arXiv subjects

Riccardo Molle

Publications and source records attributed to Riccardo Molle.

At least 19 recordsLinked to original sources

arXiv2026

Qualitative analysis of energy ground states for magnetic focusing Gross-Pitaevskii equations

We prove the uniqueness, asymptotics, symmetry, and orbital stability of energy ground states for 3D magnetic Gross-Pitaevskii equations under mild conditions on the electric potential and magnetic field. In particular, both the electric potential and the magnetic field are allowed to have singularities, and we cover in a unified approach the physically relevant Aharonov-Bohm magnetic field as well as the constant magnetic field. Since we are in the 3D case, the unique energy ground state (up to a phase factor) is obtained as a local minimizer, rather than a global one, by restricting the kinetic energy of candidate critical points within a suitable range. The qualitative analysis of the energy ground state we carry on is mainly based on a related Pohozaev identity, the implicit function theorem, and variational methods.

math.AP↗

arXiv2025

Normalized Schrödinger equations with mass-supercritical nonlinearity in exterior domains

We consider the problem $-Δu+λu=u^{p-1}$, where $u\in H^1_0(Ω)$ verifies $\|u\|_{L^2}=m>0$, and $λ\in [0,+\infty)$. Here, $\mathbb{R}^N\setminusΩ$ is nonempty and compact. We prove the existence of a solution with a constrained Morse index lower than or equal to $N+1$, both in the case $m$ fixed and $\mathbb{R}^N\setminusΩ$ in a small ball and in the case $Ω$ fixed and $m$ large.

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arXiv2023

Positive solutions to nonlinear elliptic problems involving Sobolev exponent

In this paper we consider nonlinear elliptic PDEs of the type $$-Δ_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }Ω,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of the weight function $a$ to be sensitive to the direction. We provide a unified variational approach to obtain existence of distinct solutions in either the unbounded case $Ω=\mathbb{R}^N$ or when $Ω$ is a smooth bounded domain. A key point is a precise description of the compactness properties of certain sequences of approximating solutions (Palais-Smale sequences), for which we use novel observations on nonexistence in certain regimes. Most of our main results are new in the case of the classical Laplace operator, $p=2$.

math.AP↗

arXiv2023

Normalised solutions and limit profiles of the defocusing Gross-Pitaevskii-Poisson equation

We study normalised solutions of the stationary Gross-Pitaevskii-Poisson (GPP) equation with a defocusing local nonlinear term, $$-Δu+λu+|u|^2u =(I_α*|u|^2)u\quad\text{in $\mathbb R^3$},\qquad\int_{\mathbb R^3}u^2dx=ρ^2,$$ where $ρ^2>0$ is the prescribed mass of the solutions, $λ\in\mathbb R$ is an a-priori unknown Lagrange multiplier, and $I_α(x)=A_α|x|^{3-α}$ is the Riesz potential of order $α\in(0,3)$. When $α=2$ this problem appears in the models of self-gravitating Bose-Einstein condensates, which were proposed in cosmology and astrophysics to describe Cold Dark Matter and Boson Stars. We establish the existence of branches of normalised solutions to the GPP equation, paying special attention to the shape of the associated mass-energy relation curves and to the limit profiles of solutions at the endpoints of these curves. The main novelty in this work is in the derivation of sharp asymptotic estimates on the mass-energy curves. These estimates allow us to show that after appropriate rescalings, the constructed normalized solutions converge either to a ground state of the Choquard equation or to a compactly supported radial ground state of the integral Thomas-Fermi equation. The behaviour of normalized solutions depends sensitively on whether $α$ is greater than, equal to, or less than one.

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arXiv2022

Standing waves for two-component elliptic system with critical growth in $\mathbb{R}^{4}$: the attractive case

In this paper, we consider the following two-component elliptic system with critical growth \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ)u=μ_1u^{3}+βuv^{2}, \ \ x\in \mathbb{R}^4, -Δv+(V_2(x)+λ)v=μ_2v^{3}+βvu^{2}, \ \ x\in \mathbb{R}^4 , % u\geq 0, \ \ v\geq 0 \ \text{in} \ \R^4. \end{cases} \end{equation*} where $V_j(x) \in L^{2}(\mathbb{R}^4)$ are nonnegative potentials and the nonlinear coefficients $β,μ_j$, $j=1,2$, are positive. Here we also assume $λ>0$. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of positive solutions under the hypothesis $β>\max\{μ_1,μ_2\}$. These results generalize the results for semilinear Schrödinger equation on half space by Cerami and Passaseo (SIAM J. Math. Anal., 28, 867-885, (1997)) to the above elliptic system, while extending the existence result from Liu and Liu (Calc. Var. Partial Differential Equations, 59:145, (2020)).

math.AP↗

arXiv2022

Normalized positive solutions for Schrödinger equations with potentials in unbounded domains

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation $$ -Δu+λu+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(Ω),\quad\int_Ωu^2dx=ρ^2,\quadλ\in\mathbb{R}, $$ where $Ω=\mathbb{R}^N$ or $\mathbb{R}^N\setminusΩ$ is a compact set, $ρ>0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in \left(2,2+\frac 4 N\right)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption $(D_ρ)$, or if $\|V\|_{L^q}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption $(D_ρ)$ is fulfilled. There are no assumptions on the size of $\mathbb{R}^N\setminusΩ$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $Ω=\mathbb{R}^N$.

