Volume 10, Issue 1

In this paper, we study the effect of domain shape on the existence of 2-nodal solutions for a semilinear elliptic equation involving non-odd nonlinearities.

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic infuences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo- stable and pseudo-unstable manifolds for a class of random partial differential equations and stochastic partial differential equations is shown. Unlike the in- variant manifold theory for stochastic ordinary differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with linear multiplicative noise.

Given a Lorentzian manifold (M, g), a geodesic γ in M and a timelike Jacobi field γ along γ, we introduce a special class of instants along γ that we call γ- pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the γ-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field γ is obtained as the restriction of a globally defined timelike Killing vector field.

We consider nonlinear elliptic Neumann problems driven by the p-Laplacian. Using variational techniques together with Morse theory (in particular, critical groups and the Poincaré-Hopf formula), we prove some multiplicity results: either three or four distinct nontrivial solutions are guaranteed, depending on the geometry and smoothness of the nonlinear term.

In this paper we show the existence of multiple solutions to a class of quasilinear elliptic equations with singular weights when the continuous nonlinearity satisfies a superlinear condition only at zero. In particular, our approach allows us to consider superlinear, critical and supercritical nonlinearities.

In this note we consider the notion of nonuniform exponential contraction for delay difference equations with infinite delay (one can argue that the only delay difference equations are those with infinite delay, since otherwise we can always bring them to the standard recurrence form in some higher-dimensional space). We consider the general case of a nonautonomous dynamics. Our main objective is to show that exponential contractions persist under sufficiently small linear and nonlinear perturbations. This includes establishing the continuous dependence with the perturbation of the constants in the notion of contraction. We also characterize the nonuniform exponential contractions in terms of strict Lyapunov sequences, in particular by constructing explicitly a strict Lyapunov sequence for each exponential contraction.

In this paper we prove existence and blow-up results for a class of filtration equations in presence of suitable reaction terms. In particular we cover the case of principal part of fast diffusion type.

In this paper, a new, variational concept of asymptotical stability of zero solution to an ordinary differential system of the second order, considered in Sobolev space, is presented. Sufficient conditions for an asymptotical stability in a variational sense with respect to initial condition and functional parameter (control) are given. Relation to the classical asymptotical stability is illustrated.

This paper concerns existence and multiplicity of ground states and bound states of the time-independent Schrödinger system where n = 2, 3, N ≥ 2, λ j > 0 for j = 1, ... , N, β jj > 0 for j = 1, ... , N, and β ij = β ji . In the attractive case we give sufficient condition for existence of co-existing ground states with large couplings, and in the repulsive case we prove existence of infinitely many co-existing bound state solutions with arbitrary couplings.

We proceed by variational method to prove the existence of solution of a fourth order partial differential equation on complete non compact manifolds. Sufficient conditions, including the geometry of the manifold, are given to make a minimizing sequence converge to a non trivial solution belonging to C loc 4,α (M) for some α ∈ (0, 1).

Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. What we propose here is a Sturm type oscillation theorem for indefinite systems with Dirichlet boundary conditions of the form where p i is a smooth path of matrices on the complex n-dimensional vector space ℂ n , p 2m is the symmetry represented by the diagonal block matrix diag (I n-v, -I v ), and where v is an integer between 0 and n and I is the identity matrix.

The purpose of the paper is to establish the existence of ground state solutions to -Δu = ηa(x)f(u)+γb(x)g(u), where the locally Hölder continuous non-negative functions a and b and the non-linearities f and g satisfy some general conditions.

In this paper we improve Theorem 7 in [1] which deals with the existence and uniqueness of solutions of the three dimensional globally modified Navier-Stokes equations.