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Marco Squassina

Publications and source records attributed to Marco Squassina.

At least 19 recordsLinked to original sources
arXiv2026

Power law convergence and concavity for the Logarithmic Schrödinger equation

We study concavity properties of positive solutions to the Logarithmic Schrödinger equation $-Δu=u\, \log u^2$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane-Emden problems $-Δu = σ\, (u^q-u)$ and build, for any $σ>0$ and $q>1$, solutions $u_q$ such that $u_q^{(1-q)/2}$ is convex. By choosing $σ_q=2/(q-1)$ and letting $q \to 1^+$ we eventually construct a solution $u$ of the Logarithmic Schrödinger equation such that $\log u$ is concave. This seems to be one of the few attempts at studying concavity properties for superlinear, sign changing sources. To get the result, we both make inspections on the constant rank theorem and develop Liouville theorems on convex epigraphs, which might be useful in other frameworks.

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arXiv2025

Quantitative and exact concavity principles for parabolic and elliptic equations

Goal of this paper is to study classes of Cauchy-Dirichlet problems which include parabolic equations of the type $$u_t -Δu= a(x,t)f(u)\quad\hbox{in $Ω\times(0,T)$}$$ with $Ω\subset\mathbb{R}^N$ bounded, convex domain and $T\in(0,+\infty]$. Under suitable assumptions on $a$ and $f$, we show logarithmic or power concavity (in space, or in space-time) of the solution $u$; under some relaxed assumptions on $a$, we show moreover that $u$ enjoys concavity properties up to a controlled error. The results include relevant examples like the torsion $f(u)=1$, the Lane-Emden equation $f(u)=u^q$, $q\in(0,1)$, the eigenfunction $f(u)=u$, the logarithmic equation $f(u)=u\log(u^2)$, and the saturable nonlinearity $f(u)=\frac{u^2}{1+u}$. The logistic equation $f(x,u)=a(x)u-u^2$ can be treated as well. Some exact results give a different approach, as well as generalizations, to [Ishige-Salani2013, Ishige-Salani2016]. Moreover, some quantitative results are valid also in the elliptic framework $-Δu=a(x)f(u)$ and refine [Bucur-Squassina2019, Gallo-Squassina2024].

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arXiv2025

Concavity and perturbed concavity for $p$-Laplace equations

In this paper we study convexity properties for quasilinear Lane-Emden-Fowler equations of the type $$\begin{cases} -Δ_p u = a(x) u^q & \quad \hbox{ in $Ω$},\\ u >0 & \quad \hbox{ in $Ω$}, \\ u =0 & \quad \hbox{ on $\partial Ω$}, \end{cases}$$ when $Ω\subset \mathbb{R}^N$ is a convex domain. In particular, in the subhomogeneous case $q \in [0,p-1]$, the solution $u$ inherits concavity properties from $a$ whenever assumed, while it is proved to be concave up to an error if $a$ is near to a constant. More general problems are also taken into account, including a wider class of nonlinearities. These results generalize some contained in [Kennington, Indiana Univ. Math. J., 1985] and [Sakaguchi, Ann. Sc. Norm. Super. Pisa, 1987]. Additionally, some results for the singular case $q \in [-1,0)$ and the superhomogeneous case $q>p-1$, $q \approx p-1$ are obtained. Some properties for the $p$-fractional Laplacian $(-Δ)^s_p$, $s\in (0,1)$, $s \approx 1$, are shown as well. We highlight that some results are new even in the semilinear framework $p=2$; in some of these cases, we deduce also uniqueness (and nondegeneracy) of the critical point of $u$.

