Positive solutions of semilinear elliptic equation \(\Delta u+ hu^{(n+ 2)/(n- 2)} =0\) (Q1903055)

The author considers the elliptic equation \[ \Delta u+ h(x) u^p= 0,\quad u> 0\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega \] with critical nonlinearity \(p= (n+ 2)/(n- 2)\), and \(h(x)\geq 0\). Existence of a nontrivial solution of (1) is shown for the following choices of \(h\) and \(\Omega\) the unit ball in \(\mathbb{R}^n\): -- \(h= 0\) in \(B_\delta\subset \Omega\), \(h= 1\) in \(\Omega\backslash B_\delta\), where \(B_\delta\) is the ball of radius \(\delta< 1\) centered at 0; -- \(h= 0\) in \(B_\delta(x_0)\), \(x_0\in \Omega\), \(\delta< 1- |x_0|\), \(h= 1\) in \(\Omega\backslash B_\delta(x_0)\); -- \(h= 0\) in \(\Omega_0\), \(h= 1\) in \(\Omega\backslash \Omega_0\), if \(\Omega_0\) is a small, symmetric and starshaped domain in \(\Omega\). The proof of these results is based on variational methods and a truncation procedure.