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Zbl 1017.35033
Souplet, Philippe; Zhang, Qi S.
Stability for semilinear parabolic equations with decaying potentials in $\bbfR^n$ and dynamical approach to the existence of ground states. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19, No.5, 683-703 (2002). ISSN 0294-1449

The elliptic problem: $\Delta u-Vu+u^p=0$ in $\bbfR^n$, with $1<p<(n+2)/(n-2)$, $n\ge 2$, and nonnegative bounded potential $V(x)$ which may decay to 0 at infinity is considered. The main result is that if $V$ satisfies $a_1/(1+ |x|^b)\le V(x)\le a_2$ with $0\le b<2(n-1) (p-1)/(p+3)$ and $V$ radial, then it admits a (ground state) positive solution. The result relies on the study of global solutions of the associated parabolic problem. Indeed, the authors show that, under suitable conditions on $V$ (not necessarily radial), this problem admits global positive solutions and that when $V$ and $u(\cdot,0)$ are radial some global solutions have $\omega$-limit sets containing a positive equilibrium.
[Petr Girg (Plzen)]

MSC 2000:: *35J15 Second order elliptic equations, general
35K10 Second order parabolic equations, general
35B05 General behavior of solutions of PDE
35A05 General existence and uniqueness theorems (PDE)

Keywords: decaying potentials; elliptic equations; equilibria for parabolic equations

Cited in: Zbl pre06129499

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