Multiple solutions to a perturbed Neumann problem

Giuseppe Cordaro¹

¹ Department of Mathematics University of Messina 98166 Sant'Agata-Messina, Italy

Studia Mathematica, Tome 178 (2007) no. 2, pp. 167-175

Résumé

We consider the perturbed Neumann problem $$ \cases{ -{\mit\Delta} u + \alpha(x)u=\alpha(x)f(u)+\lambda g(x,u) \hbox{a.e. in ${\mit\Omega}$},\cr {\partial u}/{\partial \nu}=0\hbox{on $\partial {\mit\Omega}$,}} $$ where ${\mit\Omega}$ is an open bounded set in $\mathbb{R}^N$ with boundary of class $C^2$, $\alpha\in L^\infty({\mit\Omega})$ with $\mathop{\rm ess\,inf}_{\mit\Omega} \alpha >0$, $f:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and $g:{\mit\Omega}\times \mathbb{R}\rightarrow \mathbb{R}$, besides being a Carathéodory function, is such that, for some $p>N$, $\sup_{|s|\leq t}|g(\cdot,s)|\in L^p({\mit\Omega})$ and $g(\cdot,t)\in L^\infty({\mit\Omega})$ for all $t\in \mathbb{R}$. In this setting, supposing only that the set of global minima of the function $\frac{1}{2}\xi^2-\int_0^\xi f(t)\,dt$ has $M\geq 2$ bounded connected components, we prove that, for all $\lambda\in \mathbb{R}$ small enough, the above Neumann problem has at least $M+{}$integer part of ${M}/{2}$ distinct strong solutions in $W^{2,p}({\mit\Omega})$.

Détail

DOI : 10.4064/sm178-2-3

Keywords: consider perturbed neumann problem cases mit delta alpha alpha lambda hbox mit omega partial partial hbox partial mit omega where mit omega bounded set mathbb boundary class alpha infty mit omega mathop ess inf mit omega alpha mathbb rightarrow mathbb continuous function mit omega times mathbb rightarrow mathbb besides being carath odory function sup leq cdot mit omega cdot infty mit omega mathbb setting supposing only set global minima function frac int has geq bounded connected components prove lambda mathbb small enough above neumann problem has least integer part distinct strong solutions mit omega