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Abstract

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 $({\rm P}_{\lambda})$.