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- W2953192579 abstract "Let $Omega$ be a domain in $mathbb{R}^d$, $dgeq 2$, and $1<p<infty$. Fix $Vin L_{mathrm{loc}}^infty(Omega)$. Consider the functional $Q$ and its G^{a}teaux derivative $Q^prime$ given by $$Q(u):=int_Omega (|nabla u|^p+V|u|^p)dx, frac{1}{p}Q^prime (u):=-nablacdot(|nabla u|^{p-2}nabla u)+V|u|^{p-2}u.$$ If $Qge 0$ on $C_0^{infty}(Omega)$, then either there is a positive continuous function $W$ such that $int W|u|^p mathrm{d}xleq Q(u)$ for all $uin C_0^{infty}(Omega)$, or there is a sequence $u_kin C_0^{infty}(Omega)$ and a function $v>0$ satisfying $Q^prime (v)=0$, such that $Q(u_k)to 0$, and $u_kto v$ in $L^p_mathrm{loc}(Omega$). In the latter case, $v$ is (up to a multiplicative constant) the unique positive supersolution of the equation $Q^prime (u)=0$ in $Omega$, and one has for $Q$ an inequality of Poincar'e type: there exists a positive continuous function $W$ such that for every $psiin C_0^infty(Omega)$ satisfying $int psi v mathrm{d}x neq 0$ there exists a constant $C>0$ such that $C^{-1}int W|u|^p mathrm{d}xle Q(u)+C|int u psi mathrm{d}x|^p$ for all $uin C_0^infty(Omega)$. As a consequence, we prove positivity properties for the quasilinear operator $Q^prime$ that are known to hold for general subcritical resp. critical second-order linear elliptic operators." @default.
- W2953192579 created "2019-06-27" @default.
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- W2953192579 date "2005-11-02" @default.
- W2953192579 modified "2023-09-26" @default.
- W2953192579 title "Ground state alternative for p-Laplacian with potential term" @default.
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- W2953192579 doi "https://doi.org/10.48550/arxiv.math/0511039" @default.
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