### Non-collision solutions for a second order singular hamiltonian system with weak force

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In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on ${\mathbb{R}}^{N}$. The main difficulties to overcome are the lack of a priori bounds for Palais–Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the “Problem at infinity” is autonomous, in contrast to just periodic, can be used in order...

We consider a singularly perturbed elliptic equation ${\u03f5}^{2}\Delta u-V\left(x\right)u+f\left(u\right)=0,u\left(x\right)>0$ on ${\mathbb{R}}^{N}$, ${\mathrm{\U0001d695\U0001d692\U0001d696}}_{\left|x\right|\to \infty}u\left(x\right)=0$, where $V\left(x\right)>0$ for any $x\in {\mathbb{R}}^{N}$. The singularly perturbed problem has corresponding limiting problems $\Delta U-cU+f\left(U\right)=0,U\left(x\right)>0$ on ${\mathbb{R}}^{N}$, ${\mathrm{\U0001d695\U0001d692\U0001d696}}_{\left|x\right|\to \infty}U\left(x\right)=0,c>0$. Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions on $f$. In...

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on ${}^{N}$. The main difficulties to overcome are the lack of bounds for Palais–Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the “Problem at infinity” is autonomous, in contrast to just periodic, can be used in order to regain...

We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When $V\left(x\right)$ has multiple critical points, (1.1) has a wide variety of positive solutions for small $\epsilon $ and the number of positive solutions increases to $\infty $ as $\epsilon \to 0$. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of $V\left(x\right)$. Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

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