NONEXISTENCE AND EXISTENCE OF POSITIVE SOLUTIONS FOR 2nTH-ORDER SINGULAR SUPERLINEAR PROBLEMS WITH STRUM-LIOUVILLE BOUNDARY CONDITIONS

This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C 2 n- 2 [0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C 2 n-1 [0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.     

A necessary and sufficient condition for 2nth-order singular super-linear m-point boundary value problems

In this paper, we study the existence of positive solutions for singular super-linear m-point boundary value problems of 2nth-order ordinary differential equations. A necessary and sufficient condition for the existence of C2n−2[0,1] positive solutions as well as C2n−1[0,1] positive solutions is given by means of the fixed point theorems on cones.     

Large time behavior of solutions to a degenerate parabolic equation not in divergence form

In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: ut=upΔu+auq−bur, subject to the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then study the large time behavior for the global solutions. When the positive source dominates the model, we prove that the global solutions uniformly tend to the positive steady state of the problem as t→∞. In particular, we establish the uniform asymptotic profiles for the decay solutions when the problem is governed by the nonlinear diffusion or absorption.     

Positive solutions for m-point boundary-value problems with one-dimensional p-Laplacian

In this paper, we investigate the existence of positive solutions for a classes of m-point boundary value problems with p-Laplacian. By applying a monotone iterative technique, some sufficient conditions for the existence of twin positive solutions are established.     

Solutions to the system of operator equations AXB=C=BXA

In this paper, we present some necessary and sufficient conditions for the existence of solutions, hermitian solutions and positive solutions to the system of operator equations AXB=C=BXA in the setting of bounded linear operators on a Hilbert space. Moreover, we obtain the general forms of solutions, hermitian solutions and positive solutions to the system above.     

Nonexistence of bounded energy solutions for a critical exponent problem

In this paper, we discuss positive solutions for certain weighted elliptic equations with critical Sobolev exponent in R^N. The weights depend on a positive parameter γ, which is allowed to increase to infinity. While for small values of γ solutions are completely classified, an attempt to such a classification is much more difficult for large values of the parameter. In the present work we prove the nonexistence of solutions with bounded energy as γ increases to infinity. We also prove a multiplicity result for high energy solutions.     

On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function

In this paper, we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign-changing weight function. With the help of the Nehari manifold, we prove that there are at least two positive solutions for Eq. (Eλ,f) in bounded domains.     

Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems

We study the existence and asymptotic behavior of positive and sign-changing multipeak solutions for the equation $$ -\varepsilon^2\Delta v+V(x)v=f(v)\quad{\rm in}\,\,\,\mathbb{R}^N, $$ where ?? is a small positive parameter, f a superlinear, subcritical and odd nonlinearity, V a uniformly positive potential. No symmetry on V is assumed. It is known (Kang and Wei in Adv Differ Equ 5:899?C928, 2000) that this equation has positive multipeak solutions with all peaks approaching a local maximum of V. It is also proved that solutions alternating positive and negative spikes exist in the case of a minimum (see D??Aprile and Pistoia in Ann Inst H. Poincaré Anal Non Linéaire 26:1423?C1451, 2009). The aim of this paper is to show the existence of both positive and sign-changing multipeak solutions around a nondegenerate saddle point of V.     

Constrained core solutions for totally positive games with ordered players

In many applications of cooperative game theory to economic allocation problems, such as river-, polluted river- and sequencing games, the game is totally positive (i.e., all dividends are nonnegative), and there is some ordering on the set of the players. A totally positive game has a nonempty core. In this paper we introduce constrained core solutions for totally positive games with ordered players which assign to every such a game a subset of the core. These solutions are based on the distribution of dividends taking into account the hierarchical ordering of the players. The Harsanyi constrained core of a totally positive game with ordered players is a subset of the core of the game and contains the Shapley value. For special orderings it coincides with the core or the Shapley value. The selectope constrained core is defined for acyclic orderings and yields a subset of the Harsanyi constrained core. We provide a characterization for both solutions.     

