## Existence results for some functional elliptic equations.(English)Zbl 1324.35195

Let $$\Omega$$ be a bounded open subset of $$\mathbb {R}^{d}$$ and let $$A: \Omega \times L^{p}(\Omega)\rightarrow \mathbb {R}$$, $$p\geq 1$$, such that $$x\in \Omega \mapsto A(x,u)$$, be measurable for all $$u\in L^{p}(\Omega)$$ and $$u\in L^{p}(\Omega)\mapsto A(x,u)$$ be continuous for a.e. $$x\in \Omega$$. Moreover, let $$a_{0}\leq A(x,u)\leq a_{\infty}$$ for a.e. $$x\in \Omega$$ and for every $$u\in L^{p}(\Omega)$$, where $$a_{0},\, a_{\infty}>0$$. Finally, let $$f: [0,+\infty [\,\rightarrow [0,+\infty [$$ be a Lipschitz continuous function such that $$f(\theta_{1})=f(\theta_{2})=0$$ and $$f(\theta)>0$$ for $$\theta \in \,]\theta_{1},\theta_{2}[\,$$, where $$0\leq \theta_{1}<\theta_{2}$$.
Theorem 1. For $$\lambda >0$$ sufficiently large there exists a weak solution $$u_{\lambda}\in H_{0}^{1}(\Omega)$$ of $$-A(\cdot ,u_{\lambda})\Delta u_{\lambda}=\lambda f\circ u_{\lambda}$$ such that $$0<u_{\lambda}(x)\leq \theta_{2}$$ for a.e. $$x\in \Omega$$, $$| u_{\lambda}|_{\infty}>\theta_1$$; if $$f$$ has $$n$$ loops the problem has at least $$n$$ distinct solutions.
Theorem 2. Let $J_{\lambda}(u)=\frac {1}{2}\int_{\Omega}| \nabla u| ^{2}\,dx-\lambda \int_{\Omega}^{u(x)} {1}_{]\theta_{1},\theta_{2}[}f\, ds\, dx$ for every $$u\in H_{0}^{1}(\Omega)$$ and $$\psi_{\lambda}$$ a minimizer of $$J_{\lambda}$$. Then, $$\psi_{\lambda}\rightarrow \theta_{2}$$ and $$u_{\lambda}\rightarrow \theta_{2}$$ in $$L^{p}(\Omega)$$ as $$\lambda \rightarrow \infty$$.

### MSC:

 35R10 Partial functional-differential equations 35J60 Nonlinear elliptic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35B40 Asymptotic behavior of solutions to PDEs