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arXiv2022

On a planar Schrödinger-Poisson system involving a non-symmetric potential

We prove the existence of a ground state positive solution of Schrödinger-Poisson systems in the plane of the form $$ -Δu + V(x)u + \fracγ{2π} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \qquad\text{in}\ \mathbb{R}^2, $$ where $p>4$, $γ,b>0$ and the potential $V$ is assumed to be positive and unbounded at infinity. On the potential we do not require any symmetry or periodicity assumption, and it is not supposed it has a limit at infinity. We approach the problem by variational methods, using a variant of the mountain pass theorem and the Cerami compactness condition.

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arXiv2022

Variational properties of the first curve of the Fuč\'ık spectrum for elliptic operators

In this paper we present a new variational characteriztion of the first nontrival curve of the Fuč\'ık spectrum for elliptic operators with Dirichlet boundary conditions. Moreover, we describe the asymptotic behaviour and some properties of this curve and of the corresponding eigenfunctions. In particular, this new characterization allows us to compare the first curve of the Fuč\'ık spectrum with the infinitely many curves we obtained in previous works (see R. Molle, D. Passaseo, New properties of the Fuč\'ık spectrum. C. R. Math. Acad. Sci. Paris 351 (2013), no. 17/18, 681--685 and R. Molle, D. Passaseo, Infinitely many new curves of the Fuč\'ık spectrum. Ann. I. H. Poincaré - AN (2014), http://dx.doi.org/10.1016/j.anihpc.2014.05.007): for example, we show that these curves are all asymptotic to the same lines as the first curve, but they are all distinct from such a curve.

math.AP↗

arXiv2021

Infinitely many solutions for elliptic equations with non-symmetric nonlinearities

We deal with the existence of infinitely many solutions for a class of elliptic problems with non-symmetric nonlinearities. Our result, which is motivated by a well known conjecture formulated by A. Bahri and P.L. Lions, suggests a new approach to tackle these problems. The proof is based on a method which does not require to use techniques of deformation from the symmetry and may be applied to more general non-symmetric problems.

math.AP↗

arXiv2021

Normalized solutions to mass supercritical Schrodinger equations with negative potential

We study the existence of positive solutions with prescribed $L^2$-norm for the Schrödinger equation \[ -Δu-V(x)u+λu=|u|^{p-2}u\qquadλ\in \mathbb{R},\quad u\in H^1(\mathbb{R}^N), \] where $V\ge 0$, $N\ge 1$ and $p\in\left(2+\frac 4 N,2^*\right)$, $2^*:=\frac{2N}{N-2}$ if $N\ge 3$ and $2^*:=+\infty$ if $N=1,2$. We treat two cases. Firstly, under an explicit smallness assumption on $V$ and no condition on the mass, we prove the existence of a mountain pass solution at positive energy level, and we exclude the existence of solutions with negative energy. Secondly, requiring that the mass is smaller than some explicit bound, depending on $V$, and that $V$ is not too small in a suitable sense, we find two solutions: a local minimizer with negative energy, and a mountain pass solution with positive energy. Moreover, a nonexistence result is proved.

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arXiv2020

Normalized solutions of mass supercritical Schrödinger equations with potential

This paper is concerned with the existence of normalized solutions of the nonlinear Schrödinger equation \[ -Δu+V(x)u+λu = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in the mass supercritical and Sobolev subcritical case $2+\frac{4}{N}<p<2^*$. We prove the existence of a solution $(u,λ)\in H^1(\mathbb{R}^N)\times\mathbb{R}^+$ with prescribed $L^2$-norm $\|u\|_2=ρ$ under various conditions on the potential $V:\mathbb{R}^N\to\mathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.

math.AP↗

arXiv2020

Multiple positive bound state solutions of a critical Choquard equation

IIn this paper we consider the problem $$ \left\{ \begin{array}{rcl} -Δu+V_λ(x)u=(I_μ*|u|^{2^{*}_μ})|u|^{2^{*}_μ-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right.\leqno{(P_λ)} $$ where $V_λ=λ+V_{0}$ with $λ\geq 0$, $V_0\in L^{N/2}(\R^N)$, $I_μ=\frac{1}{|x|^μ}$ is the Riesz potential with $0<μ<\min\{N,4\}$ and $2^{*}_μ=\frac{2N-μ}{N-2}$ with $N\geq 3$. Under some smallness assumption on $V_0$ and $λ$ we prove the existence of two positive solutions of $(P_λ)$. In order to prove the main result, we used variational methods combined with degree theory.

