arXiv subjects

Donato Passaseo

Publications and source records attributed to Donato Passaseo.

6 recordsLinked to original sources

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↗

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

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↗

arXiv2013

Multiple Solutions for Scalar Field Equations with Potentials having "Subsidences"

In this paper the question of finding infinitely many solutions to the problem $-Δu+a(x)u=|u|^{p-2}u$, in $\mathbb{R}^N$, $u \in H^1(\mathbb{R}^N)$, is considered when $N\geq 2$, $p \in (2, 2N/(N-2))$, and the potential $a(x)$ is a positive function which is not required to enjoy symmetry properties. Assuming that $a(x)$ satisfies a suitable "slow decay at infinity" condition and, moreover, that its graph has some "dips", we prove that the problem admits either infinitely many nodal solutions either infinitely many constant sign solutions. The proof method is purely variational and allows to describe the shape of the solutions.

math.AP↗