Detalles de Evento

Título: Bifurcation in a Quasilinear Schrödinger-Type Equation with Two
Parameters: Theory and Applications to Multiplicity of Solutions.

Conferenciante: Miguel Martínez Teruel (Universidad de Granada) (mmteruel@ugr.es).

Resumen:

In this talk we deal with the following family of equations,

\begin{equation}
\begin{cases}
-\Delta u-\lambda m(x) u \Delta(u^2)=f(\mu, x, u)\quad \text{in}\quad \Omega,\\
u=0\qquad \text{on}\qquad \Omega.
\end{cases}
\tag{\(P(\lambda,\, \mu)\)}
\end{equation}

where \(\lambda,\, \mu\in \mathbb{R}\), \(\Omega\) is an open and bounded subset of \(\mathbb{R}^N\) with smooth boundary, \(m(x)\) is a continuous function with \(0\leq m(x)\leq M\) and \(f\, \colon\, \mathbb{R}\times\Omega\times\mathbb{R}^+ \to \mathbb{R}\) is a \(C^1\) function such that \(f(\mu,\, x,\, 0)=0\) for all \(x\in\Omega\) and \(\mu\in\mathbb{R}\) and satisfies:

For every \(\Gamma\) bounded subset of \(\mathbb{R} - \{0\}\) and \(\mu \in \Gamma\),

\[
\lim_{s \to 0^+} \frac{f(\mu, x, s)}{s} = \mu f'_+(x,0), \quad \text{uniformly in } (\mu, x) \in \Gamma \times \Omega,
\]

with \(0 \leq f'_+(x,0) \in L^\infty(\Omega)\) not identically zero.

With these conditions we prove the existence of a continuum of solutions, we study the laterality of the continuum and we give explicit examples in which the number of solutions changes depending on the parameter \(\lambda\).

Descargar resumen aquí.

References:

  1. [1] Ambrosetti, A., Arcoya, D., An introduction to nonlinear functional analysis and elliptic problems, in: Progress in Nonlinear Differential Equations and their Applications, Birkäuser, 2011.
  2. [2] Arcoya, D., Carmona, J., Pellacci, B., Bifurcation for some quasi-linear operators, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 733–765.
  3. [3] Cintra, W., Medeiros, E., Severo, U. On positive solutions for a class of quasilinear elliptic equations. Z. Angew. Math. Phys. 70, 79 (2019).
  4. [4] Figueiredo, G.M., Santos J´unior, J.R., Su´arez, A. Structure of the set of positive solutions of a non-linear Schr¨odinger equation. Isr. J. Math. 227, 485–505 (2018).

Toda la información puede encontrarse en:
Seminario de Ecuaciones Diferenciales.