Critical exponent and discontinuous nonlinearities (Q1308524)

We prove existence of a positive solution to the problem \[ -\Delta u=| u|^{2^*- 2} u+bh (u-a), \quad u(x)>0 \;\;\text{in }\Omega, \quad u(x)=0\;\;\text{on } \partial\Omega, \tag{1} \] where \(\Omega\) is a bounded regular open set \(\subset \mathbb{R}^ N\), \(2^*= {{2N} \over {n-2}}\) is the critical Sobolev exponent, \(h\) is the Heaviside function and \(a\), \(b\) are strictly positive parameters. The nonlinearity in (1) has critical power-like behavior and a discontinuity jump of length \(b\) in the point \(u=a\). So in problem (1) there are two difficulties to be overcome: the presence of a discontinuity and the lack of compactness due to the critical exponent.