arXiv subjects
Publications and source records attributed to Vincenzo Ambrosio.
In this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional $p$-Laplacian operator and a Berestycki-Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity.
In this paper, we study the regularity of weak solutions for a class of nonlinear Choquard equations driven by the $p$-Laplacian operator. We also establish a Pohozaev type identity.
In this paper, we are concerned with the following fractional relativistic Schrödinger equation with critical growth: \begin{equation*} \left\{ \begin{array}{ll} (-Δ+m^{2})^{s}u + V(\varepsilon x) u= f(u)+u^{2^{*}_{s}-1} \mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 \, \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical exponent, $(-Δ+m^{2})^{s}$ is the fractional relativistic Schrödinger operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential $V$, we construct a family of positive solutions $u_{\varepsilon}\in H^{s}(\mathbb{R}^{N})$, with exponential decay, which concentrates around a local minimum of $V$ as $\varepsilon\rightarrow 0$.
In this paper we study the following class of fractional relativistic Schrödinger equations: \begin{equation*} \left\{ \begin{array}{ll} (-Δ+m^{2})^{s}u + V(\varepsilon x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $(-Δ+m^{2})^{s}$ is the fractional relativistic Schrödinger operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential satisfying a local condition, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous subcritical nonlinearity. By using a variant of the extension method and a penalization technique, we first prove that, for $\varepsilon>0$ small enough, the above problem admits a weak solution $u_{\varepsilon}$ which concentrates around a local minimum point of $V$ as $\varepsilon\rightarrow 0$. We also show that $u_{\varepsilon}$ has an exponential decay at infinity by constructing a suitable comparison function and by performing some refined estimates. Secondly, by combining the generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate the number of positive solutions with the topology of the set where the potential $V$ attains its minimum value.
In this paper we study the multiplicity and concentration of positive solutions for the following $(p, q)$-Laplacian problem: \begin{equation*} \left\{ \begin{array}{ll} -Δ_{p} u -Δ_{q} u +V(\varepsilon x) \left(|u|^{p-2}u + |u|^{q-2}u\right) = f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in W^{1, p}(\mathbb{R}^{N})\cap W^{1, q}(\mathbb{R}^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $1< p<q<N$, $Δ_{r}u=\mbox{div}(|\nabla u|^{r-2}\nabla u)$, with $r\in \{p, q\}$, is the $r$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous function satisfying the global Rabinowitz condition, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik-Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where $V$ attains its minimum for small $\varepsilon$.
We study the multiplicity and concentration of complex valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik-Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.
In this paper we study the following class of fractional Kirchhoff problems: \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}M(\varepsilon^{2s-N}[u]^{2}_{s})(-Δ)^{s}u + V(x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $N\geq 2$, $(-Δ)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive continuous function, $M: [0, \infty)\rightarrow \mathbb{R}$ is a Kirchhoff function satisfying suitable conditions and $f:\mathbb{R}\rightarrow \mathbb{R}$ fulfills Berestycki-Lions type assumptions of subcritical or critical type. Using suitable variational arguments, we prove the existence of a family of positive solutions $(u_{\varepsilon})$ which concentrates at a local minimum of $V$ as $\varepsilon\rightarrow 0$.
The aim of this paper is to investigate the existence, multiplicity and concentration of positive solutions for the following nonlocal system of fractional Schrödinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-Δ)^{s}u+V(x)u=Q_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}, \varepsilon^{2s} (-Δ)^{s}v+W(x)v=Q_{v}(u, v) &\mbox{ in } \mathbb{R}^{N}, u, v>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-Δ)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $W:\mathbb{R}^{N}\rightarrow \mathbb{R}$ are positive continuous potentials, $Q$ is a homogeneous $C^{2}$-function with subcritical growth. In order to relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values, we apply penalization techniques, Nehari manifold arguments and Ljusternik-Schnirelmann theory.
We prove the existence of infinitely many nontrivial weak periodic solutions for a class of fractional Kirchhoff problems driven by a relativistic Schrödinger operator with periodic boundary conditions and involving different types of nonlinearities.
