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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.

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