arXiv subjects

Annamaria Canino

Publications and source records attributed to Annamaria Canino.

8 recordsLinked to original sources

arXiv2026

Quasilinear Elliptic Cooperative and Competitive Systems

We study the existence and multiplicity of weak solutions for the following quasilinear elliptic system: \[ \begin{cases} -\mathrm{div}(A_1(x,u_1)\nabla u_1) + \displaystyle\frac{1}{2} D_{u_1}A_1(x,u_1)\nabla u_1 \cdot \nabla u_1 = λ_1 u_1 + g_{β,1}(u) & \text{in } Ω, \\[3mm] -\mathrm{div}(A_2(x,u_2)\nabla u_2) + \displaystyle\frac{1}{2} D_{u_2}A_2(x,u_2)\nabla u_2 \cdot \nabla u_2 = λ_2 u_2 + g_{β,2}(u) & \text{in } Ω, \\[2mm] u_1 = u_2 = 0 & \text{on } \partialΩ, \end{cases} \] where $λ_1, λ_2 < μ_1$, $μ_1$ is the first Dirichlet eigenvalue of the Laplacian, and $Ω$ is a bounded domain. The nonlinearity is derived from a potential $G_β$ with subcritical growth. We prove the existence of least energy solutions in both the cooperative ($β> 0$) and competitive ($β< 0$) regimes. Due to the lack of differentiability of the associated energy functional, we employ nonsmooth critical point theory and variational methods based on the concept of weak slope.

math.AP↗

arXiv2026

Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian

We prove existence and regularity results for the following elliptic system: \[ \begin{cases} -\textbf{div}(|D\boldsymbol{u}|^{p-2}D\boldsymbol{u})=\boldsymbol{f}(x,\boldsymbol{u}) & \text{in } Ω\\ \boldsymbol{u}=0 & \text{on } \partialΩ, \end{cases} \] where $\boldsymbol{u}=(u^1,\dots,u^m)$, $p>1$, and $Ω\subset\mathbb{R}^N$ is a bounded domain. We also consider the special case \[\boldsymbol{f}(x,\boldsymbol{u})=λ|\boldsymbol{u}|^{p-2}\boldsymbol{u}+|\boldsymbol{u}|^{q-2}\boldsymbol{u},\] and we prove a classification result. In particular, we show that any least energy solution is of the form $(c^1ω,\dots,c^mω)$, where $\boldsymbol{c}=(c^1,\dots,c^m)\in S^{m-1}$ (the $(m-1)$-sphere in $\mathbb R^m$) and $ω$ is a positive solution of the corresponding scalar equation.

math.AP↗

arXiv2026

On the multiplicity of weak solutions for a class of coupled quasilinear elliptic systems

We study the existence and regularity of weak solutions to the following quasilinear elliptic system: \[ -\mathrm{div}(A_k(x, u_k) |\nabla u_k|^{p_k - 2} \nabla u_k) + \dfrac{1}{p_k} D_s A_k(x, u_k) |\nabla u_k|^{p_k} = g_k(x, u) \quad \text{in } Ω,\quad u_k = 0 \quad \text{on } \partialΩ, \] where $k=1,\dots,d$, $ Ω\subset \mathbb{R}^N $ is a bounded domain with $ N \geq 2 $, $ \boldsymbol{p} = (p_1, \dots, p_d) $, $ p_k > 1 $. Using tools from nonsmooth critical point theory, we prove the existence of infinitely many weak solutions in $ W_0^{1,\boldsymbol{p}}(Ω) \cap L^\infty(Ω; \mathbb{R}^d) $, where $W_0^{1,\boldsymbol p}(Ω)=W_0^{1,p_1}(Ω)\times\dots\times W_0^{1,p_d}(Ω)$.

math.AP↗

arXiv2025

Quasilinear Equations with Neumann Boundary Conditions

We prove a multiplicity result for non-constant weak solutions $u \in H^1(Ω)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) - λu & \text{in } Ω\\ A(x,u)\nabla u \cdot η= 0 & \text{on } \partial Ω\end{cases} \] where $λ\in \mathbb{R}$, $ Ω$ is a bounded lipschitz domain, $ η$ is the outward normal to the boundary $ \partial Ω$, and $g(x,u)$ is a Carathéodory function that satisfies a general subcritical (and superlinear) growth condition. We also prove that any weak solution is bounded under a stronger growth assumption.

math.AP↗

arXiv2018

A variational approach to nonlocal singular problems

We provide a suitable variational approach for a class of nonlocal problems involving the fractional laplacian and singular nonlinearities for which the standard techniques fail. As a corollary we deduce a characterization of the solutions.

math.AP↗

arXiv2016

The moving plane method for singular semilinear elliptic problems

We consider positive solutions to semilinear elliptic problems with singular nonlinearities, under zero Dirichlet boundary condition. We exploit a refined version of the moving plane method to prove symmetry and monotonicity properties of the solutions, under general assumptions on the nonlinearity.

math.AP↗

arXiv2016

Nonlocal problems with singular nonlinearity

We investigate existence and uniqueness of solutions for a class of nonlinear nonlocal problems involving the fractional $p$-Laplacian operator and singular nonlinearities.

math.AP↗

arXiv2014

A uniqueness result for some singular semilinear elliptic equations

Given $Ω$ a bounded open subset of $\mathbb{R}^N$, we consider nonnegative solutions to the singular semilinear elliptic equation $-Δ\,u\,=\,\frac{f}{u^β}$ in $H^1_{loc}(Ω)$, under zero Dirichlet boundary conditions. For $β>0$ and $f\in L^1(Ω)$, we prove that the solution is unique.

math.AP↗