### A bifurcation theory for some nonlinear elliptic equations

We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ (${P}_{\lambda}$) ⎩ ${u}_{\mid \partial \Omega}=0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem (${P}_{\lambda}$) admits a non-zero, non-negative strong solution ${u}_{\lambda}\in {\bigcap}_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0\u207a}\left|\right|{u}_{\lambda}{\left|\right|}_{{W}^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda \mapsto {I}_{\lambda}\left({u}_{\lambda}\right)$ is negative and decreasing in ]0,λ*[, where ${I}_{\lambda}$ is the energy functional related to (${P}_{\lambda}$).