We study the existence of positive solutions on $\R^{N+1}$ to semilinear elliptic equation $-\Delta u+u=f(u)$ where $N\geq 1$ and $f$ is modeled on the power case $f(u)=|u|^{p-1}u$. Denoting with $c$ the mountain pass level of $\f(u)=\tfrac 12\|u\|^{2}_{H^{1}(\R^{N})}-\int_{\R^{N}}F(u)\, dx$, $u\in H^{1}(\R^{N})$ ($F(s)=\int_{0}^{s}f(t)\, dt$), we show that for any $b\in [0,c)$ there exists a positive bounded solution $v_{b}\in C^{2}(\R^{N+1})$ such that $E_{v_{b}}(y)=\tfrac 12\|\partial_{y}v_{b}(\cdot,y)\|^{2}_{L^{2}(\R^{N})}-V(v_{b}(\cdot,y))=-b$. We also characterize the monotonicity, symmetry and periodicity properties of $v_{b}$.

An energy constrained method for the existence of layered type solutions of NLS equations / Alessio, FRANCESCA GEMMA; Montecchiari, Piero. - In: ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE. - ISSN 0294-1449. - STAMPA. - 31:4(2014), pp. 725-749. [10.1016/j.anihpc.2013.07.003]

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

ALESSIO, FRANCESCA GEMMA;MONTECCHIARI, Piero
2014-01-01

Abstract

We study the existence of positive solutions on $\R^{N+1}$ to semilinear elliptic equation $-\Delta u+u=f(u)$ where $N\geq 1$ and $f$ is modeled on the power case $f(u)=|u|^{p-1}u$. Denoting with $c$ the mountain pass level of $\f(u)=\tfrac 12\|u\|^{2}_{H^{1}(\R^{N})}-\int_{\R^{N}}F(u)\, dx$, $u\in H^{1}(\R^{N})$ ($F(s)=\int_{0}^{s}f(t)\, dt$), we show that for any $b\in [0,c)$ there exists a positive bounded solution $v_{b}\in C^{2}(\R^{N+1})$ such that $E_{v_{b}}(y)=\tfrac 12\|\partial_{y}v_{b}(\cdot,y)\|^{2}_{L^{2}(\R^{N})}-V(v_{b}(\cdot,y))=-b$. We also characterize the monotonicity, symmetry and periodicity properties of $v_{b}$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/83761
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