An optimal Liouville-type theorem for radial entire solutions of the porous medium equation with source

We consider nonnegative (continuous) weak solutions of the porous medium equation with source ut−Δum=up, with p>m>1. We address the question of existence of nontrivial entire solutions, that is, solutions defined for all x∈Rn and t∈R. Such solutions do exist for critical and supercritical p (positive bounded stationary solutions). Our main result asserts that for subcritical p there are no bounded radial entire solutions u?0. This parabolic Liouville-type theorem is the first of its kind for reaction-diffusion equations involving porous medium operators. On the other hand, it will be the main tool in the study of universal bounds for global and nonglobal solutions in the forthcoming article [K. Ammar, Ph. Souplet, Liouville-type results and universal bounds for positive solutions of the porous medium equation with source, in preparation]. The proof is based on intersection-comparison arguments. A key step is to first show the positivity of possible bounded radial entire solutions. Among other auxiliary results, we establish pointwise gradient estimates of possible independent interest.     

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

Structure of a class of singular boundary value problem with superlinear effect

This paper investigates the structure of solutions of singular boundary value problem with superlinear effect. It is proved that the closure of positive solution set possesses a maximal subcontinuum C (i.e., a maximal closed connected subset of solutions), which comes from (0,θ) and tends to (0,+∞) finally. As a corollary, the existence of multiple positive solutions and the behavior of solutions according to parameter λ are obtained.     

Positive solutions for three-point one-dimensional p-Laplacian boundary value problems with advanced arguments

This paper considers the existence of positive solutions for advanced differential equations with one-dimensional p-Laplacian. To obtain the existence of at least three positive solutions we use a fixed point theorem due to Avery and Peterson.     

Positive solution of singular Dirichlet boundary value problems for second order differential equation system

This paper investigates the existence of positive solutions of singular Dirichlet boundary value problems for second order differential system. A necessary and sufficient condition for the existence of C[0,1]×C[0,1] positive solutions as well as C¹[0,1]×C¹[0,1] positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems. Our nonlinearity fi(t,x₁,x₂) may be singular at x₁=0, x₂=0, t=0 and/or t=1, i=1,2.     

Existence and multiplicity of positive solutions for singular quasilinear problems

In this work we combine perturbation arguments and variational methods to study the existence and multiplicity of positive solutions for a class of singular p-Laplacian problems. In the first two theorems we prove the existence of solutions in the sense of distributions. By strengthening the hypotheses, in the third and last result, we establish the existence of two ordered positive weak solutions.     

A variational inequality index for condensing maps in Hilbert spaces and applications to semilinear elliptic inequalities

A variational inequality index for γ-condensing maps is established in Hilbert spaces. New results on existence of nonzero positive solutions of variational inequalities for such maps are proved by using the theory of variational inequality index. Applications of such a theory are given to existence of nonzero positive weak solutions for semilinear second order elliptic inequalities, where previous results of variational inequalities for S-contractive maps cannot be applied.     

Properties of positive solutions to an elliptic equation with negative exponent

In this paper, we study some quantitative properties of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics as well as differential geometry. We can get some useful local gradient estimate and L¹ lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given.     

A uniform estimate for scalar curvature equation on manifolds of dimension 4

We consider the solutions to the prescribed scalar curvature equation on a four-dimensional Riemannian manifold M. We prove an upper bound for the supremum of all the solutions on every compact subset K of M, provided that all the solutions on M are bounded from below by a positive number.     

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

We investigate a semilinear elliptic equation with a logistic nonlinearity and an indefinite nonlinear boundary condition, both depending on a parameter λ. Overall, we analyze the effect of the indefinite nonlinear boundary condition on the structure of the positive solutions set. Based on variational and bifurcation techniques, our main results establish the existence of three nontrivial non-negative solutions for some values of λ, as well as their asymptotic behavior. These results suggest that the positive solutions set contains an S-shaped component in some case, as well as a combination of a C-shaped and an S-shaped components in another case.     

Multiple solutions and their limiting behavior of coupled nonlinear Schrödinger systems

We study the multiplicity of positive solutions and their limiting behavior as ? tends to zero for a class of coupled nonlinear Schrödinger system in R^N. We relate the number of positive solutions to the topology of the set of minimum points of the least energy function for ? sufficiently small. Also, we verify that these solutions concentrate at a global minimum point of the least energy function.