Existence, non-existence and asymptotic behavior of blow-up entire solutions of semilinear elliptic equations

We are mainly concerned with existence, non-existence and the behavior at infinity of non-negative blow-up entire solutions of the equation Δu=ρ(x)f(u) in RN. No monotonicity condition is assumed upon f and, in fact, we obtain solutions with a prescribed behavior both at infinity and at the origin. The method used to get existence is based upon lower and upper solutions techniques while for non-existence we explore radial symmetry, estimates on an associated integral equation and the Keller-Osserman condition.     

Sign-Changing and Multiple Solutions Theorems for Semilinear Elliptic Boundary Value Problems with Jumping Nonlinearities

In this paper, we use the ordinary differential equation theory of Banach spaces and minimax theory, and in particular, the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems, at last we get up to six nontrivial solutions. Received April 21, 1998, Revised November 2, 1998, Accepted January 14, 1999     

Existence of multi-valued solutions with asymptotic behavior of Hessian equations

In this paper, we use the Perron method to prove the existence of multi-valued solutions with asymptotic behavior at infinity of Hessian equations.     

Traveling wave solutions in a delayed lattice competition-cooperation system

In this paper, we first reduce the existence of traveling wave solutions in a delayed lattice competition-cooperation system to the existence of a pair of upper and lower solutions by means of Schauder’s fixed point theorem and the cross iteration scheme, and then we construct a pair of upper and lower solutions to obtain the existence and nonexistence of traveling wave solutions. We also consider the asymptotic behaviour of any nonnegative traveling wave solutions at negative infinity.     

Conditions at infinity for boltzmann's equation.

We consider here realistic conditions at infinity for solutions of the Boltzmann's equation, such as a pure Maxwellian equilibrium at infinity possibly with suitable boundary conditions on an exterior domain, different Maxwellian equilibria at +∞ and -;∞ in a tube-like situation and more generally conditions at infinity obtained from a fixed solution. In order to adapt the recent global existence and compactness results due to R.J. DiPerna and the author, we have to obtain some local a priori estimates on the mass, kinetic energy and entropy. And this is precisely what we achieve here by two different and new methods. The first one consists in using the relative entropy of solutions with respect to a fixed, possibly local, Maxwellian. This method allows to treat general collision kernels with angular cut-off and some of the conditions at infinity mentioned above. The second method is based upon a L¹ estimate and an extension of the entropy identity which uses a truncated H-functional. This method requires a “uniform integrability” condition on the collision kernel but allows to consider the most general conditions at infinity.     

Rotating periodic solutions for asymptotically linear second‐order Hamiltonian systems with resonance at infinity

In this paper, we consider a class of asymptotically linear second‐order Hamiltonian system with resonance at infinity. We will use Morse theory combined with the technique of penalized functionals to obtain the existence of rotating periodic solutions.     

Computations of critical groups and applications to some differential equations at resonance

In this paper, we first consider the computations of the critical groups of a functional with generalized Ahmad–Lazer–Paul type conditions both at zero and infinity. Then the abstract results are applied to investigate some new cases of the existence of nontrivial solutions of the second order Hamiltonian systems and elliptic boundary value problems, which may be resonant both at zero and infinity.     

Discrete problems associated to elliptic equations

This paper deals with the existence of positive solutions of the discrete counterpart of nonlinear elliptic problems. We apply our methods to the study of positive solutions under different hypotheses about the nonlinearities. For example, we consider the cases that are superlinear and sublinear at infinity. Copyright © 2016 John Wiley & Sons, Ltd.     

Fucik spectrum, sign-changing and multiple solutions for semilinear elliptic boundary value problems with jumping nonlinearities at zero and infinity

In this paper, Fucik spectrum, ordinary differential equation theory of Banach spaces and Morse theory are used to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, and some new results on the existence of nontrivial solutions, multiple solutions and sign-changing solutions are obtained. In one case seven nontrivial solutions are got. The techniques have independent interest.     

MULTIPLE SOLUTIONS TO AN ASYMPTOTICALLY LINEAR ROBIN BOUNDARY VALUE PROBLEM

Under some weaker conditions,we prove the existence of at least two solutions to an asymptotically linear elliptic problem with Robin boundary value condition,using truncation arguments.Our results are also valid for the case of the so-called resonance at infinity.     

