115 research outputs found

### 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]

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