math.AP↗

arXiv2019

Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth

We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }Ω,\qquad u=0\ \mbox{ on }\partialΩ, $$ where $Ω$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1 0$ small enough, there exists a unique solution of the Dirichlet problem in the domain $Ω=Ω^Γ_\varepsilon=\{(x_1,x_2)\in\mathbb{R}^2\ :\ \mathop{\rm dist}\big((x_1,x_2),Γ\big)<\varepsilon\}$, where $Γ=\{γ(t)\ :\ t\in[a,b]\}$. Moreover, we extend this uniqueness result to the case where $n>2$ and $Ω$ is, for example, a domain of the type $$ Ω=\widetildeΩ^Γ_{\varepsilon,s}=\{(x_1,x_2,y)\ :\ (x_1,x_2)\inΩ^Γ_\varepsilon, \ y\in\mathbb{R}^{n-2},\ |y|<s\}. $$

math.AP↗

arXiv2019

Infinitely many positive solutions of nonlinear Schrodinger equations

The paper deals with the equation $-Δu+a(x) u =|u|^{p-1}u $, $u \in H^1(\mathbb{R}^N)$, with $N\ge 2$, $p>1,\ p<{N+2\over N-2}$ if $N\ge 3$, $a\in L^{N/2}_{loc}(\mathbb{R}^N)$, $\inf a>0$, $\lim_{|x| \to \infty} a(x)= a_\infty$. Assuming on the potential that $\lim_{|x| \to \infty}[a(x)-a_\infty] e^{η|x|}= \infty \ \ \forall η>0$ and $ \lim_{ρ\to \infty} \sup \left\{a(ρθ_1) - a(ρθ_2) \ :\ θ_1, θ_2 \in \mathbb{R}^N,\ |θ_1|= |θ_2|=1 \right\} e^{\tildeηρ} = 0 \mbox{ for some } \ \tildeη>0$, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers [8,12,15,17,28].

math.AP↗

arXiv2019

Nonexistence of solutions for Dirichlet problems with supercritical growth in tubular domains

We deal with Dirichlet problems of the form $$ Δu+f(u)=0 \mbox{ in }Ω,\qquad u=0\ \mbox{ on }\partial Ω$$ where $Ω$ is a bounded domain of $\mathbb{R}^n$, $n\ge 3$, and $f$ has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where $Ω$ is a tubular domain $T_\varepsilon(Γ_k)$ with thickness $\varepsilon>0$ and centre $Γ_k$, a $k$-dimensional, smooth, compact submanifold of $\mathbb{R}^n$. Our main result concerns the case where $k=1$ and $Γ_k$ is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for $\varepsilon>0$ small enough. When $k\ge 2$ or $Γ_k$ is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on $k$ and $f$.

math.AP↗

arXiv2019

Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in exterior domains

The paper concerns with positive solutions of problems of the type $-Δu+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $Ω\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2 0$; in particular $a\equiv {\rm const}$ is allowed. First, some existence results of ground state solutions are proved. Then the case $a(x)\ge a_\infty$ is considered, with $a(x)\not\equiv a_\infty$ or $Ω\neq\mathbb{R}^N$. In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small $\varepsilon$. No hypotheses are assumed on the size of $\mathbb{R}^N\setminusΩ$ and on $\|a-a_\infty\|_{L^{N/2}}$.

math.AP↗

arXiv2019

Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains

We deals with nonlinear elliptic Dirichlet problems of the form $${\rm div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }Ω,\qquad u\in H^{1,p}_0(Ω) $$ where $Ω$ is a bounded domain in $\mathbb{R}^n$, $n\ge 2$, $p> 1$ and $f$ has supercritical growth from the viewpoint of Sobolev embedding. Our aim is to show that there exist bounded contractible non star-shaped domains $Ω$, arbitrarily close to domains with nontrivial topology, such that the problem does not have nontrivial solutions. For example, we prove that if $n=2$, $1 {2p\over 2-p}$ and $Ω=\{(ρ\cosθ,ρ\sinθ)\ :\ |θ|<α,\ |ρ-1| {2p\over 2-p}$ there exists $\bar s>0$ such that the problem has only the trivial solution $u\equiv 0$ for all $α\in (0,π)$ and $s\in (0,\bar s)$.

math.AP↗

arXiv2018

Infinitely many positive standing waves for Schrödinger equations with competing coefficients

The paper deals with the equation $-Δu+a(x) u +b(x)u^q -u^p = 0$, $u \in H^1(\R^N)$, whith $N\ge 2$, $1 0$, $a(x)\to a_\infty$ and $b(x)\to 0$ as $|x|\to\infty$. When $a(x)\le a_\infty$ and $b(x) = 0$ only a finite number of positive solutions to the problem is reasonably expected. Here we prove that the presence of a nonzero term $b(x)u^q $ with $b(x)\geq 0, \ b(x)\neq 0,$ under suitable assumptions on the decay rates of $a$ and $b,$ allows to obtain infinitely many positive solutions.

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