## Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems.(English)Zbl 0984.35066

The authors study some class of nonlinear nonlocal elliptic and parabolic problems. Precisely, let $$\Omega$$ be a bounded domain in $$\mathbb{R}^n$$ with smooth boundary $$\partial Q$$, and let $$\Gamma_0$$ be a subset of $$\partial \Omega$$ having a positive superficial measure. Set $$V=\{v\in H^1 (\Omega)\mid v=0$$ on $$\Gamma_0\}$$, and consider $$m$$ functionals $$q_1,\dots, q_m:V\to \mathbb{R}$$, where each $$q_i$$ is a positive homogeneous function of degree $$\alpha_i (\in \mathbb{R})$$. For a positive function $$a:\mathbb{R}^m \to\mathbb{R}$$, consider the problem: $-a\bigl(q_1(u), \dots,q_m(u) \bigr)Au= f\text{ in }\Omega, \tag{1}$
$u=0\text{ on }\Gamma_0,\;\partial_\nu u=0\text{ on }\partial\Omega \setminus\Gamma_0, \tag{2}$ where $$f$$ is an element of $$V'$$, the dual of $$V$$, $$A$$ is a linear elliptic operator in divergence form, and $$\partial_\nu u$$ denotes the conormal derivative of $$u$$. It is assumed that the bilinear form canonically associated to $$A$$ is coercive on $$V$$. A typical example of $$A$$ is the Laplacian $$\Delta$$. It is shown that this problem has as many solutions as the system of equations in $$\mu=(\mu_1, \dots,\mu_m) \in\mathbb{R}^m$$: $a^{\alpha_i} (\mu)\mu_i=q_i (\varphi),\;i=1, \dots,m,$ where $$\varphi\in V$$ is the unique solution of the problem: $-A\varphi= f\text{ in }\Omega, \quad \varphi=0 \text{ on }\Gamma_0,\;\partial_\nu \varphi=0 \text{ on }\partial \Omega\setminus \Gamma_0.$ When $$\Gamma_0= \partial\Omega$$ and $$V=H^1_0 (\Omega)$$, also the parabolic problem associated to problem (1)-(2) is solved and in particular it is shown that the solutions of the parabolic problem can quench in the sense that they can vanish identically at some finite time.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations