MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF NONLINEAR SCHRODINGER EQUATIONS

This paper proves the multiplicity of positive solutions for the following class of quasilinear problems: {-epsilon(p)Delta(p)u+(lambda A(x) + 1)vertical bar u vertical bar(p-2)u = f(u), R(N) u(x)>0 in R(N), where Delta(p) is the p-Laplacian operator, N > p >= 2, lambda and epsilon are positive parameters, A is a nonnegative continuous function and f is a continuous function with subcritical growth. Here, we use variational methods to get multiplicity of positive solutions involving the Lusternick-Schnirelman category of intA(-1)(0) for all sufficiently large lambda and small epsilon.

Micro Far-Infrared Reflectivity of CaNb(2)O(6) Single Crystal Fibers Grown by the Laser-Heated Pedestal Growth Technique

CaNb(2)O(6) single crystal fibers were grown by the laser-heated pedestal growth technique, directly from the starting reagents. Optically transparent fibers were obtained in the form of rods with elliptical cross-section, free from cracks, impurities, and secondary phases, with an average diameter of 0.4 mm and about 20 mm of length. The fibers grew within the orthorhombic Pbcn columbite structure, with the growth axis nearly parallel to the crystallographic a-direction. The parameters b and c were parallel to the shorter and larger ellipsis axes. A special setup using a microscope was developed to obtain the far-infrared reflectivity spectra of these micrometer-sized fibers, allowing the identification and assignment of 34 of the 38 polar phonons foreseen for the material. From these phonons, the intrinsic dielectric constant ( = 18.2) and quality factor ( of 185 THz) could be estimated, showing the potential of the material for applications in microwave circuitry. These results, along with previous polarized Raman data (Cryst. Growth Des. 2010, 10, 1569), allow us to present a comprehensive set of optical phonon modes and to discuss the potential use of designed CaNb(2)O(6) microcrystals in compact optical devices.