In this paper we obtain the local Hölder regularity of the gradients of weak solutions for a class of non-uniformly nonlinear variable exponent elliptic equations in divergence form

including the following special model

under some proper assumptions on Ai and the Hölder continuous functions f, pi(x) for i = 1, 2.

The problem of nonlinear self-focusing of elliptic Gaussian laser beam in collisionless magnetized plasma is studied using variation approach. The dynamics of the combined effects of nonlinearity and spatial diffraction is presented. With a and b as the beam width parameters of the beam along x and y directions, respectively, the phenomenon of cross-focusing is observed where focusing of a results in defocusing of b and vice versa. Although no stationary self-trapping is observed, oscillatory self-trapping occurs far below the threshold. The regularized phase is always negative.


In recent years several papers have been devoted to stabilityand smoothing properties in maximum-norm offinite element discretizations of parabolic problems.Using the theory of analytic semigroups it has been possibleto rephrase such properties as bounds for the resolventof the associated discrete elliptic operator. In all thesecases the triangulations of the spatial domain has beenassumed to be quasiuniform. In the present paper weshow a resolvent estimate, in one and two space dimensions,under weaker conditions on the triangulations than quasiuniformity.In the two-dimensional case, the bound for the resolvent containsa logarithmic factor.