Some discontinuous problems with a quasilinear operator. (English) Zbl 0815.35018

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 \(I\).


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
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