Existence theorems for a semilinear elliptic boundary value problem. (English) Zbl 0826.35146

The author gives sufficient conditions under which the problem \[ f(x,u) \leq Lu \leq g (x,u) \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega \] has a strong solution \(u \in W^{2,p} (\Omega) \cap W_0^{1,p} (\Omega)\). Here \(L\) is a second-order linear elliptic operator, and \(f,g : \Omega \times \mathbb{R} \to \mathbb{R}\) are Carathéodory functions.


35R70 PDEs with multivalued right-hand sides
35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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