Neumann problems of a class of elliptic equations with doubly critical Sobolev exponents (Q812466)

The paper deals with the following Neumann problem \[ -\Delta u+u=| u| ^{2^*-2}u+\mu | u| ^{q-2}u\;\;in\;\;\Omega,\;\;\frac{\partial u}{\partial\gamma}=| u| ^{s^*-2}u\;\;on\;\;\partial\Omega, \] where \(\Omega\) is a bounded smooth domain in \(R^N\), \(N\geq 3\), \(2^*=2N/(N-2)\), \(s^*=2(N-1)/(N-2)\), \(1<q<2\), \(\mu>0\) and \(\gamma\) denotes the outward normal to the boundary \(\partial\Omega\). The non trivial solutions of this problem are non zero critical points of the functional \[ J(u)=\int_\Omega\biggl(\frac{1}{2}| \nabla u| ^2+\frac{1}{2}u^2-\frac{1}{2^*}| u| ^{2^*}-\frac{\mu}{q}| u| ^q\biggr)\,dx-\frac{1}{s^*}\int_{\partial\Omega} | u| ^{s^*}\,ds,\quad u\in H^1(\Omega). \] By using variational methods, the author proves that there exists \(\mu^*>0\) such that, for every \(0<\mu<\mu^*\) this problem has a sequence of solutions \(u_k\in H^1(\Omega)\) such that \(J(u_k)<0\) and \(J(u_k)\to 0\) as \(k\to\infty\).