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

Détail

DOI : 10.4064/cm95-1-12

Keywords: 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