arXiv subjects
Publications and source records attributed to Simone Secchi.
We consider the semilinear fractional equation $ (I-Δ)^s u = a(x) |u|^{p-2}u$ in $\mathbb{R}^N$, where $N \geq 3$, $0<s<1$, $2<p<2N/(N-2s)$ and $a$ is a bounded weight function. Without assuming that $a$ has an asymptotic profile at infinity, we prove the existence of a ground state solution.
We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in $\mathbb{R}^N$ \begin{equation*} \sqrt{\strut -Δ+ m^2} u - mu + V(x)u = \left( \int_{\mathbb{R}^N} \frac{|u(y)|^p}{|x-y|^{N-α}} \, dy \right) |u|^{p-2}u - Γ(x) |u|^{q-2}u, \end{equation*} where $m > 0$ and the potential $V$ is decomposed as the sum of a $\mathbb{Z}^N$-periodic term and of a bounded term that decays at infinity. The result is proved by variational methods applied to an auxiliary problem in the half-space $\mathbb{R}_{+}^{N+1}$.
In this paper we discuss the existence and non--existence of weak solutions to parametric equations involving the Laplace-Beltrami operator $Δ_g$ in a complete non-compact $d$--dimensional ($d\geq 3$) Riemannian manifold $(\mathcal{M},g)$ with asymptotically non--negative Ricci curvature and intrinsic metric $d_g$. Namely, our simple model is the following problem $$ \left\{ \begin{array}{ll} -Δ_gw+V(σ)w=λα(σ)f(w) & \mbox{ in } \mathcal{M}\\ w\geq 0 & \mbox{ in } \mathcal{M} \end{array}\right. $$ where $V$ is a positive coercive potential, $α$ is a positive bounded function, $λ$ is a real parameter and $f$ is a suitable continuous nonlinear term. The existence of at least two non--trivial bounded weak solutions is established for large value of the parameter $λ$ requiring that the nonlinear term $f$ is non--trivial, continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational methods. No assumptions on the sectional curvature, as well as symmetry theoretical arguments, are requested in our approach.
We prove that the generalized pseudorelativistic equation $$ \left( -c^2 Δ+ m^{2} c^{\frac{2}{1-s}} \right)^{s} u - m^{2s} c^{\frac{2s}{1-s}} u +μu = |u|^{p-1} u $$ can be solved for large values of the "light speed" $c$ even when $p$ crosses the critical value for the fractional Sobolev embedding.
We study a boundary value problem related to the search of standing waves for the nonlinear Schrödinger equation (NLS) on graphs. Precisely we are interested in characterizing the standing waves of NLS posed on the {\it double-bridge graph}, in which two semi-infinite half-lines are attached at a circle at different vertices. At the two vertices the so-called Kirchhoff boundary conditions are imposed. The configuration of the graph is characterized by two lengths, $L_1$ and $L_2$, and we are interested in the existence and properties of standing waves of given frequency $ω$. For every $ω>0$ only solutions supported on the circle exist (cnoidal solutions), and only for a rational value of $L_1/L_2$; they can be extended to every $ω\in \mathbb{R}$. We study, for $ω<0$, the solutions periodic on the circle but with nontrivial components on the half-lines. The problem turns out to be equivalent to a nonlinear boundary value problem in which the boundary condition depends on the spectral parameter $ω$. After classifying the solutions with rational $L_1/L_2$, we turn to $L_1/L_2$ irrational showing that there exist standing waves only in correspondence to a countable set of frequencies $ω_n$. Moreover we show that the frequency sequence $\{ω_n\}_{n \geq 1}$ has a cluster point at $-\infty$ and it admits at least a finite limit point, in general non-zero. Finally, any negative real number can be a limit point of a set of admitted frequencies up to the choice of a suitable irrational geometry $L_1/L_2$ for the graph. These results depend on basic properties of diophantine approximation of real numbers.
We prove the existence of a solution to the semirelativistic Hartree equation $$\sqrt{-Δ+m^2}u+ V(x) u = A(x)\left( W * |u|^p \right) |u|^{p-2}u $$ under suitable growth assumption on the potential functions $V$ and $A$. In particular, both can be unbounded from above.
We prove some existence results for a class of nonlinear fractional equations driven by a nonlocal operator.
