Let $W$ be a non-negative function of class ${\mathrm{C}}^3$ from ${ℝ}^2$ to $ℝ$, which vanishes exactly at two points $𝐚$ and $𝐛$. Let $S^1(𝐚,𝐛)$ be the set of functions of a real variable which tend to $𝐚$ at $-\infty$ and to $𝐛$ at $+\infty$ and whose one dimensional energy $$E_1\left(v\right)={\int }_{ℝ}\left[W\left(v\right)+|v^{\text{'}}{|}^2/2\right]\mathrm{d}x$$ is finite. Assume that there exist two isolated minimizers $z_{+}$ and $z_{-}$ of the energy $E_1$ over $S^1(𝐚,𝐛)$. Under a mild coercivity condition on the potential $W$ and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator...

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left({\nabla }_{\alpha }{u}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_3{u}_{\epsilon }\right)$ bounded in $L^p(\Omega ;{ℝ}^9),1\<p\<+\infty .$ Here it is shown that, up to a subsequence, $u_{\epsilon }$ may be decomposed as $w_{\epsilon }+z_{\epsilon },$ where $z_{\epsilon }$ carries all the concentration effects, i.e. $\left\{{\left|\left({\nabla }_{\alpha }{w}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_3{w}_{\epsilon }\right)\right|}^p\right\}$ is equi-integrable, and $w_{\epsilon }$ captures the oscillatory behavior, i.e. $z_{\epsilon }\to 0$ in measure. In addition, if $\left\{u_{\epsilon }\right\}$ is a recovering sequence then $z_{\epsilon }=z_{\epsilon }\left(x_{\alpha }\right)$ nearby $\partial \Omega .$

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V\left(x\right)u={K\left(x\right)\left|u\right|}^{2^{*}-2}u+g(x,u),u\in W^{1,2}\left({𝐑}^N\right)$, where $N\ge 4;V,K,g$ are periodic in $x_j$ for $1\le j\le N$ and 0 is in a gap of the spectrum of $-\Delta +V$; $K\>0$. If $0\<g(x,u)u\le {c\left|u\right|}^{2^{*}}$ for an appropriate constant $c$, we show that this equation has a nontrivial solution.

In this paper, we discuss the conditions for a center for the generalized Liénard system $$\frac{\mathrm{d}x}{\mathrm{d}t}=\varphi \left(y\right)-F\left(x\right),\frac{\mathrm{d}y}{\mathrm{d}t}=-g\left(x\right),$$ or $$\frac{\mathrm{d}x}{\mathrm{d}t}=\psi \left(y\right),\frac{\mathrm{dy}}{\mathrm{d}t}=-f\left(x\right)h\left(y\right)-g\left(x\right),$$ with $f\left(x\right)$, $g\left(x\right)$, $\varphi \left(y\right)$, $\psi \left(y\right)$, $h\left(y\right)ℝ\to ℝ$, $F\left(x\right)={\int }_0^{x}f\left(x\right)\mathrm{d}x$, and $xg\left(x\right)>0$ for $x\ne 0$. By using a different technique, that is, by introducing auxiliary systems and using the differential inquality theorem, we are able to generalize and improve some results in [1], [2].