A singularly perturbed elliptic partial differential equation with an almost periodic term
Driven by an analogy with a result in Hamiltonian systems \textit{E. Serra, M. Tarallo} and \textit{S. Terracini} [Ann. Inst. Henri Poincaré, Anal. Non Lineaire 13, No.~6, 783-812 (1996; Zbl 0873.58032)] the author considers partial differential equations of the form \(-\varepsilon^2\Delta u +V(x)u=f(u)\) for \(x\in\mathbb{R}^n\), \(V\) almost periodic and \(f\) satisfying certain growth conditions. He shows that for sufficiently small \(\varepsilon\) there exits a nontrivial homoclinic solution (\(|u(x)|+|\nabla u|\rightarrow 0\) as \(|x|\rightarrow \infty\)) under certain other technical restrictions, which are automatic in the two-dimensional case. The analogy with the Hamiltonian situation in one dimension does not supply a general proof for the higher dimensional case for topological reasons. The result follows from elliptic regularity theory applied to an appropriate functional for the equation.
scientific article; zbMATH DE number 1383817