions for the Lane–Emden system

In this article, we consider (component-wise) positive radial solutions of a weakly coupled system of elliptic equations in a ball with homogeneous nonlinearities. The existence is well-known in general: We give a result for the remaining cases. The uniqueness is less studied: We complement the known results.     

Existence of positive solutions for a system of second-order m-point BVPs with variable parameters

In this paper, we study the existence of positive solutions for a class of second-order nonlinear m-point boundary value problems of differential system. By using the fixed point theory of cone expansion and compression with norm type, we show the sufficient conditions for the existence results.     

Multiple positive solutions of the stationary Keller–Segel system

We consider the stationary Keller–Segel equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta v+v=\lambda e^v, \quad v>0 \quad &{} \text {in }\Omega ,\\ \partial _\nu v=0 &{}\text {on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \) is a ball. In the regime \(\lambda \rightarrow 0\), we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given \(n\in \mathbb {N}_0\), we build a solution having multiple layers at \(r_1,\ldots ,r_n\) by which we mean that the solutions concentrate on the spheres of radii \(r_i\) as \(\lambda \rightarrow 0\) (for all \(i=1,\ldots ,n\)). A remarkable fact is that, in opposition to previous known results, the layers of the solutions do not accumulate to the boundary of \(\Omega \) as \(\lambda \rightarrow 0\). Instead they satisfy an optimal partition problem in the limit.

     

Multiplicity of positive periodic solutions to a generalized delayed predator–prey system with stocking

In this paper, by using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a generalized delayed predator–prey system with stocking is established. When our result is applied to a delayed predator–prey system with nonmonotonic functional response and stocking, we establish the sufficient condition for the existence of multiple positive periodic solutions for the system.     

A uniqueness theorem of radially symmetric positive stationary solutions for a competition–diffusion system

In this paper, as a first step to study the global structure of solution, we treat a class of reaction–diffusion systems with competitive interaction, and discuss a uniqueness theorem of radially symmetric solutions for the system. To do this, the comparison principle and the shooting method are employed, and the spatial profile is investigated.     

Properties of positive solutions to a nonlinear parabolic problem

This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of solutions are described in terms of different parameters appearing in this problem. Moreover, by a result of Chasseign and Vazquez and the comparison principle, we deduce that the blow-up occurs only on the boundary (?)Ω. In addition, for a bounded Lipschitz domainΩ, we establish the blow-up rate estimates for the positive solution to this problem with a= 0.     

Existence and uniqueness of positive solutions of semilinear elliptic equations

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.     

Infinitely many positive solutions for a Schrödinger–Poisson system

We are interested in the existence of infinitely many positive non-radial solutions of a Schrödinger–Poisson system with a positive radial bounded external potential decaying at infinity.     

Multiplicity of positive solutions of a nonlinear Schrödinger equation

We consider the multiple existence of positive solutions of the following nonlinear Schrödinger equation: where if N3 and p(1, ) if N=1,2, and a(x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well := int a^–1(0) consisting of k components and the first eigenvalues of –+b(x) on _j under Dirichlet boundary condition are positive for all . Under these conditions we show that (PM) has at least 2^k–1 positive solutions for large . More precisely we show that for any given non-empty subset , (P) has a positive solutions u(x) for large . In addition for any sequence _n we can extract a subsequence _ni along which u_ni converges strongly in H¹(R^N). Moreover the limit function u(x)=lim_iu_ni satisfies (i) For jJ the restriction u|_j of u(x) to _j is a least energy solution of –v+b(x)v=v^p in _j and v=0 on _j. (ii) u(x)=0 for .Mathematics Subject Classifications (2000):35Q55, 35J20     

Existence of positive solutions to an elliptic system with coupled nonlinearity†

In this article, we study weakly coupled systems of elliptic equations without Hamiltonian structure in both bounded domain and the whole space. By using a priori estimates, Pohozaev identity and degree theory, we are able to show the existence of positive solutions.     

Integrability and continuity of solutions to Fokker–Planck–Kolmogorov equations

New results concerning the local integrability of any order and continuity of solution densities of Fokker–Planck–Kolmogorov equations with nondifferentiable coefficients are obtained.     

Uniqueness and stability of positive steady state solutions for a ratio-dependent predator–prey system with a crowding term in the prey equation

This paper deals with a ratio-dependent predator–prey system with a crowding term in the prey equation, where it is assumed that the coefficient of the functional response is less than the coefficient of the intrinsic growth rates of the prey species. We demonstrate some special behaviors of solutions to the system which the coexistence states of two species can be obtained when the crowding region in the prey equation only is designed suitably. Furthermore, we demonstrate that under some conditions, the positive steady state solution of the predator–prey system with a crowding term in the prey equation is unique and stable. Our result is different from those ones of the predator–prey systems without the crowding terms.     

A sufficient and necessary condition for the existence of positive solutions for a prey–predator system with Ivlev-type functional response

We deal with a predator–prey interaction model with Ivlev-type functional response which is subject to the homogeneous Dirichlet boundary condition. On the basis of the fixed point index theory, we give a sufficient and necessary condition for the existence of positive solutions of the model.