Arcoya, David; Calahorrano, Marco Some discontinuous problems with a quasilinear operator. (English) Zbl 0815.35018 J. Math. Anal. Appl. 187, No. 3, 1059-1072 (1994). The authors study the boundary value problem \[ - \Delta_ p u = f(u) + q(x) \quad \text{in } \Omega, \quad u = 0 \quad \text{on } \partial \Omega \tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\), \(\Delta_ pu = \text{div} (| \nabla u |^{p-2} \nabla u)\) \((p > 1)\), is the \(p\)-Laplacian, \(q \in L^{p/p-1} (\Omega)\) and \(f \in C(\mathbb{R} \backslash \{a\}, \mathbb{R})\). Moreover, \(f\) is assumed to have an upward discontinuity at \(a\). In the study of (1), if the variational method is used, the main problem is that the corresponding functional \(I\) is not Fréchet differentiable. The authors overcome this difficulty observing that \(I\) is locally Lipschitz and then, by using the generalized critical point theory developed by K.-C. Chang [J. Math. Anal. Appl. 80, 102-129 (1981; Zbl 0487.49027)] for this kind of functionals. In this way, the critical points of \(I\) are solutions of (1) in a certain multivalued sense. As in the work by A. Ambrosetti and M. Badiale when \(p = 2\) [J. Math. Anal. Appl. 140, No. 2, 363-373 (1989; Zbl 0687.35033)], with some additional restrictions, involving the function \(q\) and the lateral limits of \(f\) at \(a\), they obtain a solution of (1) a.e.; the same conclusion is obtained if the critical point is a local minimum of \(I\). Reviewer: A.Cañada (Granada) Cited in 26 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35R05 PDEs with low regular coefficients and/or low regular data 35J20 Variational methods for second-order elliptic equations Keywords:quasilinear operators; discontinuous nonlinearity; generalized critical point theory Citations:Zbl 0487.49027; Zbl 0687.35033 PDFBibTeX XMLCite \textit{D. Arcoya} and \textit{M. Calahorrano}, J. Math. Anal. Appl. 187, No. 3, 1059--1072 (1994; Zbl 0815.35018) Full Text: DOI