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Simone Secchi

Publications and source records attributed to Simone Secchi.

At least 19 recordsLinked to original sources
arXiv2026

Nonlinear Schr\"odinger equations with a critical, inverse-square potential

We study the existence of solutions of the following nonlinear Schr\"odinger equation $$ -\Delta u+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic with respect to $x\in\mathbb{R}^N.$ We assume that $V$ has positive essential infimum, $f$ satisfies weak growth conditions and $N\geq 3$. The approach to the problem uses variational methods with nonstandard functional setting. We obtain the existence of the ground state solution using the new profile decomposition.

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arXiv2026

A note on critical problems involving the $p$-Grushin Operator: existence of infinitely many solutions

We consider a critical problem in a bounded domain involving the $p$-Grushin operator $\Delta_\alpha^p$. After a truncation argument, we obtain infinitely many solutions to our problem via Krasnoselskii's genus, extending a previous result of Garc\'ia Azorero and Peral Alonso to the $p$-Grushin operator. A central part of our analysis is the verification of the Palais-Smale condition of the associated functional under a certain level.

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arXiv2025

Regularizing effect of the interplay between coefficients in linear and semilinear $X$-elliptic equations

We study the regularizing effect arising from the interaction between the coefficient \(a\) of the zero order term and the datum \(f\) in the problem $$ \left\lbrace \begin{array}{ll} -\mathcal{L}u + a(x) g(u) = f(x) \quad &\mbox{in} \;\; \Omega, u = 0 \quad &\mbox{on} \;\; \partial\Omega, \end{array} \right. $$ where $\Omega\subseteq\mathbb{R}^N$ is a bounded domain and $\mathcal{L}$ is an $X$-elliptic operator introduced by Lanconelli and Kogoj. If $f \in L^1(\Omega)$, we prove that the \(Q\)-condition introduced by Arcoya and Boccardo is sufficient to ensure the existence and boundedness of solutions in the framework of $X$-elliptic operators as well. Finally, we prove the existence of a bounded solution for linear problems under a more general condition between $f$ and $a$.

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arXiv2025

Existence and decay for a Grushin problem in $\mathbb{R}^N$ with singular, convective, critical reaction

We establish an existence result for a problem set in the whole Euclidean space involving the Grushin operator and featuring a critical term perturbed by a singular, convective reaction. Our approach combines variational methods, truncation techniques, and concentration-compactness arguments, together with set-valued analysis and fixed point theory. Additionally, we prove the decay at infinity of solutions in the absence of the convective term. The result is new even in the case where more than one feature between singularity, convectivity and criticality is taken into account.

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arXiv2025

Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator

In this article we prove a bifurcation and multiplicity result for a critical problem involving a degenerate nonlinear operator $\Delta_\gamma^p$. We extend to a generic $p>1$ a result which was proved only when $p=2$. When $p\neq 2$, the nonlinear operator $-\Delta_\gamma^p$ has no linear eigenspaces, so our extension is nontrivial and requires an abstract critical theorem which is not based on linear subspaces. We also prove a new abstract result based on a pseudo-index related to the $\mathbf{Z}_2$-cohomological index that is applicable here. We provide a version of the Lions' Concentration-Compactness Principle for our operator.

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arXiv2024

On the embedding of weighted Sobolev spaces with applications to a planar nonlinear Schr\"{o}dinger equation

In this paper we study the embedding properties for the weighted Sobolev space $H^1_V(\mathbb{R}^N)$ into the Lebesgue weighted space $L^\tau_W(\mathbb{R}^N)$. Here $V$ and $W$ are diverging weight functions. The different behaviour of $V$ with respect to $W$ at infinity plays a crucial role. Particular attention is paid to the case $V=W$. This situation is very delicate since it depends strongly on the dimension and, in particular, $N=2$ is somewhat a limit case. As an application, an existence result for a planar nonlinear Schr\"odinger equation in presence of coercive potentials is provided.

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arXiv2024

Mountain Pass Solutions for an entire semipositone problem involving the Grushin Subelliptic Operator

For $N\ge 3$ we study the following semipositone problem $$ -\Delta_\gamma u = g(z) f_a(u) \quad \hbox{in $\mathbb{R}^N$}, $$ where $\Delta_\gamma$ is the Grushin operator $$ \Delta_ \gamma u(z) = \Delta_x u(z) + \vert x \vert^{2\gamma} \Delta_y u (z) \quad (\gamma\ge 0), $$ $g\in L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ is a positive function, $a>0$ is a parameter and $f_a$ is a continuous function on $\mathbb{R}$ that coincides with $f(t) -a$ for $t\in\mathbb{R}^+$, where $f$ is a continuous function with subcritical and Ambrosetti-Rabinowitz type growth and which satisfies $f(0) = 0$. Depending on the range of $a$, we obtain the existence of positive mountain pass solutions in $D_\gamma(\mathbb{R}^N)$

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arXiv2024

A note on nonlinear critical problems involving the Grushin Subelliptic Operator: bifurcation and multiplicity results

We consider the boundary value problem $$ \cases{ -\Delta_\gamma u = \lambda u + \left\vert u \right\vert^{2^*_\gamma-2}u &in $\Omega$\cr u = 0 &on $\partial\Omega$,\cr } $$ where $\Omega$ is an open bounded domain in $\mathbb{R}^N$, $N \geq 3$, while $\Delta_\gamma$ is the Grushin operator $$ \Delta_ \gamma u(z) = \Delta_x u(z) + \vert x \vert^{2\gamma} \Delta_y u (z) \quad (\gamma\ge 0). $$ We prove a multiplicity and bifurcation result for this problem, extending the results of Cerami, Fortunato and Struwe and of Fiscella, Molica Bisci and Servadei.

