arXiv subjects

Marino Badiale

Publications and source records attributed to Marino Badiale.

14 recordsLinked to original sources

arXiv2026

Computations for the first Lyapunov coefficient

These notes are a supplementary file to the paper Hopf bifurcations for HANDY-type models (M. Badiale and I. Cravero, under submission), providing full details of the computations developed in Section 4.2. The purpose of this supplement is to derive explicitly the first Lyapunov coefficient associated with a Hopf bifurcation, following the framework of Yu. A. Kuznetsov (Elements of Applied Bifurcation Theory, Springer, 4th ed., 2023). We compute the multilinear forms $B$ and $C$, the right and left eigenvectors and their normalization, and the resolvents $A^{-1}$ and $(2iω_0 I - A)^{-1}$. Using asymptotic expansions with respect to the small parameter $\varepsilon$, we derive explicit formulas for $μ(\varepsilon)$, $ω_0$, and the Lyapunov coefficient $a(μ(\varepsilon),\varepsilon)$, which characterize the criticality of the Hopf bifurcation in the main model.

math.DS↗

arXiv2022

A note on quasilinear Schrödinger equations with singular or vanishing radial potentials

In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation \begin{equation*} -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \end{equation*} with singular or vanishing continuous radial potentials $V(r)$, $K(r)$. In our previuos study we assumed, for technical reasons, that $K(r)$ was vanishing as $r \rightarrow 0$, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables $w=f(u)$ and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in $\mathbb{R}^{N} \setminus \{0\}$. The nonlinearity $g$ has a double-power behavior, whose standard example is $g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$ ($t>0$), recovering the usual case of a single-power behavior when $q_1 = q_2$.

math.AP↗

arXiv2022

Existence results for a class of quasilinear Schrödinger equations with singular or vanishing potentials

Given two continuous functions $V\left(r \right)\geq 0$ and $K\left(r\right)> 0$ ($r>0$), which may be singular or vanishing at zero as well as at infinity, we study the quasilinear elliptic equation \[ -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \] where $N\geq3$. To study this problem we apply a change of variables $w=f(u)$, already used by several authors, and find existence results for nonnegative solutions by the application of variational methods. The main features of our results are that they do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and that they essentially rely on power type estimates of the relative growth of $V$ and $K$, not of the potentials separately. Our solutions satisfy a weak formulations of the above equation, but we are able to prove that they are in fact classical solutions in $\mathbb{R}^{N} \backslash \{ 0\}$. To apply variational methods, we have to study the compactness of the embedding of a suitable function space into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. The nonlinearity $g$ has a double-power behavior, whose standard example is $g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$, recovering the usual case of a single-power behavior when $q_1 = q_2$.

math.AP↗

arXiv2022

Radial quasilinear elliptic problems with singular or vanishing potentials

In this paper we continue the work that we began in arXiv:1912.07537. Given $1 0$, and a continuous function $A(r) >0\ (r>0)$, we consider the quasilinear elliptic equation \[ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) +V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}, \] where all the potentials $A,V,K$ may be singular or vanishing, at the origin or at infinity. We find existence of nonnegative solutions by the application of variational methods, for which we need to study the compactness of the embedding of a suitable function space $X$ into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$. The nonlinearity has a double-power super $p$-linear behavior, as $f(t)= \min \left\{ t^{q_1 -1}, t^{q_2 -1} \right\}$ with $q_1,q_2>p$ (recovering the power case if $q_1=q_2$). With respect to \cite{AVK_I}, in the present paper we assume some more hypotheses on $V$, and we are able to enlarge the set of values $q_1 , q_2$ for which we get existence results.

math.AP↗

arXiv2019

Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials

Given $N\geq 3$, $1 0$ and a continuous function $A(r) >0$ ($r>0$), we study the quasilinear elliptic equation \[ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) u+V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}. \] We find existence of nonegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space $X$ into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. Our results do not require any compatibility between how the potentials $A$, $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of $V$ and $K$, not of the potentials separately. The nonlinearity $f$ has a double-power behavior, whose standard example is $f(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$, recovering the usual case of a single-power behavior when $q_1 = q_2$.

math.AP↗

arXiv2018

Radial solutions for the bilaplacian equation with vanishing or singular radial potentials

Given three measurable functions $V\left(r \right)\geq 0$, $K\left(r\right)> 0$ and $Q\left(r \right)\geq 0$, $r>0$, we consider the bilaplacian equation \[ Δ^2 u+V(|x|)u=K(|x|)f(u)+Q(|x|) \quad \text{in }\,\mathbb{R}^N \] and we find radial solutions thanks to compact embeddings of radial spaces of Sobolev functions into sum of weighted Lebesgue spaces.

math.AP↗

arXiv2017

Radial nonlinear elliptic problems with singular or vanishing potentials

In this paper we prove existence of radial solutions for the nonlinear elliptic problem \[ -\mathrm{div}(A(|x|)\nabla u)+V(|x|)u=K(|x|)f(u) \quad \text{in }\mathbb{R}^{N}, \] \noindent with suitable hypotheses on the radial potentials $A,V,K$. We first get compact embeddings of radial weighted Sobolev spaces into sum of weighted Lebesgue spaces, and then we apply standard variational techniques to get existence results.

