Sesión Ecuaciones Diferenciales y aplicacionesOn the existence of solutions for a quasilinear nonlocal equation
Lisbeth Carrero
Universidad de O'Higgins, Chile - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
We introduce a new class of quasilinear operators, which represents a nonlocal version of Stuart's operator [1], inspired by models in nonlinear optics.
We aim to extend the results of Jeanjean and Radulescu [2] to a nonlocal framework. As a first step, we consider a simplified version of the problem, where the operator depends only on $\nabla^{s} u$. Our primary objective is to investigate the existence of solutions, in dimensions $d \geq 2$, to the following quasilinear nonlocal partial differential equation. \[ \left\{ \begin{array}{rcll} -\text{div}^{s} \left( \gamma \left(\frac{|\nabla^{s} u|^2}{2}\right)\nabla^{s} u \right)&=&f(u)+h & \text{ in }\Omega \\ u &=&0 & \text{ in } \mathbb{R}^d\setminus\Omega \end{array} \right. \] where \(\gamma: [0, \infty) \to\mathbb{R} \) is a nonlinear heterogeneity in the fractional diffusivity with \(s \in (0,1) \), \(h \in L^2(\Omega)\) is a non-negative right side.
We study the existence of at least one or two solutions in the cone \( X := \{ u \in H_{0}^{s} (\Omega) : u \geq 0 \} \) by employing variational methods. To this end, we consider two distinct cases: asymptotically sublinear and asymptotically linear growth. Additionally, in the sublinear case, we establish a nonexistence result.
Trabajo en conjunto con: Alexander Quaas (Universidad Técnica Federico Santa María, Chile) y Andrés Zuñiga (Universidad de O'Higgins, Chile).
Referencias
[1] C. Stuart, Two positive solutions of a quasilinear elliptic dirichlet problem, Milan Journal of Mathematics 79 (2011), 327–341.
[2] L. Jeanjean and V. Radulescu, Nonhomogeneous quasilinear elliptic problems: linear and sublinear cases, Journal d’Analyse Mathématique (2021).