A bifurcation theory for some nonlinear elliptic equations

Biagio Ricceri¹

¹ Department of Mathematics University of Catania Viale A. Doria 6 95125 Catania, Italy

Colloquium Mathematicum, Tome 95 (2003) no. 1, pp. 139-151.

Résumé

We deal with the problem $$\cases {-{\mit\Delta} u= f(x,u)+\lambda g(x,u) in $\Omega ,$\cr u_{|\partial {\mit\Omega}}=0,\cr} \tag*{$({\mathrm{P}}_{\lambda })$} $$ where $\Omega \subset {\mathbb{R}}^n$ is a bounded domain, $\lambda \in \mathbb{R}$, and $f,g:\Omega \times \mathbb{R}\to \mathbb{R}$ are two Carathéodory functions with $f(x,0)=g(x,0)=0$. Under suitable assumptions, we prove that there exists ${\lambda }^{\ast }>0$ such that, for each $\lambda \in (0,{\lambda }^{\ast })$, problem $({\mathrm{P}}_{\lambda })$ admits a non-zero, non-negative strong solution $u_{\lambda }\in \bigcap _{p\ge 2}{W}^{2,p}(\Omega )$ such that $\underset{\lambda \to 0^{+}}{lim}\Vert u_{\lambda }{\Vert }_{W^{2,p}(\Omega )}=0$ for all $p\ge 2$. Moreover, the function $\lambda \mapsto I_{\lambda }(u_{\lambda })$ is negative and decreasing in $]0,{\lambda }^{\ast }[$, where $I_{\lambda }$ is the energy functional related to $({\mathrm{P}}_{\lambda })$.

Détail

DOI : 10.4064/cm95-1-12

Mots-clés : problem cases mit delta lambda mit omega partial mit omega tag* lambda where mit omega subset mathbb bounded domain lambda mathbb mit omega times mathbb mathbb carath odory functions under suitable assumptions prove there exists lambda * each lambda lambda * problem lambda admits non zero non negative strong solution lambda bigcap geq mit omega lim lambda lambda mit omega geq moreover function lambda mapsto lambda lambda negative decreasing lambda * where lambda energy functional related lambda