Volume 10, Issue 2

Let Ω be a bounded domain with smooth boundary in ℝ N and h ∈ C((0,∞), (0,∞)) with lim s→0 + h(s) = Υ ∈ (0,∞). By the perturbation method, which is due to García Melián, and nonlinear transformations and comparison principles, we derive the exact boundary behavior of solutions to a singular Dirichlet problem . Then, applying the result, combining two kinds of nonlinear transformations, we derive the exact boundary behavior of solutions to a boundary blow-up elliptic problem and a singular Dirichlet problem, where the weight b is positive in Ω and may be (rapidly) vanishing or blow up on the boundary.

In this article, let ∑ ⊂R 2n be a compact convex hypersurface which is (r,R)-pinched with . Then ∑ carries at least two strictly elliptic closed characteristics; moreover, ∑ carries at least non-hyperbolic closed characteristics.

We study existence and asymptotic behavior of positive solutions for quasilinear elliptic equations of the form -ε p Δ p v + V(x)|v| p-2 v = f(v) in ℝ N , 1 < p < N, as ε → 0. The potential V : ℝ N → ℝ has a positive infimum, inf ∂Ω V > inf Ω V for some bounded domain Ω in ℝ N and f is a subcritical nonlinearity without some growth conditions such as Ambrosetti-Rabinowitz.

Positive periodic solutions of the equation ẍ + x = g(t)x p-1 are considered with continuous and T-periodic g. Some existence results of solutions are obtained by variational method.

In this paper, we apply a cross-constrained variational method to study the classic nonlinear Klein-Gordon equation with cubic nonlinearity in three space dimensions. By constructing a type of cross-constrained variational problem and establishing the so-called cross invariant manifolds, we obtain a sharp threshold for blowup and global existence of the solution to the equation under study which is different from that in [10] . On the other hand, we give an answer to the question that how small the initial data have to be for the global solutions to exist.

In this paper we work with a vastly analyzed tritrophic food chain model. We provide a complete characterization of their Darboux polynomials and of their exponential factors. We also show the non-existence of polynomial first integrals, of rational first integrals, of local analytic first integrals in a neighborhood of the origin, of first integrals that can be described by formal series and of Darboux first integrals.

A mathematical problem arising in the modelling of travelling waves in self- focusing waveguides is studied. It involves a nonlinear Schrödinger equation in ℝ 1+1 . In the paraxial approximation, the travelling waves are interpreted as standing waves of the NLS. Local and global bifurcation results are established for the corresponding \stationary" equation. The combination of variational arguments with an implicit function theorem yields smooth branches of positive solutions parametrized by the \frequency" of the standing wave of NLS. Precise informations on the asymptotic behaviour of various norms of the solutions along the branches are given. In the case of even symmetry, a global bifurcation result is proved, providing a branch of positive even solutions parametrized by frequencies in the whole half-line (0, ∞). The stability of the standing waves of the NLS relies on the monotonicity of their L 2 -norm as a function of the frequency. This criterion, precisely going back to formal arguments in the physical literature on self-focusing waveguides, is proved to hold along the whole global branch of solutions in the even case. The mathematical results are applied to the study of a non-homogeneous planar self-focusing waveguide with a Kerr-type nonlinear response, yielding existence of TE travelling waves and their stability among the set of TE modes. The bifurcation analysis gives informations on the power of the beam. In particular, it provides the possibility of low power cut-off.

We find coexistence states for systems in three dimensions with cyclic dynamic in the boundary of ℝ + 3 . This problem had been already studied for competitive systems. In this paper we consider any interaction. This extension seems to be meaningful since the cyclic dynamic on the boundary can appear in natural way in many interactions.

The rotation-modified Kadomtsev-Petviashvili equation is a modification of the Kadomtsev-Petviashvili (KP) equation widely used to describe the effects of rotation on small-amplitude, long waves propagating in one primary direction. In this paper we investigate conditions for the finite-time blow-up solutions and the uniform bounds of the solution of the generalized rotation-modified Kadomtsev-Petviashvili equation. The conditions are expressed in terms of the energy and the best constant of the anisotropic Sobolev inequality.

We study existence and multiplicity results for solutions of elliptic problems of the type -Δu = g(x; u) in a bounded domain Ω with Dirichlet boundary conditions. The function g(x; s) is asymptotically linear as |s| → +∞. Also resonant situations are allowed. We also prove some perturbation results for Dirichlet problems of the type -Δu = g Ɛ (x, u) where g ε (x, s) → g(x, s) as ε → 0. The previous results find an application in the study of Dirichlet problems of the type -Δu = g(x, u)+μ where μ is a Radon measure. To properly set the above mentioned problems in a variational framework we also study existence and properties of critical points of a class of abstract nonsmooth functional defined on Banach spaces and extend to this nonsmooth framework some classical linking theorems.

In this paper we study existence and orbital stability for solitary waves of the nonlinear Klein-Gordon equation. The energy of these solutions travels as a localized packet, hence they are a particular type of solitons. In particular we are interested in sufficient conditions on the potential for the existence of solitons. Our proof is based on the study of the ratio energy/charge of a function, which turns out to be a useful approach for many field equations.

We consider embeddings of a class of fractional Sobolev spaces E s (Ω) (s ≥ 0) into their limiting L p (Ω) spaces, for a bounded smooth domain Ω ⊂ ℝ N , and study the relationship between the best constants for these embeddings and other best constants corresponding to the case Ω = ℝ N .