On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary.
This paper concerns the nonlinear Neumann problem \[ -\Delta u=Q(x)| u| ^{2^*-2}u\quad \text{in }\Omega, \] \[ \frac{\partial}{\partial\nu}u(x)=P(x)| u| ^{q-2}u\quad\text{on }\partial\Omega,\qquad u>0\quad\text{on }\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with a smooth boundary \(\partial\Omega\), \(\nu\) is the outward normal to \(\partial\Omega,\) \(2^*=2N/(N-2)\), \(N\geq 3\), is the usual critical Sobolev exponent, \(q=2(N-1)/(N-2)\) is the critical Sobolev exponent for the trace embedding of \(H^1\) into \(L^q(\partial\Omega)\), \(Q(x)\) and \(P(x)\) are smooth functions. The author observes that if both \(Q\) and \(P\) are positive then the problem does not have solution. First, the case \(Q\equiv 0\) is discussed. \(P\) is assumed to change sign on \(\partial\Omega\) and moreover \[ \int_{\partial\Omega}P(x)\,dS<0. \] Furthermore, \(P\) is assumed to attain its maximum value at a point \(y\in\partial\Omega\) with \(H(y)>0\), and moreover \(| P(x)-P(y)| =o(| x-y| )\) for \(x\) near \(y\). Here \(H(y)\) denotes the mean curvature of \(\partial\Omega\) at the point \(y\), and \(o\) has the usual meaning. Under all the above assumptions, by using a constrained variational problem, it is proved that the Neumann problem has a solution. Next, the case \(Q(x)\geq 0\), \(Q(x)\not\equiv0\) is discussed. It is assumed that \(Q(x)=0\) for \(x\in\{\partial\Omega:P(x)>0\}\). Under some suitable additional conditions on \(P\) and \(Q\), by using the mountain-pass principle, it is proved the existence of a solution of the Neumann problem.
scientific article; zbMATH DE number 2056389