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)