Abstract
Abstract In this article, we are interested in multi-bump solutions of the singularly perturbed problem - ε 2 Δ v + V ( x ) v = f ( v ) in ℝ N . -\varepsilon^{2}\Delta v+V(x)v=f(v)\quad\text{in }\mathbb{R}^{N}. Extending previous results, we prove the existence of multi-bump solutions for an optimal class of nonlinearities f satisfying the Berestycki–Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as ε → 0 {\varepsilon\rightarrow 0} . Examples of potential wells include the following: the union of two compact smooth submanifolds of ℝ N {\mathbb{R}^{N}} where these two submanifolds meet at the origin and an embedded topological submanifold of ℝ N {\mathbb{R}^{N}} .
Highlights
One of the best-known nonlinear partial differential equations is the nonlinear Schrodinger equation, which appears in a variety of physical contexts including Bose-Einsten condensation [40], nonlinear atom optics
This paper is devoted to the study of the standing wave solutions to the nonlinear
We examine standing waves of the nonlinear Schrodinger equation (1) for small > 0, which are referred to as semi-classical states
Summary
One of the best-known nonlinear partial differential equations is the nonlinear Schrodinger equation, which appears in a variety of physical contexts including Bose-Einsten condensation [40], nonlinear atom optics [35, 36] and many others. In [10], Byeon and Tanaka proved the existence of a family of solutions of (2) with clustering peaks around an isolated set of critical points of V that are non-minimal when the nonlinearity f satisfies (f1)-(f3) as well as the following condition (f4) f ∈ C1(R, R) and there exists a q0 ∈ (0, 1) such that f (t) lim. We prove the existence of multi-bump solutions of (2) in which the centers of bumps converge toward general sets M under the conditions (f1)-(f4). To the best of our knowledge, Theorem 1.1 is the first result concerning the existence of multi-bump solutions of (2) in which the centers of bumps converge toward general classes of potential wells.
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