arXiv subjects

Simone Mauro

Publications and source records attributed to Simone Mauro.

6 recordsLinked to original sources

arXiv2026

Multiplicity and Regularity Results for Quasilinear Elliptic Systems via Nonsmooth Critical Point Theory

We study the quasilinear elliptic system \[ -\textbf{div}(A(x,\boldsymbol u)|D\boldsymbol u|^{p-2}D\boldsymbol u) +\frac{1}{p}\nabla_{\boldsymbol s}A(x,\boldsymbol u)|D\boldsymbol u|^p = \boldsymbol g(x,\boldsymbol u) \quad \text{in } \Omega, \qquad \boldsymbol u = 0 \text{ on } \partial\Omega, \] where $p>1$, $\Omega\subset\mathbb R^N$ is a bounded domain with $N>1$, and $\boldsymbol g$ satisfies a subcritical growth condition. In this setting, the associated energy functional is, in general, neither differentiable nor locally Lipschitz in the natural Sobolev space. By exploiting a nonsmooth critical point theory, we prove the existence of infinitely many weak solutions by means of an Equivariant Mountain Pass Theorem. In addition, we establish $L^\infty$-bounds for weak solutions by adapting a Moser-type iteration.

math.AP↗

arXiv2025

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: \[ -\mathrm{div}(A_k(x, u_k) |\nabla u_k|^{p_k - 2} \nabla u_k) + \dfrac{1}{p_k} D_s A_k(x, u_k) |\nabla u_k|^{p_k} = g_k(x, u) \quad \text{in } \Omega,\quad u_k = 0 \quad \text{on } \partial\Omega, \] where $k=1,\dots,d$, $ \Omega \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}}(\Omega) \cap L^\infty(\Omega; \mathbb{R}^d) $, where $W_0^{1,\boldsymbol p}(\Omega)=W_0^{1,p_1}(\Omega)\times\dots\times W_0^{1,p_d}(\Omega)$.

math.AP↗

arXiv2025

Quasilinear Elliptic Cooperative and Competitive Systems

We study the existence and multiplicity of weak solutions for the following quasilinear elliptic system: \[ \begin{cases} -\mathrm{div}(A_1(x,u_1)\nabla u_1) + \displaystyle\frac{1}{2} D_{u_1}A_1(x,u_1)\nabla u_1 \cdot \nabla u_1 = \lambda_1 u_1 + g_{\beta,1}(u) & \text{in } \Omega, \\[3mm] -\mathrm{div}(A_2(x,u_2)\nabla u_2) + \displaystyle\frac{1}{2} D_{u_2}A_2(x,u_2)\nabla u_2 \cdot \nabla u_2 = \lambda_2 u_2 + g_{\beta,2}(u) & \text{in } \Omega, \\[2mm] u_1 = u_2 = 0 & \text{on } \partial\Omega, \end{cases} \] where $\lambda_1, \lambda_2 < \mu_1$, $\mu_1$ is the first Dirichlet eigenvalue of the Laplacian, and $\Omega$ is a bounded domain. The nonlinearity is derived from a potential $G_\beta$ with subcritical growth. We prove the existence of least energy solutions in both the cooperative ($\beta > 0$) and competitive ($\beta < 0$) 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↗

arXiv2025

Quasilinear Equations with Neumann Boundary Conditions

We prove a multiplicity result for non-constant weak solutions $u \in H^1(\Omega)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) - \lambda u & \text{in } \Omega \\ A(x,u)\nabla u \cdot \eta = 0 & \text{on } \partial \Omega \end{cases} \] where $\lambda \in \mathbb{R}$, $ \Omega$ is a bounded lipschitz domain, $ \eta $ is the outward normal to the boundary $ \partial \Omega $, and $g(x,u)$ is a Carath\'eodory 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↗

arXiv2025

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

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

math.AP↗

arXiv2025

Least Energy Solutions for Cooperative and Competitive Schr\"odinger Systems with Neumann Boundary Conditions

We study the following gradient elliptic system with Neumann boundary conditions \begin{equation*} -\Delta u + \lambda_1 u = u^3 + \beta uv^2, \ -\Delta v + \lambda_2 v = v^3 + \beta u^2 v \ \text{in } \Omega,\qquad \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 \ \text{on } \partial \Omega, \end{equation*} where $\Omega \subset \mathbb{R}^N $ is a bounded $ C^2$ domain with $ N \leq 4 $, and $ \nu $ denotes the outward unit normal on the boundary. We investigate the existence of non-constant least energy solutions in both the cooperative ($\beta > 0 $) and the competitive ($ \beta < 0 $) regimes, considering both the definite and the indefinite case, namely $\lambda_1,\lambda_2\in\mathbb R$. We emphasize that our analysis includes both the subcritical case $ N \leq 3 $ and the critical case $ N = 4 $. Depending on the values of $\beta,\lambda_1,\lambda_2$, the least energy solution is obtained either via a linking theorem, by minimizing over a suitable Nehari manifold, or by direct minimization on the set of all non-trivial weak solutions. Our results and techniques can be also adapted to cover some previously untreated cases for Dirichlet conditions.

math.AP↗