A value function and applications to translation-invariant semilinear elliptic equations on unbounded domains

Abstract

We combine results from nonlinear functional analysis relating nonlinear eigenvalues of the type $ g'(u) = \rho u $ to the derivatives of the critical value function $ \gamma (t) := \sup_{\|u\|^2=t} g(u) $ with concentration compactness techniques to study the Dirichlet boundary value problem on $\Omega$, $$ -\Delta u + u = \lambda f(x,u), \tag 0.1 $$ where $\Omega$ is an unbounded cylindrical domain and the dependence on $x$ in the unbounded direction is periodic. We give sufficient conditions on $f$ to obtain an interval in which the $\lambda$'s for which (0.1) has a weak solution are dense.