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Quasilinear elliptic equations with discontinuous coefficients. (English) Zbl 0679.35035
Summary: We prove an existence result for equations of the form \[ -D_ i(a_{ij}(x,u)D_ ju)=f\quad in\quad \Omega,\quad u\in H^ 1_ 0(\Omega), \] where the coefficients \(a_{ij}(x,s)\) satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable s. Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients \(a_{ij}(x,s)\) are supposed only Borel functions.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
49J20 Existence theories for optimal control problems involving partial differential equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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