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- W4386408733 abstract "The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation $$-Delta u+lambda u+V(x)u=|u|^{p-2}u,quad uin H^1_0(Omega),quadint_Omega u^2{rm d},x=rho^2,quadlambdainmathbb{R},$$ where $Omega =mathbb {R}^N$ or $mathbb {R}^Nsetminus Omega$ is a compact set, $rho >0$ , $Vge 0$ (also $Vequiv 0$ is allowed), $pin (2,2+frac 4 N)$ . The existence of a positive solution $bar u$ is proved when $V$ verifies a suitable decay assumption ( D ρ ), or if $|V|_{L^q}$ is small, for some $qge frac N2$ ( $q>1$ if $N=2$ ). No smallness assumption on $V$ is required if the decay assumption ( D ρ ) is fulfilled. There are no assumptions on the size of $mathbb {R}^Nsetminus Omega$ . The solution $bar u$ is a bound state and no ground state solution exists, up to the autonomous case $Vequiv 0$ and $Omega =mathbb {R}^N$ ." @default.
- W4386408733 created "2023-09-05" @default.
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- W4386408733 date "2023-09-04" @default.
- W4386408733 modified "2023-10-18" @default.
- W4386408733 title "Normalized positive solutions for Schrödinger equations with potentials in unbounded domains" @default.
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- W4386408733 doi "https://doi.org/10.1017/prm.2023.78" @default.
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