Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces

In this paper, we are interested in the existence of infinitely many weak solutions for a non-homogeneous eigenvalue Dirichlet problem. By using variational methods, in an appropriate Orlicz–Sobolev setting, we determine intervals of parameters such that our problem admits either a sequence of non-negative weak solutions strongly converging to zero provided that the non-linearity has a suitable behaviour at zero or an unbounded sequence of non-negative weak solutions if a similar behaviour occurs at infinity.     

Learning a function from noisy samples at a finite sparse set of points

In learning theory the goal is to reconstruct a function defined on some (typically high dimensional) domain Ω, when only noisy values of this function at a sparse, discrete subset ω⊂Ω are available.In this work we use Koksma–Hlawka type estimates to obtain deterministic bounds on the so-called generalization error. The resulting estimates show that the generalization error tends to zero when the noise in the measurements tends to zero and the number of sampling points tends to infinity sufficiently fast.     

Existence and nonexistence of ground state solutions for elliptic equations with a convection term

We deal with the existence of positive solutions u decaying to zero at infinity, for a class of equations of Lane-Emden-Fowler type involving a gradient term. One of the main points is that the differential equation contains a semilinear term σ(u) where σ:(0,∞)→(0,∞) is a smooth function which can be both unbounded at infinity and singular at zero. Our technique explores symmetry arguments as well as lower and upper solutions.     

Asymptotic behavior of solutions for initial–boundary value problems for strongly damped nonlinear wave equations

We study initial–boundary value problems for strongly damped nonlinear wave equations. By using improved integral estimates, it is proven that the solutions of the problems decay to zero exponentially as time t$t$ approaches infinity, under a very simple and general assumption regarding the nonlinear term.     

Poisson–Dirichlet distribution with small mutation rate

A large deviation principle is established for the Poisson–Dirichlet distribution when the mutation rate θ$\theta$ converges to zero. The rate function is identified explicitly, and takes on finite values only on states that have finite number of alleles. This result is then applied to the study of the asymptotic behavior of the homozygosity, and the Poisson–Dirichlet distribution with selection. The latter shows that several alleles can coexist when selection intensity goes to infinity in a particular way as θ$\theta$ approaches zero.     

Self-similar solutions of a generalized Burgers equation with nonlinear damping

The nonlinear ordinary differential equation resulting from the self-similar reduction of a generalized Burgers equation with nonlinear damping is studied in some detail. Assuming certain asymptotic conditions at plus infinity or minus infinity, we find a wide variety of solutions—(positive) single hump, monotonic (bounded or unbounded) or solutions with a finite zero. The existence or non-existence of positive bounded solutions with exponential decay to zero at infinity for specific parameter ranges is proved. The analysis relies mainly on the shooting argument.     

Computations of critical groups and applications to asymptotically linear wave equation and beam equation

In this paper, by using the Morse index theory for strongly indefinite functionals developed in [Nonlinear Anal. TMA, in press], we compute precisely the critical groups at the origin and at infinity, respectively. The abstract theorems are used to study the existence (multiplicity) of nontrivial periodical solutions for asymptotically wave equation and beam equation with resonance both at infinity and at zero.     

MULTIPLICITY RESULTS FOR THE TWO-POINT BOUNDARY VALUE PROBLEMS AT RESONANCE

In this article several existence theorems on multiple solutions for the two-point boundary value problem with resonance at both infinity and zero are proved.     

Solutions to a gradient system with resonance at both zero and infinity

In this paper, we study the existence and multiplicity of nontrivial solutions for a gradient system with resonance at both zero and infinity via Morse theory.     

Nontrivial solutions of elliptic semilinear equations¶at resonance

We find nontrivial solutions for semilinear boundary value problems having resonance both at zero and at infinity. Received: 14 January 1999 / Revised version: 17 May 1999     

Periodic problems with double resonance

We consider a second order periodic problem with resonance both at infinity and at zero. Combining variational methods together with Morse theory, we produce six nontrivial solutions for the periodic problem.     

Solution of nonlinear equations having asymptotic limits at zero and infinity

We obtain nontrivial solutions for semilinear elliptic boundary value problems having asymptotic limits both at zero and at infinity. Received September 28, 1999 / Accepted May 9, 2000 / Published online: December 8, 2000     

Multiple solutions for boundary value problems of second-order difference equations with resonance

In this paper, we study the existence of multiple solutions for boundary value problems of second-order difference equations with resonance at both infinity and zero by using Morse theory, critical point theory, minimax methods and bifurcation theory.     

Nontrivial solutions of Kirchhoff type problems

In this work, we study Kirchhoff type problems on a bounded domain. We consider the cases where the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity, and the nonlinearity is asymptotically linear near zero but 4-superlinear at infinity. By computing the relevant critical groups, we obtain nontrivial solutions via Morse theory.     

Algebraic topological methods for the supercritical Q-curvature problem

We study the problem of existence of conformal metrics with prescribed Q-curvature on closed four-dimensional Riemannian manifolds. This problem has a variational structure, and in the case of interest here, it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated Euler–Lagrange functional, the so-called true critical points at infinity of the associated variational problem, occur. Using the characterization of the critical points at infinity of the associated variational problem which is established in [42], combined with some arguments from Morse theory, some algebraic topological methods, and some tools from dynamical system originating from Conley's isolated invariant sets and isolated blocks theory, we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold, stable and unstable manifolds of some of the critical points at infinity of the associated Euler–Lagrange functional.     

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.     

On sign-changing solution for a fourth-order asymptotically linear elliptic problem

In this paper we deal with a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity. By using the variational method, we obtain an existence result of sign-changing solutions as well as positive and negative solutions.