An energy constrained method for the existence of layered type solutions of NLS equations

Alessio, Francesca; Montecchiari, Piero

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 4, pp. 725-749.

We study the existence of positive solutions on ${ℝ}^{N+1}$ to semilinear elliptic equation $-\Delta u+u=f\left(u\right)$ where $N⩾1$ and f is modeled on the power case $f\left(u\right)={\left|u\right|}^{p-1}u$. Denoting with c the mountain pass level of $V\left(u\right)=\frac{1}{2}{\parallel u\parallel }_{H^1\left({ℝ}^N\right)}^2-{\int }_{{ℝ}^N}F\left(u\right)dx$, $u\in H^1\left({ℝ}^N\right)$ ($F\left(s\right)={\int }_0^{s}f\left(t\right)dt$), we show, via a new energy constrained variational argument, that for any $b\in [0,c)$ there exists a positive bounded solution $v_b\in C^2\left({ℝ}^{N+1}\right)$ such that $E_{v_b}\left(y\right)=\frac{1}{2}{\parallel {\partial }_y{v}_b(·,y)\parallel }_{L^2\left({ℝ}^N\right)}^2-V\left(v_b(·,y)\right)=-b$ and $v(x,y)\to 0$ as $\left|x\right|\to +\infty$ uniformly with respect to $y\in ℝ$. We also characterize the monotonicity, symmetry and periodicity properties of $v_b$.

Classification : 35J60, 35B08, 35B40, 35J20, 34C37
Mots clés : Semilinear elliptic equations, Locally compact case, Variational methods, Energy constraints

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