Volume 8, Issue 4

In this paper we will establish two different classes of ellipticity criteria, called the L p criteria and the L p -L q criteria respectively, for planar linear Hamiltonian systems with periodic coefficients. The criteria are explicitly expressed using the L p and L q norms of coefficients and some known Sobolev constants. These results can be considered as the extensions of the famous Lyapunov stability criterion for Hill’s equations.

We consider the boundary value problem u″ + g(u)u = h(t, u, u′) u(0) = 0, u(π) = 0 with g continuous, periodic, and positive and h continuous and bounded. It is shown that there is a solution if the mean value of g is not the square of an integer.

We characterize a compactness property for a quasilinear equation with critical growth and singular term. Some applications of the compactness property are also pointed out.

For a wide class of systems of nonlinear parabolic inequalities with Chipot-Weissler nonlinearity in the gradient term which depend on two nonnegative parameters λ 1 and λ 1 , we study the nonexistence of global solutions according to the \test function" method. We establish theorems that, when λ 1 = λ 1 = 0, include and, in some cases, improve, or, in part, yield the results obtained by Es- cobedo and Herrero; Escobedo and Levine; Galaktionov;Levine, Lieberman and Meier; Mitidieri and Pohozaev; Pohozaev and Tesei.

In this paper, we are interested in the following biharmonic equation: with Navier boundary conditions on ∂Ω; where Ω is an open bounded domain of ℝ n , with n ≥ 5, having a smooth boundary, and ∑ ⊂ Ω is either a set of finite number of points of Ω or a smooth closed submanifold of Ω without boundary. We construct positive solutions to this problem which are singular at ∑.

In this paper we will study the dynamics of the periodic asymmetric oscillator xʺ + q + (t)x + + q – (t)x – = 0, where q + ; q – ∊ L 1 (ℝ / 2πℤ) and x + = max(x; 0), x – = min(x; 0) for x ∊ ℝ. It will be proved that the exponential growth rate does exist for each non-zero solution x(t) of the oscillator. The properties of these rates, or the Lyapunov exponents, will be given using the induced circle di®eomorphism of the oscillator. The proof is extensively based on the Denjoy theorem in topological dynamics and the unique ergodicity theorem in ergodic theory.

We study a system of equations arising from angiogenesis which contains a non- regular term that vanishes below a certain threshold. We are forced to modify the usual methods of bifurcation theory because of this loss of regularity. Nevertheless, we obtain results on the existence, uniqueness and permanence of a positive solution for the time-dependent problem; and the existence and uniqueness of a positive solution for the stationary one.

Consider the problem -∆u = N(N - 2)u p + εk(x)u in Ω, u > 0 in Ω, u|∂Ω = 0 where Ω­ is a smooth bounded domain in ℝ N (N ≥ 6), p = (N +2)/(N - 2), ε > 0 and k ∈ C 2 (Ω̅). Under certain assumptions, we prove the nondegeneracy of least energy solutions to the above problem as Ɛ →0. This is an extension of the recent work of Grossi [9].

We investigate the bipolar quantum drift-diffusion model, a fourth order parabolic system, in two or there space dimensions. First, we establish the global weak solution with large initial value and periodic boundary conditions. Then we show the semiclassical limit by using new methods based on delicate interpolation estimates and compactness argument. Furthermore, when the doping profile is a constant, we find that the weak solution approaches its mean value exponentially as time increases to infinity.

We prove the existence of rotating solitary waves (vortices) with enough largely prescribed L 2 norm for the nonlinear Klein-Gordon equation with nonnegative potential, which makes the equation suitable for physical models and guarantees the well-posedness of the corresponding Cauchy problem. This is done by finding nonnegative cylindrical solutions to the standing equation where x = (y, z) ∈ ℝ k × R N–k , N > k ≥ 0 and μ > 0 is the fundamental eigenvalue, namely, the Lagrange multiplier of the minimization problem (with infimum over a suitable subspace of H 1 (ℝ N )).

We consider nonlinear Neumann problems driven by p-Laplacian-type operators which are not homogeneous in general. We prove an existence and a multiplicity result for such problems. In the existence theorem, we assume that the right hand side nonlinearity is p-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. In the multiplicity result, when specialized to the case of the p-Laplacian, we allow strong resonance at infinity and resonance at 0.

The well-known Belousov-Zhabotinskii system can be written as ẋ = s(x + y - qx 2 - xy), ẏ = s -1 (-y + fz - xy), ż = w(x - z) with f, q, s, w ∈ ℝ and s ≠ 0. In this paper we characterize its global analytic first integrals.