Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.32629

Anisotropic quasilinear elliptic systems involving nonlinearities with negative critical terms

Elisandra Gloss·Artur Jorge Marinho·Kanishka Perera
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we investigate existence and multiplicity of solutions for a critical growth anisotropic quasilinear elliptic system, depending on two real parameters $λ,\,μ$, that is coupled through a subcritical perturbation term and involves a negative critical part. We identify a specific scaling for the system and introduce a parameter $γ$ associated with this scaling, which governs the geometry of the corresponding variational functional. We establish three distinct types of multiplicity results, depending on the regimes $γ= 1$, $γ> 1$, and $γ< 1$. The multiple results of multiplicity of solutions are based on a recent abstract critical point theorem for symmetric functionals on product spaces when $γ\geq1$ and on a truncation argument for $γ<1$.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Elisandra Gloss, Artur Jorge Marinho, Kanishka Perera. 2026-09-26. Anisotropic quasilinear elliptic systems involving nonlinearities with negative critical terms. https://arxiv.org/abs/2609.32629

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Optimizers in Sobolev-curl inequalities

We study a Sobolev-type inequality involving the $p$-curl operator in $\mathbb{R}^3$. We prove the existence of a minimizer $u:\mathbb{R}^3\to \mathbb{R}^3$ which yields a solution to the $p$-curl-curl equation in the critical case $$ \nabla\times (|\nabla\times u|^{p-2}\nabla\times u)= |u|^{p^*-2}u\quad \hbox{in }\mathbb{R}^3.$$ The cases $p=2$ and $p=3/2$ are motivated, respectively, by nonlinear Maxwell equations and zero modes of three-dimensional Dirac operators. The infinite-dimensional kernel of curl and the critical exponent prevent a direct application of standard compactness arguments. Our proof combines local compactness under the nonlinear constraint $\operatorname{div}(|u|^{p^*-2}u)=0$ and a new variational approach that allows to treat quasilinear strongly indefinite problems by direct minimization on a Nehari-type constraint. We also establish existence and compactness results in axially symmetric classes, including two types of solutions for $p=3/2$ distinct from the explicit Loss--Yau fields. Finally, the same minimization strategy gives a new proof of compactness modulo translations and dilations for minimizing sequences in the classical critical Sobolev inequality.

math.AP↗
arXiv2026

A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation

We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by $$ dρ\propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x - \frac 12\int_{\mathbb T} |\nabla u|^2 d x - \frac 12\int_{\mathbb T} |u|^2 d x\big) \mathbf 1_{\| u \|_{L^2(\mathbb T)}^2 \le K}dud\overline{u}. $$ When $2 \le p \le 4$, we show that these measures indeed satisfy a log-Sobolev inequality. When $p> 4$, we show a lower bound for the Hessian of the potential, which implies that the known techniques to show these inequalities cannot apply to the measure $ρ$.

math.AP↗
arXiv2026

On the inhomogeneous discounted Hamilton-Jacobi equations

In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations \begin{equation}\label{hjs1} λ(x)u+h(x,d_x u)=c \quad \tag{$\ast$} \end{equation} on a closed manifold $M$ with a non-identically vanishing discount factor $λ(x)$. There is a critical value $c_0\in[-\infty,\infty)$ such that \eqref{hjs1} admits a viscosity solution if $c>c_0$ and no solution if $c c_0$. In this case, we determine the basin of the stable solution and investigate the long-time behavior of the solution semigroup associated with \eqref{hjs1}. In particular, we obtain a formula relating the lowest convergence rate of the solution semigroup near a stable solution to a minimizing problem concerning the integral of $λ$ over Mather measures supporting on the $1$-graph of the stable solution. This formula has two direct consequences: 1) it yields a description of their asymptotic behavior as $c$ tends to infinity; 2) it helps to classify the ergodic Mather measures and locate their distribution in the phase space. The second consequence leads to a dynamical necessary condition for $c=c_0$.

math.AP↗