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arXiv2023

Compact embeddings for weighted fractional Sobolev spaces and applications to Nonlinear Schr\"odinger Equations

The aim of this work is to prove a compact embedding for a weighted fractional Sobolev spaces. As an application, we use this embedding to prove, via variational methods, the existence of solutions for the following Schr\"odinger equation $$ (-\Delta)^su + V(|x|)u = K(|x|)f(u), \quad \text{ in } \mathbb{R}^N, $$ where the two measurable functions $K > 0$ and $V \geq 0$ could vanish at infinity.

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arXiv2023

A note on the NLS equation on Cartan-Hadamard manifolds with unbounded and vanishing potentials

We study the semilinear equation $-\Delta_g u + V(\sigma) u = f(u)$ on a Cartan-Hadamard manifold ${\cal M}$ of dimension $N \geq 3$, and we prove the existence of a nontrivial solution under suitable assumptions on the potential function $V \in C({\cal M})$. In particular, the decay of $V$ at infinity is allowed, with some restrictions related to the geometry of ${\cal M}$. We generalize some results proved in $\mathbb{R}^N$ by Alves \emph{et al.}

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arXiv2022

Schr\"odinger equation on Cartan-Hadamard manifolds with oscillating nonlinearities

We study the equation $-\Delta_g w+w=\lambda \alpha(\sigma) f(w)$ on a $d$-dimensional homogeneous Cartan-Hadamard Manifold $\mathcal{M}$ with $d \geq 3$. Without using the theory of topological indices, we prove the existence of infinitely many solutions for a class of nonlinearities $f$ which have an oscillating behavior either at zero or at infinity.

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arXiv2022

Multiple solutions for Schr\"odinger equations on Riemannian manifolds via $\nabla$-theorems

We consider a smooth, complete and non-compact Riemannian manifold $(\mathcal{M},g)$ of dimension $d \geq 3$, and we look for positive solutions to the semilinear elliptic equation $$ -\Delta_g w + V w = \alpha f(w) + \lambda w \quad\hbox{in $\mathcal{M}$}. $$ The potential $V \colon \mathcal{M} \to \mathbb{R}$ is a continuous function which is coercive in a suitable sense, while the nonlinearity $f$ has a subcritical growth in the sense of Sobolev embeddings. By means of $\nabla$-Theorems introduced by Marino and Saccon, we prove that at least three solution exists as soon as the parameter $\lambda$ is sufficiently close to an eigenvalue of the operator $-\Delta_g$.

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arXiv2021

On Critical Kirchhoff problems driven by the fractional Laplacian

We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the sense of Sobolev embeddings. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of some recent contributions already present in the literature.

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arXiv2021

Semirelativistic Choquard equations with singular potentials and general nonlinearities arising from Hartree-Fock theory

We are interested in the general Choquard equation \begin{multline*} \sqrt{\strut -\Delta + m^2} \ u - mu + V(x)u - \frac{\mu}{|x|} u = \left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-\alpha}} \, dy \right) f(x,u) - K (x) |u|^{q-2}u \end{multline*} under suitable assumptions on the bounded potential \(V\) and on the nonlinearity \(f\). Our analysis extends recent results by the second and third author on the problem with $\mu = 0$ and pure-power nonlinearity $f(x,u)=|u|^{p-2}u$. We show that, under appropriate assumptions on the potential, whether the ground state does exist or not. Finally, we study the asymptotic behaviour of ground states as $\mu \to 0^+$.

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arXiv2020

Concentration phenomena for the Schr\"odinger-Poisson system in $\mathbb{R}^2$

We perform a semiclassical analysis for the planar Schr\"odinger-Poisson system \[ \cases{ -\varepsilon^{2} \Delta\psi+V(x)\psi= E(x) \psi \quad \text{in $\mathbb{R}^2$},\cr -\Delta E= |\psi|^{2} \quad \text{in $\mathbb{R}^2$}, \cr } \tag{$SP_\varepsilon$} \] where $\varepsilon$ is a positive parameter corresponding to the Planck constant and $V$ is a bounded external potential. We detect solution pairs $(u_\varepsilon, E_\varepsilon)$ of the system $(SP_\varepsilon)$ as~$\ge \rightarrow 0$.

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arXiv2020

Normalized solutions for the fractional NLS with mass supercritical nonlinearity

We investigate the existence of solutions to the fractional nonlinear Schr\"{o}dinger equation $(-\Delta)^s u = f(u)$ with prescribed $L^2$-norm $\int_{\mathbb{R}^N} |u|^2 \, dx =m$ in the Sobolev space $H^s(\mathbb{R}^N)$. Under fairly general assumptions on the nonlinearity $f$, we prove the existence of a ground state solution and a multiplicity result in the radially symmetric case.

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arXiv2019

Non-local to local transition for ground states of fractional Schr\"{o}dinger equations on bounded domains

We show that ground state solutions to the nonlinear, fractional problem \begin{align*} \left\{ \begin{array}{ll} (-\Delta)^{s} u + V(x) u = f(x,u) &\quad \mathrm{in} \ \Omega, \newline u = 0 &\quad \mathrm{in} \ \mathbb{R}^N \setminus \Omega, \end{array} \right. \end{align*} on a bounded domain $\Omega \subset \mathbb{R}^N$, converge (along a subsequence) in $L^2 (\Omega)$, under suitable conditions on $f$ and $V$, to a solution of the local problem as $s \to 1^-$.

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