We are interested in problems as follows \begin{equation} \tag{$P_{\mu,\beta}$} \left\{ \begin{array}[c]{ll} - \Delta u = \lambda_1 u + \mu g(x,u) + W(x)f(u) + u^{\beta-1}\, \, \, \mbox{in} \, \, \Omega, & \\ u= 0, \, \, \, \mbox{on}\, \, \partial \Omega, &\\ u(x)\geq 0\, \, \, \mbox{in}\, \, \Omega, \end{array} \right. \end{equation} where $\beta\geq 2^*$, $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with regular boundary $\partial \Omega$, $N\geq 3$, $\mu > 0$ is a parameter, $\lambda_1$ is the first eigenvalue of the operator $-\Delta$, $W$ is a weight function that changes signal and under suitable conditions on functions $f$ and $g$. We apply variational and sub-supersolutions methods to obtain a non-negative and nontrivial solution for problem $(P_{\mu,s})$. Our results are related to the critical and supercritical cases of the work [13].

In 1983, Antontsev, Kazhikhov, and Monakhov published a proof of the existence and uniqueness of solutions to the 3D Euler equations in which on certain inflow boundary components fluid is forced into the domain while on other outflow components fluid is drawn out of the domain. A key tool they used was the linearized Euler equations in vorticity form. We extend their result on the linearized problem to multiply connected domains and establish compatibility conditions on the initial data that allow higher regularity solutions.

The initial value problem is considered for a higher order nonlinear Schrödinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional damping large-time decay of solutions without any smallness assumptions is also established. The main difficulty of the study is the non-smooth character of the nonlinearity.

Aim of this paper is to discuss the existence of multiple solutions to double phase anisotropic variational problems for the case of a combined effect of concave-convex nonlinearities. Especially the superlinear (convex) term to the given problem substantially fulfills a weaker condition as well as Ambrosetti-Rabinowitz condition. To achieve these results, we apply the variational methods such as the famous mountain pass theorem and Ekeland's type variational principle when an energy functional corresponding to our problem satisfies the compactness condition of the Palais-Smale type. In particular, we establish several existence results of a sequence of infinitely many solutions by employing the Cerami compactness condition. The key tools for obtaining these results are the fountain theorem and the dual fountain theorem.

This paper deals with the Stackelberg-Nash strategies for boundary control problems for linear and semilinear wave equations. Assuming that we can act on the system through a hierarchy of controls, to each leader, we associate a Nash equilibrium (two followers) corresponding to a bi-objective optimal control problem. Then we look for a leader that solves an exact boundary controllability problem. We mainly consider the case of a semilinear hyperbolic PDE where all the controls (leader and followers) act on small parts of the boundary. In view of the lack of regularity, the arguments usually invoked to prove controllability do not work here. Accordingly, we carry out a fixed-point approach based on weak compactness in $L^1$. To this purpose, we apply Dunford-Pettis' Theorem. We also consider the case where the followers act in small subsets of the domain.