Multiple condensations for a nonlinear elliptic equation with sub-critical growth and critical behaviour (Q2758152)

The author studies the following nonlinear elliptic equationNEWLINENEWLINE\[NEWLINE\begin{cases} -\Delta u=u_+^p,\\ u=\mu\text{ on }\partial\Omega\;&(\mu\text{ is an unknown constant}),\\ -\int_{\partial\Omega} \frac{\partial u}{\partial n}=M, \end{cases} \tag{1}NEWLINE\]NEWLINENEWLINEwhere \(N\geq 3\), \(1<p<2^*:= \frac{N+2}{N-2}\) and \(u_+= \max(u,0)\). Here \(M\) is a prescribed constant, and \(\Omega\) is a bounded and smooth domain in \(\mathbb R^N\). The exponent \(p_*= \frac{N}{N-2}\) turns out to be a natural critical exponent for (1). Even though it has subcritical growth, it has critical behaviour. When \(\Omega= B_1(0)\) and \(M=M_*^{(N)}\) it is known that one can find a continuum of solutionso f(1). Here the author shows that for \(M\) near \(KM_*^{(N)}\), \(K>1\), there exist solutions with multiple concentrations in \(\Omega\). These concentration points are non-degenerate critical points of a function related to the Green function.