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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