In this paper, we study the semilinear elliptic problem with critical nonlinearity and an indefinite weight function, namely $- \Delta u =\lambda u + h (x) u^{\frac{n+2}{n-2}} $ in a smooth domain bounded (respectively, unbounded) $\Omega\subseteq\,\mathbb R^n , \ n > 4 $, for $\lambda \geq 0 $. Under suitable assumptions on the weight function, we obtain the positive solution branch, bifurcating from the first eigenvalue $\lambda_1(\Omega)$ (respectively, the bottom of the essential spectrum).

This paper is devoted to the prescribed scalar curvature under minimal boundary mean curvature on the standard four-dimensional half sphere. Using topological methods from the theory of critical points at infinity, we prove some existence results.

The paper deals with initial boundary-value problems for the Zakharov system arising in plasma physics in two-dimensional domains under various boundary conditions. We prove the global well-posedness of these problems in some Sobolev-type classes and study properties of the solutions. In the dissipative case the existence of a global attractor is established.

The Kadomtsev-Petviashvili equation is a universal model for the evolution of surface waves of small amplitude propagating in one direction and with weak variations in the transverse direction. The pure initial-value problem was and is extensively studied. This paper deals with the initial-and-boundary-value problem for this equation on a strip with a Dirichlet left boundary condition and two kinds of conditions on the right boundary. Moreover, we treat the case of the half plane and we show a result of convergence.