The approximation of a mixed formulation of elliptic variational inequalities is studied. Mixed formulation is defined as the problem of finding a saddle-point of a properly chosen Lagrangian $\mathcal{2}$ on a certain convex set $Kx\Lambda$. Sufficient conditions, guaranteeing the convergence of approximate solutions are studied. Abstract results are applied to concrete examples.

In this Note we consider the following problem $$\left\{\begin{array}{cc}-\mathrm{△}u=N\left(N-2\right)u^{p_{ϵ}}-\lambda u\hfill & \text{in }\mathrm{\Omega }\hfill \\ u>0\hfill & \text{in }\mathrm{\Omega }\hfill \\ u=0\hfill & \text{on }\partial \mathrm{\Omega }.\hfill \end{array}\right.$$ where $\mathrm{\Omega }$ is a bounded smooth starshaped domain in ${\mathbb{R}}^N$, $N\ge 3$, $p_{ϵ}=\frac{N+2}{N-2}-ϵ$, $ϵ>0$, and $\lambda \ge 0$. We prove that if $u_{ϵ}$ is a solution of Morse index $m>0$ than $u_{ϵ}$ cannot have more than $m$ maximum points in $\mathrm{\Omega }$ for $ϵ$ sufficiently small. Moreover if $\mathrm{\Omega }$ is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for $ϵ$ sufficiently small.

We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S^2$, on a bounded domain $\Omega \subset {ℝ}^2$ and on all of ${ℝ}^2$. In some cases we also address the question of existence of minimizers.

In this work we consider the magnetic NLS equation$${(\frac{\hslash }{i}\nabla -A\left(x\right))}^{2}u+V\left(x\right)u-{f\left(\right|u|}^2)u=0\text{in}{ℝ}^N(0.1)$$where $N\ge 3$, $A:{ℝ}^N\to {ℝ}^N$ is a magnetic potential, possibly unbounded, $V:{ℝ}^N\to ℝ$ is a multi-well electric potential, which can vanish somewhere, $f$ is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u:{ℝ}^N\to ℂ$ to (0.1), under conditions on the nonlinearity which are nearly optimal.

In this work we consider the magnetic NLS equation $${(\frac{\hslash }{i}\nabla -A\left(x\right))}^{2}u+V\left(x\right)u-{f\left(\right|u|}^2)u=0\text{in}{ℝ}^N(0.1)$$ where $N\ge 3$, $A:{ℝ}^N\to {ℝ}^N$ is a magnetic potential, possibly unbounded, $V:{ℝ}^N\to ℝ$ is a multi-well electric potential, which can vanish somewhere, f is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u:{ℝ}^N\to ℂ$ to (0.1), under conditions on the nonlinearity which are nearly optimal.