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arXiv2025

On nodal solutions with a prescribed number of nodes for a Kirchhoff-type problem

We are concerned with the existence and asymptotic behavior of multiple radial sign-changing solutions with the nodal characterization for a Kirchhoff-type problem involving the nonlinearity $|u|^{p-2}u(2<p<4)$ in $\mathbb{R}^3$. By developing some useful analysis techniques and introducing a novel definition of the Nehari manifold for the auxiliary system of the equations, we show that, for any positive integer $k$, the problem has a sign-changing solution $u_k^b$ changing signs exactly $k$ times. Furthermore, the energy of $u_k^b$ is strictly increasing in $k$, as well as some asymptotic behaviors of $u_k^b$ are obtained. Our result is a complement of [Deng Y, Peng S, Shuai W, {\it J. Funct. Anal.}, {\bf269}(2015), 3500-3527], where the case $2<p<4$ is left open.

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arXiv2024

Existence of normalized solutions of a Hartree-Fock system with mass subcritical growth

In this paper, we are concerned with normalized solutions in $H_{r}^{1}(\mathbb{R}^{3}) \times H_{r}^{1}(\mathbb{R}^{3})$ for Hartree-Fock type systems with the form \be\lab{ Hartree-Fock} \left\{ \begin{array}{ll} -Δu +αϕ_{u,v} u=λ_{1} u+\left | u \right | ^{2q-2} u+β\left | v \right | ^{q} \left | u \right | ^{q-2} u , \\ -Δv +αϕ_{u,v} v=λ_{2} v+\left | v\right | ^{2q-2} v+β\left | u \right | ^{q} \left | v \right | ^{q-2} v , \\ \int_{\mathbb{R}^{3}}\left | u \right | ^{2} {\rm d}x=a_{1} , \quad \int_{\mathbb{R}^{3}}\left | v \right | ^{2} {\rm d}x=a_{2} , \nonumber\\ \end{array} where $$ ϕ_{u, v}\left(x\right):=\int_{\mathbb{R}^{3}} \frac{u^{2}(y)+v^{2}(y)}{|x-y|} {\rm d}y \in D^{1,2}\left(\mathbb{R}^{3}\right). $$ Here $α,β>0, a_1,a_2>0$ and $1 0$ when $1 0$ small when $\frac{4}{3}\le q < \frac{3}{2}$. The nonexistence of normalized solutions is also considered for $\frac{3}{2}\le q < \frac{5}{3}$. Also, the orbital stability of standing waves is obtained under local well-posedness assumptions of the evolution problem.

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arXiv2024

Concave solutions to Finsler $p$-Laplace type equations

We prove concavity properties for solutions to anisotropic quasi-linear equations, extending previous results known in the Euclidean case. We focus the attention on nonsmooth anisotropies and in particular we also allow the functions describing the anisotropies to be not even.

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arXiv2024

Generalized Chern-Simons-Schrodinger system with critical exponential growth: the zero mass case

We consider the existence of ground state solutions for a class of zero-mass Chern-Simons-Schrödinger systems \[ \left\{ \begin{array}{ll} \displaystyle -Δu +A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u)-a(x)|u|^{p-2}u, \newline \displaystyle \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,~\partial_1A_1+\partial_2A_2=0, \newline \displaystyle \partial_1A_0=A_2|u|^2,~ \partial_2A_0=-A_1|u|^2, \end{array} \right. \] where $a:\mathbb R^2\to\mathbb R^+$ is an external potential, $p\in(1,2)$ and $f\in \mathcal{C}(\mathbb R)$ denotes a nonlinearity that fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. By introducing an improvement of the version of Trudinger-Moser inequality, we are able to investigate the existence of positive ground state solutions for the given system using variational method.

