874 research outputs found
On a class of parametric -equations
We consider parametric equations driven by the sum of a -Laplacian and a
Laplace operator (the so-called -equations). We study the existence and
multiplicity of solutions when the parameter is near the principal
eigenvalue of . We
prove multiplicity results with precise sign information when the near
resonance occurs from above and from below of
Multiple solutions for a class of fractional equations
In this paper we study a class of fractional Laplace equations with asymptotically linear right-hand side. The existence results of three nontrivial solutions under the resonance and non-resonance conditions are established by using the minimax method and Morse theory
Three nontrivial solutions for the p-Laplacian Neumann problems with a concave nonlinearity near the origin
We consider a nonlinear Neumann problem driven by the p-
Laplacian, with a right-hand side nonlinearity which is concave near the
origin. Using variational techniques, combined with the method of upper-lower
solutions and with Morse theory, we show that the problem has at least three
nontrivial smooth solutions, two of which have a constant sign (one positive
and one negative).FCTPOCI/MAT/55524/200
Critical fluctuations in an optical parametric oscillator: when light behaves like magnetism
We study the nondegenerate optical parametric oscillator in a planar
interferometer near threshold, where critical phenomena are expected. These
phenomena are associated with nonequilibrium quantum dynamics that are known to
lead to quadrature entanglement and squeezing in the oscillator field modes. We
obtain a universal form for the equation describing this system, which allows a
comparison with other phase transitions. We find that the unsqueezed
quadratures of this system correspond to a two-dimensional XY-type model with a
tricritical Lifshitz point. This leaves open the possibility of a controlled
experimental investigation into this unusual class of statistical models. We
evaluate the correlations of the unsqueezed quadrature using both an exact
numerical simulation and a Gaussian approximation, and obtain an accurate
numerical calculation of the non-Gaussian correlations.Comment: Title changed. New figures adde
Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator
We discuss a parametric eigenvalue problem, where the differential operator is of (p,2)-Laplacian type. We show that, when p≠2, the spectrum of the operator is a half line, with the end point formulated in terms of the parameter and the principal eigenvalue of the Laplacian with zero Dirichlet boundary conditions. Two cases are considered corresponding to p>2 and p2, and to infinity in the case of p<2
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