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An \(L^ 1\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations. (English) Zbl 0866.35037
The authors consider the problem \[ -\Delta_pu =F(x,u)\quad \text{ on }\Omega, \quad u(x)=0\quad \text{ on } \partial\Omega \tag{1} \] where \(\Omega\) is an open, not necessarily bounded set in \(\mathbb{R}^N\), \(N\geq 2\) with boundary denoted by \(\partial\Omega\). Furthermore, \(1<p< \infty\), \(Du= (\partial_1u, \dots, \partial_Nu)\) with \(\Delta_pu =\text{div} (|Du |^{p-2} Du)\) and \(F\) a continuous function, non-decreasing in \(u\), such that \(F(x,0) \in L^1(\Omega)\) and \(F(x,c)\in L^1_{\text{loc}} (\Omega)\) for \(c\neq 0\). The authors are interested in the case when \(1<p<N\). This presents two difficulties even when \(\Omega\) is bounded. First, a suitable solution concept is required when \(p\) is close to unity since it is not possible to take the gradient of \(u\) appearing in the \(p\)-Laplacian in the usual distribution sense. This difficulty is overcome by introducing a new space in which the gradient of \(u\), which in the initial setting was not locally integrable, now does make sense. This is achieved by working with truncations of \(u\), denoted \(T_k(u)\), since their gradients turn out to be locally integrable. The second difficulty centres on the questions of existence and uniqueness of solution. This is resolved by working with a class of functions which satisfy a so-called entropy condition whose use, whilst familiar when working with conservation laws, is held to be novel when studying elliptic equations. For reasons of clarity of exposition many technical results are contained in a number of Appendices.

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