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arXiv2024

Existence and concentration of normalized solutions for $p$-Laplacian equations with logarithmic nonlinearity

We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type \[ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^pΔ_p u+V(x)|u|^{p-2}u=λ|u|^{p-2}u+|u|^{p-2}u\log|u|^p ~\text{in}~\mathbb R^N,\newline \displaystyle \int_{\mathbb R^N}|u|^pdx=a^p\varepsilon^N, \end{array} \right. \] where $a,\varepsilon> 0$, $λ\in\mathbb R$ is known as the Lagrange multiplier, $Δ_p\cdot =\text{div} (|\nabla \cdot|^{p-2}\nabla \cdot)$ denotes the usual $p$-Laplacian operator with $2\leq p < N$ and $V \in \mathcal{C}^0(\mathbb R^N)$ is the potential which satisfies some suitable assumptions. We prove that the number of positive solutions depends on the profile of $V$ and each solution concentrates around its corresponding global minimum point of $V$ in the semiclassical limit when $\varepsilon\to0^+$ using variational method. Moreover, we also get the existence of normalized solutions for some logarithmic $p$-Laplacian equations involving mass-supercritical nonlinearities.

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arXiv2024

Normalized solutions of quasilinear Schrödinger equations with a general nonlinearity

We are concerned with solutions of the following quasilinear Schrödinger equations \begin{eqnarray*} -{\mathrm{div}}\left(φ^{2}(u) \nabla u\right)+φ(u) φ^{\prime}(u)|\nabla u|^{2}+λu=f(u), \quad x \in \mathbb{R}^{N} \end{eqnarray*} with prescribed mass $$ \int_{\mathbb{R}^{N}} u^{2} \mathrm{d}x=c, $$ where $N\ge 3, c>0$, $λ\in \mathbb{R}$ appears as the Lagrange multiplier and $φ\in C ^{1}(\mathbb{R} ,\mathbb{R}^{+})$. The nonlinearity $f \in C\left ( \mathbb{R}, \, \mathbb{R} \right )$ is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case.

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arXiv2024

Infinitely many solutions for a class of fractional Schrodinger equations coupled with neutral scalar field

We study the fractional Schrödinger equations coupled with a neutral scalar field $$ (-Δ)^s u+V(x)u=K(x)ϕu +g(x)|u|^{q-2}u, \quad x\in \mathbb{R}^3,\qquad (I-Δ)^t ϕ=K(x)u^2, \quad x\in \mathbb{R}^3, $$ where $(-Δ)^s$ and $(I-Δ)^t$ denote the fractional Laplacian and Bessel operators with $\frac{3}{4} <s<1$ and $0<t<1$, respectively. Under some suitable assumptions for the external potentials $V$, $K$ and $g$, given $q\in(1,2)\cup(2,2_s^*)$ with $2_s^*:= \frac{6}{3-2s}$, with the help of an improved Fountain theorem dealing with a class of strongly indefinite variational problems approached by Gu-Zhou [Adv. Nonlinear Stud., {\bf 17} (2017), 727--738], we show that the system admits infinitely many nontrivial solutions.

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arXiv2024

Normalized solutions for a fractional Schrödinger-Poisson system with critical growth

In this paper, we study the fractional critical Schrödinger-Poisson system \[\begin{cases} (-Δ)^su +λϕu= αu+μ|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-Δ)^tϕ=u^2,&~~ \mbox{in}~{\mathbb R}^3,\end{cases} \] having prescribed mass \[\int_{\mathbb R^3} |u|^2dx=a^2,\] where $ s, t \in (0, 1)$ satisfies $2s+2t > 3, q\in(2,2^*_s), a>0$ and $λ,μ>0$ parameters and $α\in{\mathbb R}$ is an undetermined parameter. Under the $L^2$-subcritical perturbation $q\in (2, 2+\frac{4s}{3})$, we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. For the $L^2$-supercritical perturbation $q\in (2+\frac{4s}{3}, 2^*_s)$, by applying the constrain variational methods and the mountain pass theorem, we show the existence of positive normalized ground state solutions.

