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In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the LjusternikSchnirelmann category to prove that the existence of multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.

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The development of shocks in nonlinear hyperbolic conservation laws may be regularized through either diffusion or relaxation. However, we have observed surprisingly that for some physical problems, when both of the smoothing factors - diffusion and relaxation - coexist, under appropriate asymptotic assumptions, the dispersive waves are enhanced. This phenomenon is studied asymptotically in the sense of the Chapman-Enskog expansion and demonstrated numerically.

We study the critical dissipative quasi-geostrophic equations in $\bR^ 2$ with arbitrary H^ 1 initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [11] with a suitable modification. A decay in time estimate for higher order homogeneous Sobolev norms of solutions is also discussed.