Non-uniqueness of positive ground states of non-linear Schrodinger equations

作者：Davila Juan^*; del Pino Manuel; Guerra Ignacio

来源：Proceedings of the London Mathematical Society, 2013, 106: 318-344.

DOI：10.1112/plms/pds038

Existence of a positive, decaying radial solution to the problem %26lt;br%26gt;Delta u - u + u(p) + lambda u(q) = 0 in R-N, %26lt;br%26gt;when lambda %26gt; 0 and 1 %26lt; q %26lt; p %26lt;(N+2)/(N-2) has been known for a long time. For lambda=0, it is well known that this solution is unique. While uniqueness conditions for rather general non-linearities have been found, the issue has remained elusive for this problem. We prove that uniqueness is in general not true. We find that if N=3, 1 %26lt; q %26lt; 3, lambda is fixed sufficiently large, and p %26lt; 5 is taken sufficiently close to 5, then there are at least three positive decaying radial solutions.

• 出版日期2013-2