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arXiv2024

Multiplicity of normalized solutions for the fractional Schrödinger equation with potentials

We get multiplicity of normalized solutions for the fractional Schrödinger equation $$ (-Δ)^su+V(\varepsilon x)u=λu+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where $(-Δ)^s$ is the fractional Laplacian, $s\in(0,1)$, $a,\varepsilon>0$, $λ\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V,h:\mathbb{R}^N\rightarrow[0,+\infty)$ are bounded and continuous, and $f$ is continuous function with $L^2$-subcritical growth. We prove that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $\varepsilon$ is small enough.

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arXiv2024

Planar Schrödinger-Poisson system with steep potential well: supercritical exponential case

We study a class of planar Schrödinger-Poisson systems $$ -Δu+λV(x)u+ϕu=f(u) , \quad x\in{\mathbb R}^2,\qquad Δϕ=u^2, \quad x\in{\mathbb R}^2, $$ where $λ>0$ is a parameter, $V\in C({\mathbb R}^2,{\mathbb R}^+)$ has a potential well $Ω\triangleq\text{int}\, V^{-1}(0)$ and the nonlinearity $f$ fulfills the supercritical exponential growth at infinity in the Trudinger-Moser sense. By exploiting the mountain-pass theorem and elliptic regular theory, we establish the existence and concentrating behavior of ground state solutions for sufficiently large $λ$.

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arXiv2024

Normalized solutions to the Chern-Simons-Schrödinger system: the supercritical case

We are concerned with the existence of normalized solutions for a class of generalized Chern-Simons-Schrödinger type problems with supercritical exponential growth $$ -Δu +λu+A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u),\quad \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,\quad \partial_1A_1+\partial_2A_2=0,\quad \partial_1A_0=A_2|u|^2,\quad \partial_2A_0=-A_1|u|^2,\quad \int_{\mathbb{R}^2}|u|^2dx=a^2, $$ where $a\neq0$, $λ\in \mathbb{R}$ is known as the Lagrange multiplier and $f\in C^1(\mathbb{R})$ denotes the nonlinearity that fulfills the supercritical exponential growth in the Trudinger-Moser sense at infinity. Under suitable assumptions, combining the constrained minimization approach together with the homotopy stable family and elliptic regularity theory, we obtain that the problem has at least a ground state solution.

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arXiv2023

Concavity properties for quasilinear equations and optimality remarks

In this paper we study quasiconcavity properties of solutions of Dirichlet problems related to modified nonlinear Schrödinger equations of the type $$-{\rm div}\big(a(u) \nabla u\big) + \frac{a'(u)}{2} |\nabla u|^2 = f(u) \quad \hbox{in $Ω$},$$ where $Ω$ is a convex bounded domain of $\mathbb{R}^N$. In particular, we search for a function $φ:\mathbb{R} \to \mathbb{R}$, modeled on $f\in C^1$ and $a\in C^1$, which makes $φ(u)$ concave. Moreover, we discuss the optimality of the conditions assumed on the source.

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arXiv2023

Concavity principles for nonautonomous elliptic equations and applications

In the study of concavity properties of positive solutions to nonlinear elliptic partial differential equations the diffusion and the nonlinearity are typically independent of the space variable. In this paper we obtain new results aiming to get almost concavity results for a relevant class of anisotropic semilinear elliptic problems with spatially dependent source and diffusion.

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arXiv2023

Semiclassical states for coupled nonlinear Schrödinger equations with a critical frequency

In this paper, we are concerned with the coupled nonlinear Schrödinger system \begin{align*} \begin{cases} -\varepsilon^{2}Δu+a(x)u=μ_{1}u^{3}+βv^{2}u \ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\ -\varepsilon^{2}Δv+b(x)v=μ_{2}v^{3}+βu^{2}v \ \ \ \ \ \mbox{in}\ \mathbb{R}^{N}, \end{cases} \end{align*} where $1\leq N\leq3$, $μ_{1},μ_{2},β>0$, $a(x)$ and $b(x)$ are nonnegative continuous potentials, and $\varepsilon>0$ is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as $\varepsilon\rightarrow0$, when $a(x)$ and $b(x)$ achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.

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