Stability of non-constant equilibrium solutions for two-fluid Euler–Maxwell systems

In this work we consider periodic problems for two-fluid compressible Euler–Maxwell systems for plasmas. The initial data are supposed to be in a neighborhood of non-constant equilibrium states. Mainly by an induction argument used in Peng (2015), we prove the global stability in the sense that smooth solutions exist globally in time and converge to the equilibrium states as the time goes to infinity. Moreover, we obtain the global stability of solutions with exponential decay in time near the equilibrium states for two-fluid compressible Euler–Poisson systems.     

Fractional Schrödinger–Poisson system with critical growth and potentials vanishing at infinity

In this paper, we are concerned with the existence of positive solutions for a class of fractional Schrödinger–Poisson system with critical nonlinearity and multiple competing potentials, which may decay and vanish at infinity. Under some local conditions, we show the existence and concentration of positive solutions by using the modified penalization method and concentration–compactness principle.     

Stationary solitons of a three-wave model generated by Type II second-harmonic generation in quadratic media

In this paper, we prove the uniqueness of stationary standing wave solutions of an optical model generated by Type II Second Harmonic Generation (SHG) with behaviors tending to zero at infinity under certain conditions on parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established.     

Periodic solutions of singular second order equations at resonance

In this paper, we study the existence of positive periodic solutions for singular second order equations x" + n²/4x+h(x) = p(t), where h has a singularity at the origin and n is a positive integer. We give an explicit condition to ensure the existence of positive periodic solutions when h is an unbounded perturbation at infinity by using qualitative analysis and topological degree theory.     

Multiple solutions to p-Laplacian problems with concave nonlinearities

In this paper we study the p-Laplacian type elliptic problems with concave nonlinearities. Using some asymptotic behavior of f at zero and infinity, three nontrivial solutions are established.     

Dirichlet problems with double resonance and an indefinite potential

We consider semilinear Dirichlet problems with an unbounded and indefinite potential and with a Carathéodory reaction. We assume that asymptotically at infinity the problem exhibits double resonance. Using variational methods, together with Morse theory and flow invariance arguments, we prove multiplicity theorems producing three, five, six or seven nontrivial smooth solutions. In most multiplicity theorems, we provide precise sign information for all the solutions established.     

Asymptotic solutions for mixed-type equations with a small deviation

An asymptotic matrix solution is formulated for a class of mixed-type linear vector equations with a single variable deviation which is small at infinity. This matrix solution describes the asymptotic behavior of all exponentially bounded solutions. A sufficient condition is obtained for there to be no other solutions.     

Completion of R2 with a Conformal Metric as a Closed Surface

In this paper,we obtain some asymptotic behavior results for solutions to the prescribed Gaussian curvature equation.Moreover,we prove that under a con-formal metric in R2,if the total Gaussian curvature is 4π,the conformal area of R2 is finite and the Gaussian curvature is bounded,then R2 is a compact C1,α surface after completion at ∞,for any α ∈(0,1).If the Gaussian curvature has a Holder decay at in-finity,then the completed surface is C2.For radial solutions,the same regularity holds if the Gaussian curvature has a limit at infinity.     

Damped vibration problems with nonlinearities being sublinear at both zero and infinity

In this paper, we study a class of damped vibration problems with nonlinearities being sublinear at both zero and infinity, and we obtain infinitely many nontrivial periodic solutions by using a variant fountain theorem. To the best of our knowledge, there is no published result concerning this case by this method. Copyright © 2015 John Wiley & Sons, Ltd.     

Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems

In this paper we introduce the theory of dominant solutions at infinity for nonoscillatory discrete symplectic systems without any controllability assumption. Such solutions represent an opposite concept to recessive solutions at infinity, which were recently developed for such systems by the authors. Our main results include: (i) the existence of dominant solutions at infinity for all ranks in a given range depending on the order of abnormality of the system, (ii) construction of dominant solutions at infinity with eventually the same image, (iii) classification of dominant and recessive solutions at infinity with eventually the same image, (iv) limit characterization of recessive solutions at infinity in terms of dominant solutions at infinity and vice versa, and (v) Reid’s construction of the minimal recessive solution at infinity. These results are based on a new theory of genera of conjoined bases for symplectic systems developed for this purpose in this paper.