Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity

Abstract

We consider singularly perturbed elliptic equations ε 2 ∆u−V (x)u+ f (u) = 0, x ∈ R N , N ≥ 3. For small ε > 0, we glue together localized bound state solutions concentrating at isolated components of positive local minimum of V under conditions on f we believe to be almost optimal.

Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity

“…Our approach for an existence of a solution for small ε > 0 is basically variational, but takes advantage of a reduction to a compact set in a finite dimensional reduction method used powerfully when the nondegeneracy condition (f5) is satisfied. This approach was successfully carried out in the study of the standing waves for nonlinear Schrödinger equations [5], [6], [7].…”

Multi-peak solutions to two types of free boundary problems

“…On the other side, we apply the descent gradient flow in variational method to search for the critical point of the corresponding functional, which does not require any uniqueness result of the least energy solution nor isolatedness result of the least energy. We should point out that for gluing localized solutions, there have been many efforts starting from the pioneer works [10,11,15,20,30], for example, we can refer to [7,29] and the references therein.…”

“…We remark that differently from [7], we should overcome some additional difficulties. Firstly, we need to know the properties of the least energy solutions to the limit equations, including the mountain pass characterization of the least energy solutions and the property of decay of the least energy solutions at infinity.…”

“…Firstly, we need to know the properties of the least energy solutions to the limit equations, including the mountain pass characterization of the least energy solutions and the property of decay of the least energy solutions at infinity. Secondly, since the least energy solutions of the limit equations decay algebraically, we can not modify the functional corresponding to (1.1) as in [7]. Instead, considering the fact that f (u − c) = 0 for u ≤ c, we modify the original equation by multiplying the nonlinearity f (t) by a characteristic function on a suitable set, which can force the concentration phenomena not to occur outside…”

Positive Radial Solutions for a Class of Elliptic Systems Concentrating on Spheres With Potential Decay

“…Ambrosetti-Ruiz in [9] extended this result to the case of decaying potentials. See also [4], [7], [10], [12], [13], [15] and [23]. In the critical frequence, that means inf R N V (x) = 0, spike solutions have been constructed in [16], [17], [18] and [19], which concentrate on the zero of the potential V as ε → 0.…”