The paper deals with the existence of positive solutions with prescribed $L<^>2$ norm for the Schrodinger equation $$-\Delta u+\lambda u+V(x)u=|u|<^>{p-2}u,\quad u\in H<^>1_0(\Omega),\quad\int_\Omega u<^>2{\rm d}\,x=\rho<^>2,\quad\lambda\in\mathbb{R},$$ where $\Omega =\mathbb {R}<^>N$ or $\mathbb {R}<^>N\setminus \Omega$ is a compact set, $\rho >0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in (2,2+\frac 4 N)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption (D?), or if $\|V\|_{L<^>q}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption (D?) is fulfilled. There are no assumptions on the size of $\mathbb {R}<^>N\setminus \Omega$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $\Omega =\mathbb {R}<^>N$.
Normalized positive solutions for Schrödinger equations with potentials in unbounded domains / Lancelotti, S.; Molle, R.. - In: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SECTION A. MATHEMATICS. - ISSN 0308-2105. - (2023), pp. 1-34. [10.1017/prm.2023.78]
Normalized positive solutions for Schrödinger equations with potentials in unbounded domains
Lancelotti S.;
2023
Abstract
The paper deals with the existence of positive solutions with prescribed $L<^>2$ norm for the Schrodinger equation $$-\Delta u+\lambda u+V(x)u=|u|<^>{p-2}u,\quad u\in H<^>1_0(\Omega),\quad\int_\Omega u<^>2{\rm d}\,x=\rho<^>2,\quad\lambda\in\mathbb{R},$$ where $\Omega =\mathbb {R}<^>N$ or $\mathbb {R}<^>N\setminus \Omega$ is a compact set, $\rho >0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in (2,2+\frac 4 N)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption (D?), or if $\|V\|_{L<^>q}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption (D?) is fulfilled. There are no assumptions on the size of $\mathbb {R}<^>N\setminus \Omega$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $\Omega =\mathbb {R}<^>N$.File | Dimensione | Formato | |
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Lancelotti_Molle_normalized_solutions_R1.pdf
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Descrizione: S. LANCELOTTI e R. MOLLE, Normalized positive solutions for Schrödinger equations with potentials in unbounded domains, Proc. R. Soc. Edinburgh Sec. A (2023).
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https://hdl.handle.net/11583/2984196