Mountain impasse theorem and spectrum of semilinear elliptic problems

Abstract:

This paper studies a minimax problem for functionals in Hilbert space in the form of $G(u) = \frac {1} {2}\rho ||u|{|^2} - g(u)$, where $g(u)$ is Fréchet differentiable with weakly continuous derivative. If $G$ has a "mountain pass geometry" it does not necessarily have a critical point. Such a case is called, in this paper, a "mountain impasse". This paper states that in a case of mountain impasse, there exists a sequence ${u_j} \in H$ such that \[ g\prime ({u_j}) = {\rho _j}{u_j},\quad {\rho _j} \to \rho ,||{u_j}|| \to \infty ,\] and $G({u_j})$ approximates the minimax value from above. If \[ \gamma (t) = \sup \limits _{||u|{|^2} = t} \;g(u)\] and \[ {J_0} = \left ( {2\inf \limits _{{t_2} > {t_1} > 0} \frac {{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}},2\sup \limits _{{t_2} > {t_1} > 0} \frac {{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}}} \right ),\] then $g\prime (u) = \rho u$ has a nonzero solution $u$ for a dense subset of $\rho \in {J_0}$.