arXiv subjects

David Arcoya

Publications and source records attributed to David Arcoya.

5 recordsLinked to original sources

arXiv2026

On the de Th\'elin eigenvalue problem and Landesman-Lazer conditions for quasilinear systems

In this paper we prove that the smallest eigenvalue $\lambda_1$ of the eigenvalue problem for a quasilinear elliptic systems introduced by de Th\'elin in \cite{DT}, is not only simple (in a suitable sense), but also isolated. Moreover, we characterize variationally a sequence $\{\lambda_k\}_k$ of eigenvalues, taking into account a suitable deformation lemma for $C^1$ submanifolds proved in \cite{BON}. Furthermore we prove the existence of a weak solution for a quasilinear elliptic systems in resonance around $\lambda_1$, under new sufficient Landesman-Lazer type conditions, extending the results by Arcoya and Orsina \cite{AO}.

math.AP↗

arXiv2025

Existence and nonexistence of solutions for singular quadratic quasilinear equations

We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -\Delta u + \frac{|\nabla u|^2}{u^{\gamma}} = f & \mbox{in } \Omega,\newline \hfill u=0 \hfill & \mbox{on } \partial \Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N $, $\gamma> 0$ and $f$ is a function which is strictly positive on every compactly contained subset of $\Omega$. As a consequence of our main results, we prove that the condition $\gamma<2$ is necessary and sufficient for the existence of solutions in $H^{1}_{0}(\Omega)$ for every sufficiently regular $f$ as above.

math.AP↗

arXiv2025

Nonlocal operators in divergence form and existence theory for integrable data

We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to $L^1(\Omega)$ and to be suitably dominated. We also prove that the solution that we find converges, as $s\nearrow 1$, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in $L^1(\Omega)$ and therefore the usual regularity theory cannot be leveraged to our benefit in this framework. The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as $s\nearrow 1$, every classical operator in divergence form.

math.AP↗

arXiv2014

Continuum of solutions for an elliptic problem with critical growth in the gradient

We consider the boundary value problem \begin{equation*} - Δu = λc(x)u+ μ(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(Ω) \cap L^{\infty}(Ω) \eqno{(P_λ)} \end{equation*} where $Ω\subset \R^N, N \geq 3$ is a bounded domain with smooth boundary. It is assumed that $c\gneqq 0$, $c,h$ belong to $L^p(Ω)$ for some $p > N/2$ and that $μ\in L^{\infty}(Ω).$ We explicit a condition which guarantees the existence of a unique solution of $(P_λ)$ when $λ<0$ and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of $(P_0)$. It crosses the axis $λ=0$ if $(P_0)$ has a solution, otherwise if bifurcates from infinity at the left of the axis $λ=0$. Assuming that $(P_0)$ has a solution and strenghtening our assumptions to $μ(x)\geq μ_1>0$ and $h\gneqq 0$, we show that the continuum bifurcates from infinity on the right of the axis $λ=0$ and this implies, in particular, the existence of two solutions for any $λ>0$ sufficiently small.

math.AP↗

arXiv2013

Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions

In this note we present some uniqueness and comparison results for a class of problem of the form \begin{equation} \label{EE0} \begin{array}{c} - L u = H(x,u,\nabla u)+ h(x), \quad u \in H^1_0(Ω) \cap L^{\infty}(Ω), \end{array} \end{equation} where $Ω\subset \R^N$, $N \geq 2$ is a bounded domain, $L$ is a general elliptic second order linear operator with bounded coefficients and $H$ is allowed to have a critical growth in the gradient. In some cases our assumptions prove to be sharp.

math.AP↗