math.AP↗

arXiv2016

Compactness and existence results for the $p$-Laplace equation

Given $1 0$, $r>0$, we define the weighted spaces \[ W=\left\{ u\in D^{1,p}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V\left( \left| x\right| \right) \left| u\right| ^{p}dx<\infty \right\} ,\quad L_{K}^{q}=L^{q}(\mathbb{R}^{N},K\left( \left| x\right| \right) dx) \] and study the compact embeddings of the radial subspace of $W$ into $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. We consider exponents $q_{1},q_{2},q$ that can be greater or smaller than $p$. Our results do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately. We then apply these results to the investigation of existence and multiplicity of finite energy solutions to nonlinear $p$-Laplace equations of the form \[ -\triangle _{p}u+V\left( \left| x\right| \right) |u|^{p-1}u=g\left( \left| x\right| ,u\right) \quad \text{in }\mathbb{R}^{N},\ 1<p<N, \] where $V$ and $g\left( \left| \cdot \right| ,u\right) $ with $u$ fixed may be vanishing or unbounded at zero or at infinity. Both the cases of $g$ super and sub $p$-linear in $u$ are studied and, in the sub $p$-linear case, nonlinearities with $g\left( \left| \cdot \right| ,0\right) \neq 0$ are also considered.

math.AP↗

arXiv2015

Compactness results for the $p$-Laplace equation

Given $1 0$, $r>0$, we define the weighted spaces \[ W=\left\{ u\in D^{1,p}(\mathbb{R}^N):\int_{\mathbb{R}^N}V\left(\left|x\right|\right) \left|u\right|^p dx<\infty \right\} , \quad L_{K}^q =L^q(\mathbb{R}^N,K\left( \left| x\right| \right) dx) \] and study the compact embeddings of the radial subspace of $W$ into $L_{K}^{q_1}+L_{K}^{q_2}$, and thus into $L_{K}^q$ ($=L_{K}^q+L_{K}^q$) as a particular case. Both exponents $q_1,q_2,q$ greater and lower than $p$ are considered. Our results do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately.

math.AP↗

arXiv2015

Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence

We prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation \[ -\triangle u+V\left( \left| x\right| \right) u=g\left( \left| x\right| ,u\right) \quad \textrm{in }Ω\subseteq \mathbb{R}^{N},\ N\geq 3, \] where $Ω$ is a radial domain (bounded or unbounded) and $u$ satisfies $u=0$ on $\partial Ω$ if $Ω\neq \mathbb{R}^{N}$ and $u\rightarrow 0$ as $\left| x\right| \rightarrow \infty $ if $Ω$ is unbounded. The potential $V$ may be vanishing or unbounded at zero or at infinity and the nonlinearity $g$ may be superlinear or sublinear. If $g$ is sublinear, the case with $g\left( \left| \cdot \right| ,0\right) \neq 0$ is also considered.

math.AP↗

arXiv2014

Non radial solutions for non homogeneous Hénon equation

In this paper we study a Hénon-like equation (see equations (1) below), where the nonlinearity f(t) is not homogeneous (i.e., it is not a power). By minimization on the Nehari manifold, we prove that for large values of the parameter $α$ there is a breaking of symmetry and non radial solutions appears. This holds for sub- and super-critical growth of the nonlinearity f.

math.AP↗

arXiv2014

Compactness and existence results in weighted Sobolev spaces of radial functions, Part I: Compactness

Given two measurable functions $V(r)\geq 0$ and $K(r)> 0$, $r>0$, we define the weighted spaces \[ H_V^1 = \{u \in D^{1,2}(\mathbb{R}^N): \int_{\mathbb{R}^N}V(|x|)u^{2}dx < \infty \}, \quad L_K^q = L^q(\mathbb{R}^N,K(|x|)dx) \] and study the compact embeddings of the radial subspace of $H_V^1$ into $L_K^{q_1}+L_K^{q_2}$, and thus into $L_K^q$ ($=L_K^q+L_K^q$) as a particular case. Both super- and sub-quadratic exponents $q_1$, $q_2$ and $q$ are considered. Our results do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately. Applications to existence results for nonlinear elliptic problems like \[ -\triangle u + V(|x|)u = f(|x|,u) \quad \text{in}\mathbb{R}^N, \quad u \in H_V^1, \] will be given in a forthcoming paper.

math.FA↗

arXiv2013

A nonexistence result for a nonlinear elliptic equation with singular and decaying potential

The paper deals with positive radial solutions to a nonlinear elliptic equation with singular and decaying potential, for which several existence and nonexistence results are known, resting upon suitable compatibility conditions between the decaying rate of the potential and the growth rate of the nonlinearity. The problem of the existence is still open for essentially three cases and we give a negative answer to one of such cases.

math.AP↗

arXiv2002

Bifurcation results for semilinear elliptic problems in R^N

In this paper we obtain, for a semilinear elliptic problem in R^N, families of solutions bifurcating from the bottom of the spectrum of $-Δ$. The problem is variational in nature and we apply a nonlinear reduction method which allows us to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.

math.AP↗