Some continuation properties via minimax arguments

This note is devotes to some remarks regarding the use of variational methods, of minimax type, to establish continuity type result

Sharp non-existence results of prescribed L^2-norm solutions for some class of Schr\"odinger-Poisson and quasilinear equations

In this paper we study the existence of minimizers for F(u) = \1/2\int_{\R^3} |\nabla u|^2 dx + 1/4\int_{\R^3}\int_{\R^3}\frac{| u(x) |^2| u(y) |^2}{| x-y |}dxdy-\frac{1}{p}\int_{\R^3}| u |^p dx on the constraint $$S(c) = \{u \in H^1(\R^3) : \int_{\R^3}|u|^2 dx = c \},$$ where $c>0$ is a given parameter. In the range $p \in [3, 10/3]$ we explicit a threshold value of $c>0$ separating existence and non-existence of minimizers. We also derive a non-existence result of critical points of $F(u)$ restricted to $S(c)$ when $c>0$ is sufficiently small. Finally, as a byproduct of our approaches, we extend some results of \cite{CJS} where a constrained minimization problem, associated to a quasilinear equation, is considered.Comment: 22 page

Multi-peak solutions for magnetic NLS equations without non--degeneracy conditions

In the work we consider the magnetic NLS equation (\frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u = 0 \quad {in} \R^N where $N \geq 3$, $A \colon \R^N \to \R^N$ is a magnetic potential, possibly unbounded, $V \colon \R^N \to \R$ 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\colon \R^N \to \C, under conditions on the nonlinearity which are nearly optimal.Comment: Important modification in the last part of the pape

Multiple normalized solutions for quasi-linear Schr\"odinger equations

In this paper we prove the existence of two solutions having a prescribed $L^2$-norm for a quasi-linear Schr\"odinger equation. One of these solutions is a mountain pass solution relative to a constraint and the other one a minimum either local or global. To overcome the lack of differentiability of the associated functional, we rely on a perturbation method developed in [27]