Positive solutions for asymptotically linear problems in exterior domains

Abstract

Abstract. The existence of a positive solution for a class of asymptotically linear problems in exterior domains is established via a linking argument on the Nehari manifold and by means of a barycenter function.

A positive bound state for an asymptotically linear or superlinear Schrödinger equation in exterior domains

“…Our goal in this paper is to show the existence of a positive bound state solution for problem (P V ) when a ground state can not be obtained. Using a new approach recently developed byÉvéquoz and Weth [20], Clapp and Maia [14] and Maia and Pellacci [25] a positive solution is found, extending the existence results obtained in the celebrated papers of Benci and Cerami [5] and Bahri and Lions [3], for general non homogeneous nonlinearities, either superlinear or asymptotically linear at infinity in an exterior domain.…”

“…Most importantly, this approach enables to avoid the use of a technical algebraic inequality (a + b) p ≥ a p +b p +(p−1)(a p−1 b+ab p−1 ) largely applied in the case f (u) = |u| p−2 u ( [2,3,12]). We follow these ideas, closely related to the arguments found in [14] and [25], for general nonlinearities f which satisfy the assumption that f (s)/s is increasing. In this setting, not all functions u = 0 are projectable on the Nehari manifold , however the class of functions which are good for projections in this environment is enough to pursue the argument.…”

A Note on Existence of a Bound State for a Non-Autonomous Nonlinear Scalar Field Equation

“…where w is the positive radial ground state solution of (P ∞ ). For additional information about (4.3), see [1,4,11,12,19] and references therein. The next result of Bahri and Lions is essential for proving the asymptotic behaviour of quantity ε R .…”

“…Our objective is to extend [5] to non homogeneous non-linearities f which are either superlinear or asymptotically linear at infinity and a(x) also satisfying an exponential asymptotic limit. We use a variational approach and a topological argument introduced in [5] and updated in [12,14,19].…”

“…Thereafter, they obtained a positive solution which is not necessarily a ground state. Inspired by the ideas in [14,19], we perform some calculations of sharp energy estimates and apply a topological argument involving the barycenter function to show that there is a critical value of the functional associated with the Euler equation in (P a ), in a suitable level of energy, giving a solution of the problem.…”