Existence of solutions of elliptic equations involving critical Sobolev exponents with Neumann boundary condition in general domains (Q1177292)

The following problem is investigated: (I) \(-\Delta u= | u|^{p- 1} u+\lambda u\) in \(\Omega\), \(\partial u/\partial n=0\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain of class \(C^ 2\) in \(\mathbb{R}^ N\), \(N\geq 3\), \(n\) is the outward pointing normal on \(\partial\Omega\), \(\lambda\in\mathbb{R}\) and \(p=(N+2)/ (N-2)\) is the critical Sobolev exponent for the embedding \(H^ 1(\Omega)\to L^{p+1}(\Omega)\) which is no longer compact. The existence of nontrivial solutions of problem (I) for every \(\lambda\in\mathbb{R}\) is studied. Theorem 1. If \(N=3\), there exists a nontrivial solution of problem (I) for every \(\lambda\) in \(\mathbb{R}^ +\setminus \sigma(-\Delta)\), where \(\sigma(-\Delta)\) is the set of the eigenvalues of \(-\Delta\) in \(\Omega\) with Neumann boundary condition. If \(N\geq 4\), there exists a non-trivial solution of problem (I) for every \(\lambda\geq 0\). Set \(\sigma(-\Delta)= \{\mu_ k\): \(k\in\mathbb{N}_ 0\}\); then: Theorem 2. If \(N\geq 3\), problem (I) admits a nontrivial solution for every \(\lambda<-\mu_ 1/(p-1)\). Lin, Ni and Tagaki have studied problem (I) in the subcritical case, when \(p<(N+2)/ (N-2)\), and they have proved, that for \(\lambda\) smaller than \(-\mu_ 1/(p-1)\) there exists a non-trivial solution of problem (I). When \(\lambda\) is close enough to zero, they have shown that no positive non-trivial solution can exist. However, in the critical case, examples of positive non-trivial solutions of problem (I) are given by several authors, for \(\lambda<0\) and arbitrarily close to zero.