In this paper we deal with the following nonlocal systems of fractional Schrödinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-Δ)^{s}u+V(x)u=Q_{u}(u, v)+γH_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}\\ \varepsilon^{2s} (-Δ)^{s}v+W(x)v=Q_{v}(u, v)+γH_{v}(u, v) &\mbox{ in } \mathbb{R}^{N} \\ u, v>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$, $s\in (0, 1)$, $N>2s$, $(-Δ)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $W:\mathbb{R}^{N}\rightarrow \mathbb{R}$ are continuous potentials, $Q$ is a homogeneous $C^{2}$-function with subcritical growth, $γ\in \{0, 1\}$ and $H(u, v)=\frac{2}{α+β}|u|^α |v|^β$ with $α, β\geq 1$ such that $α+β=2^{*}_{s}$. We investigate the subcritical case $(γ=0)$ and the critical case $(γ=1)$, and using Ljusternik-Schnirelmann theory, we relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values.
We deal with the following nonlinear problem involving fractional $p\&q$ Laplacians: \begin{equation*} (-Δ)^{s}_{p}u+(-Δ)^{s}_{q}u+|u|^{p-2}u+|u|^{q-2}u=λh(x) f(u)+|u|^{q^{*}_{s}-2}u \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $1 0$ is a parameter, $h$ is a nontrivial bounded perturbation and $f$ is a superlinear continuous function with subcritical growth. Using suitable variational arguments and concentration-compactness lemma, we prove the existence of a nontrivial non-negative solution for $λ$ sufficiently large.
In this paper we consider the following class of fractional Kirchhoff equations with critical growth: \begin{equation*} \left\{ \begin{array}{ll} \left(\varepsilon^{2s}a+\varepsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-Δ)^{\frac{s}{2}}u|^{2}dx\right)(-Δ)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u \quad &\mbox{ in } \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \quad u>0 &\mbox{ in } \mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-Δ)^{s}$ is the fractional Laplacian operator, $V$ is a positive continuous potential and $f$ is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions $u_{\varepsilon}$ which concentrates around a local minimum of $V$ as $\varepsilon\rightarrow 0$.
We deal with the existence of positive solutions for the following fractional Schrödinger equation $$ \varepsilon ^{2s} (-Δ)^{s} u + V(x) u = f(u) \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-Δ)^{s}$ is the fractional Laplacian operator, and $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous positive function. Under the assumptions that the nonlinearity $f$ is either asymptotically linear or superlinear at infinity, we prove the existence of a family of positive solutions which concentrates at a local minimum of $V$ as $\varepsilon$ tends to zero.
In this paper we deal with the following fractional Choquard equation \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{sp}(-Δ)^{s}_{p} u + V(x)|u|^{p-2}u = \varepsilon^{μ-N}\left(\frac{1}{|x|^μ}*F(u)\right)f(u) \mbox{ in } \mathbb{R}^{N},\\ u\in W^{s,p}(\R^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $p\in (1, \infty)$, $N>sp$, $(-Δ)^{s}_{p}$ is the fractional $p$-Laplacian, $V$ is a positive continuous potential, $0<μ 0$ small enough.
We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-Δ)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon$ is a small positive parameter, $a, b>0$ are fixed constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential verifying the global condition due to Rabinowitz \cite{Rab}, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinearity. Due to the presence of the magnetic field and the critical growth of the nonlinearity, several difficulties arise in the study of our problem and a careful analysis will be needed. The main results presented here are established by using minimax methods, concentration compactness principle of Lions \cite{Lions}, a fractional Kato's type inequality and the Ljusternik-Schnirelmann theory of critical points.
We consider some nonlinear fractional Schrödinger equations with magnetic field and involving continuous nonlinearities having subcritical, critical or supercritical growth. Under a local condition on the potential, we use minimax methods to investigate the existence and concentration of nontrivial weak solutions.
This paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type \begin{equation*} \varepsilon^{2s}(-Δ)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a parameter, $s,t\in (0, 1)$ are such that $2s+2t>3$, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a continuous electric potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinear term. Using variational methods, we obtain the existence, multiplicity and concentration of nontrivial solutions for $\varepsilon>0$ small enough.
In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (-Δ)_{p}^{s}u+ (-Δ)_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1< p<q<\frac{N}{s}$, $(-Δ)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $\mathcal{C}^{1}$-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that $\varepsilon$ is sufficiently small.