arXiv subjects

Paolo Malanchini

Publications and source records attributed to Paolo Malanchini.

9 recordsLinked to original sources

arXiv2026

Optimal data-driven solutions for a stationary diffusive model of population growth

We study optimal data-driven solutions for the stationary diffusive population growth model $-Δu = r u$ in a bounded domain $Ω\subset\mathbb R^N$ with Neumann boundary conditions. Instead of prescribing a functional relation between the position $x$, the net per-capita growth rate $r$ and the population size $u$, we look for a pair $(u,r)\in H^1(Ω)\times L^\infty(Ω)$ that fits a given data set in an optimal way measured by a cost functional $I$ and an additional penalty term. We characterize the relaxed cost functional sc$^- I$ by showing that its density is given as the partial lower convex envelope with respect to the variable $r$, and prove the existence of optimal data-driven solutions. Furthermore, we establish a consistency result comparing conventional solutions of $-Δu = \varrho(x,u)u$ with optimal data-driven solutions where the data set stems from the functional relation $(x,u)\mapsto \varrho(x,u)$. Finally, as data sets evolve, we prove the convergence of optimal solutions via the $Γ$-convergence of the associated cost functionals.

math.AP↗

arXiv2026

Regularizing effect of the natural growth term in quasilinear problems with sign-changing nonlinearities

We investigate the existence and nonexistence of solutions to the Dirichlet problem \begin{equation*} \tag{$P$} \label{pba} \left\{ \begin{alignedat}{2} -Δ_p u + g(u) |\nabla u|^p &= λf(u) \quad &&\mbox{in} \;\; Ω, \\ u &= 0 \quad &&\mbox{on} \;\; \partialΩ, \end{alignedat} \right. \end{equation*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $p\in (1,\infty)$, $λ>0$ and $g\in C(\mathbb{R})$. Our main assumption is that $:f \mathbb{R}\to \mathbb{R}$ is a continuous function such that $f(s)>0$ for all $s\in (α,β)$, where $0<α<β$ are two zeros of $f$. If $f(0)\geq 0$, we show that an area condition involving $f$ and $g$ is both sufficient and necessary in order to have a pair $(λ,u)\in \mathbb{R}^+\times C_0^1(\overlineΩ)$, with $u\geq 0$ and $\|u\|_{C(\overlineΩ)}\in (α,β]$, solving~\eqref{pba}. We also study how the presence of the gradient term affects the existence of solution. Roughly speaking, the more negative $g$ is, the stronger its regularizing effect on~\eqref{pba}. We prove that, regardless of the shape of $f$, for any fixed $λ$, there always exists a function $g$ such that~\eqref{pba} admits a nonnegative solution with maximum in $(α,β]$.

math.AP↗

arXiv2026

Existence and regularity for an entire Grushin-Choquard equation

We consider the following Choquard equation $$ -Δ_γu + u = \left(d(z)^{-μ} \ast |u|^p\right)|u|^{p-2}u, \text{ in } \mathbb{R}^N, $$ where $Δ_γ$ is the Grushin operator. For a suitable range of the parameter $p$ we prove the existence of a mountain pass solution of the equation and we establish that the solution belongs to $L^q(\mathbb{R}^N)$ for all $q\in [2,\infty]$ and to $C^{0,α}_{\textrm{loc}}(\mathbb{R}^N)$ for some $α\in (0,1)$. Additionally, we provide a Poho\v zaev type identity, which allows us to derive a nonexistence result for smooth solutions to our equation.

math.AP↗

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 $Δ_α^p$. After a truncation argument, we obtain infinitely many solutions to our problem via Krasnoselskii's genus, extending a previous result of García 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.

math.AP↗

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 $$ \left\lbrace \begin{array}{ll} -\mathcal{L}u + a(x) g(u) = f(x) \quad &\mbox{in} \;\; Ω, u = 0 \quad &\mbox{on} \;\; \partialΩ, \end{array} \right. $$ where $Ω\subseteq\mathbb{R}^N$ is a bounded domain and $\mathcal{L}$ is an $X$-elliptic operator introduced by Lanconelli and Kogoj. If $f \in L^1(Ω)$, 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 $a$.

math.AP↗

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.

math.AP↗

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 $Δ_γ^p$. We extend to a generic $p>1$ a result which was proved only when $p=2$. When $p\neq 2$, the nonlinear operator $-Δ_γ^p$ 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.

math.AP↗

arXiv2024

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

For $N\ge 3$ we study the following semipositone problem $$ -Δ_γu = g(z) f_a(u) \quad \hbox{in $\mathbb{R}^N$}, $$ where $Δ_γ$ is the Grushin operator $$ Δ_ γu(z) = Δ_x u(z) + \vert x \vert^{2γ} Δ_y u (z) \quad (γ\ge 0), $$ $g\in L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ 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 $D_γ(\mathbb{R}^N)$

math.AP↗

arXiv2024

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

We consider the boundary value problem $$ \cases{ -Δ_γu = λu + \left\vert u \right\vert^{2^*_γ-2}u &in $Ω$\cr u = 0 &on $\partialΩ$,\cr } $$ where $Ω$ is an open bounded domain in $\mathbb{R}^N$, $N \geq 3$, while $Δ_γ$ is the Grushin operator $$ Δ_ γu(z) = Δ_x u(z) + \vert x \vert^{2γ} Δ_y u (z) \quad (γ\ge 0). $$ 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.

math.AP↗