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Existence and uniqueness of positive solutions for some quasilinear elliptic problems. (English) Zbl 0991.35035

Let \(\Omega\subset R^d\) be a smooth bounded domain and \(1<p<d\). Let \(\Delta_p\) denote the \(p\)-Laplacian, that is \(\Delta_p u=\text{div} (|\nabla u|^{p-2} \nabla u)\) and \(f:\Omega \times R\to R\) is a Carathéodory function. The authors study the quasilinear boundary value problem \[ \begin{cases} -\Delta_pu= f(x,u)\quad &\text{in }\Omega\\ u=0\quad &\text{on } \partial \Omega\end{cases} \tag{*} \] and present a rather general version of the method of sub- and supersolutions for (*) with sublinear terms. Moreover the authors give several applications to sublinear problems.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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