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Vincenzo Ambrosio

Publications and source records attributed to Vincenzo Ambrosio.

49 records · Page 3Linked to original sources
arXiv2017

A multiplicity result for a fractional Kirchhoff equation in $\mathbb{R}^{N}$ with a general nonlinearity

In this paper we deal with the following fractional Kirchhoff equation \begin{equation*} \left(p+q(1-s) \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^{2}}{|x-y|^{N+2s}} \, dx\,dy \right)(-Δ)^{s}u = g(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $N\geq 2$, $p>0$, $q$ is a small positive parameter and $g: \mathbb{R}\rightarrow \mathbb{R}$ is an odd function satisfying Berestycki-Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that $q$ is sufficiently small.

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arXiv2017

Mountain pass solutions for the fractional Berestycki-Lions problem

We investigate the existence of least energy solutions and infinitely many solutions for the following nonlinear fractional equation (-Δ)^{s} u = g(u) \mbox{ in } \mathbb{R}^{N}, where $s\in (0,1)$, $N\geq 2$, $(-Δ)^{s}$ is the fractional Laplacian and $g: \mathbb{R} \rightarrow \mathbb{R}$ is an odd $\mathcal{C}^{1, α}$ function satisfying Berestycki-Lions type assumptions. The proof is based on the symmetric mountain pass approach developed by Hirata, Ikoma and Tanaka in \cite{HIT}. Moreover, by combining the mountain pass approach and an approximation argument, we also prove the existence of a positive radially symmetric solution for the above problem when $g$ satisfies suitable growth conditions which make our problem fall in the so called "zero mass" case.

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arXiv2017

Periodic solutions for a fractional asymptotically linear problem

We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the the pseudo-index theory developed by Bartolo, Benci and Fortunato \cite{bbf} after transforming the problem to a degenerate elliptic problem in a half-cylinder with a Neumann boundary condition, via a Caffarelli-Silvestre type extension in periodic setting. The periodic nonlocal case, considered here, presents, respect to the cases studied in literature, some new additional difficulties and a careful analysis of the fractional spaces involved is necessary.

math.AP↗
arXiv2017

Multiple solutions for a class of nonhomogeneous fractional Schrödinger equations in $\mathbb{R}^{N}$

This paper is concerned with the following fractional Schrödinger equation \begin{equation*} \left\{ \begin{array}{ll} (-Δ)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N> 2s$, $(-Δ)^{s}$ is the fractional Laplacian, $k$ is a bounded positive function, $h\in L^{2}(\mathbb{R}^{N})$, $h\not \equiv 0$ is nonnegative and $f$ is either asymptotically linear or superlinear at infinity.\\ By using the $s$-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that $|h|_{2}$ is sufficiently small.

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arXiv2016

Ground states for a fractional scalar field problem with critical growth

We prove the existence of a ground state solution for the following fractional scalar field equation $(-Δ)^{s} u= g(u)$ in $\mathbb{R}^{N}$ where $s\in (0,1), N> 2s$,$ (-Δ)^{s}$ is the fractional Laplacian, and $g\in C^{1, β}(\mathbb{R}, \mathbb{R})$ is an odd function satisfying the critical growth assumption.

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arXiv2016

Zero mass case for a fractional Berestycki-Lions type problem

In this work we study the following fractional scalar field equation \begin{equation*}\label{P} \left\{ \begin{array}{ll} (-Δ)^{s} u = g'(u) \mbox{ in } \mathbb{R}^{N} \\ u> 0 \end{array} \right. \end{equation*} where $N\geq 2$, $s\in (0,1)$, $(-Δ)^{s}$ is the fractional Laplacian and the nonlinearity $g\in C^{2}(\mathbb{R})$ is such that $g''(0)=0$. By using variational methods, we prove the existence of a positive solution which is spherically symmetric and decreasing in $r=|x|$.

math.AP↗
arXiv2016

Periodic solutions for nonlocal fractional equations

The purpose of this paper is to study the existence of (weak) periodic solutions for nonlocal fractional equations with periodic boundary conditions. These equations have a variational structure and, by applying a critical point result coming out from a classical Pucci-Serrin theorem in addition to a local minimum result for differentiable functionals due to Ricceri, we are able to prove the existence of at least two periodic solutions for the treated problems. As far as we know, all these results are new.

math.AP↗
arXiv2016

Multiple solutions for a fractional $p$-Laplacian equation with sign-changing potential

We use a variant of the fountain Theorem to prove the existence of infinitely many weak solutions for the following fractional p-Laplace equation (-Δ)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u) in R^N, where $s \in (0,1)$,$ p \geq 2$,$ N \geq 2$, $(-Δ)^{s}_{p}$ is the fractional $p$-Laplace operator, the nonlinearity f is $p$-superlinear at infinity and the potential V(x) is allowed to be sign-changing.

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arXiv2016

A fractional Landesman-Lazer type problem set on R^N

By using the abstract version of Struwe's monotonicity-trick we prove the existence of a positive solution to the problem (-Δ)^s u + K u = f(x, u) in R^N u\in H^s (R^N), K>0 where f(x, t): R^N\times R \rightarrow R is a Caratheodory function, 1-periodic in x and does not satisfy the Ambrosetti-Rabinowitz condition.

math.AP↗
arXiv2016

Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition

The purpose of this paper is to study $T$-periodic solutions to [(-Δ_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in} (0,T)^{N} (P) u(x+Te_{i})=u(x) &\mbox{for all} x \in \R^{N}, i=1, \dots, N where $s\in (0,1)$, $N>2s$, $T>0$, $m> 0$ and $f(x,u)$ is a continuous function, $T$-periodic in $x$ and satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. The nonlocal operator $(-Δ_{x}+m^{2})^{s}$ can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder $\mathcal{S}_{T}=(0,T)^{N}\times (0,\infty)$. By using a variant of the Linking Theorem, we show that the extended problem in $\mathcal{S}_{T}$ admits a nontrivial solution $v(x,ξ)$ which is $T$-periodic in $x$. Moreover, by a procedure of limit as $m\rightarrow 0$, we also prove the existence of a nontrivial solution to (P) with $m=0$.

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arXiv2016

Ground states for superlinear fractional Schrödinger equations in \R^{N}

In this paper we study ground states of the following fractional Schrödinger equation (- Δ)^{s} u + V(x) u = f(x, u) \, \mbox{ in } \, \R^{N}, u\in \H^{s}(\R^{N}) where $s\in (0,1)$, $N>2s$ and $f$ is a continuous function satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. We consider the cases when the potential $V(x)$ is $1$-periodic or has a bounded potential well.

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arXiv2015

Periodic solutions for the non-local operator (-Delta + m^2)^s - m^(2s) with m>=0

By using variational methods we investigate the existence of T-periodic solutions to [(-Delta_x + m^2)^s -m^(2s)]u= f(x,u) in (0,T)^N u(x+Te_i)=u(x) for all x in R^N, i=1,...,N where s in (0,1), N>2s, T>0, m>=0 and f(x,u) is a continuous function, T-periodic in x, verifying the Ambrosetti-Rabinowitz condition and a polynomial growth at rate p in (1, (N+2s)/(N-2s)).

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