Existence and nonexistence of solutions for some nonlinear elliptic equations. (English) Zbl 0898.35035

The authors consider the following problem \[ A(u)+G(x,u,\nabla u)=\mu, \quad x\in \Omega, \qquad u=0,\quad x\in \partial \Omega, \] where \(\Omega \subset \mathbb{R}^n\) is a bounded, open set; \(1<p<n\), \(a: \Omega \times \mathbb{R}^n \mapsto \mathbb{R}^n\) is a Caratheodory function such that: \[ a(x,\xi) \cdot \xi \geq \alpha | \xi| ^p,\quad \alpha>0,\qquad | a(x,\xi)| \leq l(x)+\beta | \xi | ^{p-1}, \quad \beta>0,\;l\in L^{p'}(\Omega), \]
\[ (a(x,\xi)-a(x,\eta)) \cdot (\xi -\eta)>0,\quad \xi\neq \eta,\qquad A(u)=-\text{div} (a(x,\nabla u)). \] \(g\) is a Caratheodory function such that \[ | g(x,s,\xi)| \leq b(| s|)(| \xi| ^p+d(x)),\quad d\geq 0, d\in L^1(\Omega) \] with increasing, continuous, real valued, positive function \(b\), and \(g(x,s,\xi)\mathit{sgn}(s)\geq \rho| \xi|^p\), for every \(s\in \mathbb{R}\) such that \(| s| \geq\sigma\), with positive numbers \(\sigma\) and \(\rho\). Denote by cap\(_p(B,\Omega)\) the \(p\)-capacity of any subset \(B\subseteq \Omega\). By \(M_b(\Omega)\) denote the space of all \(\sigma\)-additive functions \(\mu\) with values in \(\mathbb{R}\) defined on the Borel \(\sigma\)-algebra, and by \(M^p_0(\Omega)\) denote the space of all measures \(\mu\) in \(M_b(\Omega)\) such that \(\mu (E)=0\) for every set \(E\) such that \(\text{cap}_p(E,\Omega)= 0\).
The authors prove the following Theorem: Let \(\mu\) be a measure in \(M_b(\Omega)\). Then there exists a solution \(u\) of the above problem in the sense that \(u\in W^{1,p}_0(\Omega)\), \(g(x,u,\nabla u)\in L^1(\Omega)\), and \[ \int_{\Omega} a(x,\nabla u) \cdot \nabla v dx + \int _{\Omega} g(x,u,\nabla u) v dx = \int _{\Omega} v d\mu , \] for every \(v\in C^{\infty}_0\), if and only if \(\mu \in M^p_0(\Omega)\).
They also prove that if the sequence \(\{u_n\}\) of solutions of the above-mentioned problem with data \(\mu _n\in L^{\infty}(\Omega)\) converging to a nonzero measure which is singular with respect to the capacity, then \(u_n\) converges to zero as \(n \to \infty\). The last fact is proved under the additional condition \(g(x,s,\xi) s\geq 0\).


35J65 Nonlinear boundary value problems for linear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
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