Under different assumptions on the potential functions $b$ and $c$, we study the fractional equation $\left( I-Δ\right)^α u = λb(x) |u|^{p-2}u+c(x)|u|^{q-2}u$ in $\mathbb{R}^N$. Our existence results are based on compact embedding properties for weighted spaces.
In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation $\sqrt{-\varepsilon^2 Δ+ m^2}u + V u = (I_α* |u|^{p}) |u|^{p-2}u$ in $\mathbb{R}^N$ where $m>0$, $2 \leq p < \frac{2N}{N-1}$, $V \colon \mathbb{R}^N \to \mathbb{R}$ is an external scalar potential, $I_α(x) = \frac{c_{N,α}}{|x|^{N-α}}$ is a convolution kernel, $c_{N,α}$ is a positive constant and $(N-1)p-N<α<N$. For $N=3$, $α=p=2$, our equation becomes the pseudo-relativistic Hartree equation with Coulomb kernel.
We prove that the critical problem for the fractional Laplacian in an annular type domain admits a nontrivial solution provided that the inner hole is sufficiently small.
We prove existence of positive ground state solutions to the pseudo-relativistic Schrödinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-Δ+m^2} u +Vu = \left( W * |u|^θ \right)|u|^{θ-2} u \quad\text{in $\mathbb{R}^N$}\\ u \in H^{1/2}(\mathbb{R}^N) \end{array} \right. \end{equation*} where $N \geq 3$, $m >0$, $V$ is a bounded external scalar potential and $W$ is a convolution potential, radially symmetric, satisfying suitable assumptions. We also furnish some asymptotic decay estimates of the found solutions.
In this note we prove the existence of radially symmetric solutions for a class of fractional Schrödinger equation in (\mathbb{R}^N) of the form {equation*} \slap u + V(x) u = g(u), {equation*} where the nonlinearity $g$ does not satisfy the usual Ambrosetti-Rabinowitz condition. Our approach is variational in nature, and leans on a Pohozaev identity for the fractional laplacian.
We investigate the soliton dynamics for the fractional nonlinear Schrodinger equation by a suitable modulational inequality. In the semiclassical limit, the solution concentrates along a trajectory determined by a Newtonian equation depending of the fractional diffusion parameter.
We prove existence and multiplicity of symmetric solutions for the \emph{Schrödinger-Newton system} in three dimensional space using equivariant Morse theory.
We construct solutions to a class of Schrödinger equations involving the fractional laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.
By using a perturbation technique in critical point theory, we prove the existence of solutions for two types of nonlinear equations involving fractional differential operators.
This work is devoted to the Dirichlet problem for the equation (-Δu = λu + |x|^α|u|^{2^*-2} u) in the unit ball of $\mathbb{R}^N$. We assume that $λ$ is bigger than the first eigenvalues of the laplacian, and we prove that there exists a solution provided $α$ is small enough. This solution has a variational characterization as a ground state.
We study the problem (-ε\mathrm{i}\nabla+A(x)) ^{2}u+V(x)u=ε^{-2}(\frac{1}{|x|}\ast|u|^{2}) u, u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),\text{\ \ \ \}ε\nabla u+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}), where $A\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an exterior magnetic potential, $V\colon\mathbb{R}^{3}\rightarrow\mathbb{R}$ is an exterior electric potential, and $ε$ is a small positive number. If A=0 and $ε=\hbar$ is Planck's constant this problem is equivalent to the Schrödinger-Newton equations proposed by Penrose in \cite{pe2}\ to describe his view that quantum state reduction occurs due to some gravitational effect. We assume that $A$ and $V$ are compatible with the action of a group $G$ of linear isometries of $\mathbb{R}^{3}$. Then, for any given homomorphism $τ:G\rightarrow\mathbb{S}^{1}$ into the unit complex numbers, we show that there is a combined effect of the symmetries and the potential $V$ on the number of semiclassical solutions $u:\mathbb{R}% ^{3}\rightarrow\mathbb{C}$ which satisfy $u(gx)=τ(g)u(x)$ for all $g\in G$, $x\in\mathbb{R}^{3}$. We also study the concentration behavior of these solutions as $ε\rightarrow0.\medskip$