On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function (Q928406)

The author considers the problem \[ -\Delta u = \lambda f(x)| u| ^{q-2}u + | u| ^{2^*-2}u~~\text{in}~~\Omega,~~~~u \in H_0^1(\Omega), \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N (N \geq 3)\), \(1<q<2<2^*=2N/(N-2)\), \(\lambda >0\) and \(f:\overline{\Omega} \to \mathbb{R}\) is a continuous function with \(f^+(x)=\max\{f(x),0\} \not \equiv 0\) in \(\overline{\Omega}\). By using variational methods he proves that, for \(\lambda>0\) small, the problem possesses at least two positive solutions. He also studies the asymptotic behavior of the obtained solutions as \(\lambda \to 0\).