Boundary value problems for Schrodinger equations with Hardy type potentials

Abstract: We consider equations of the form− $L_V u=\tau$ where $L_V= \Delta +V$ and $\tau$ is a Radon measure in a Lipschitz bounded domain $D\subset\mathbb{R}^N$. The assumptions on $V$ include the condition $|V(x)|\leq a \delta (x)^{-2}$ - where $a >0$ and $\delta (x) = \mbox{dist} (x,\partial D)$. An additional condition guarantees the existence of a ground state $\Phi_V$. The model example is $V=\gamma \delta (x)^{-2}$ where $\gamma< cH$ ($cH$=Hardy constant).

For positive solutions of the equation we define the $L_V$ boundary trace. If $\int_D \Phi_V d\tau<\infty$, the boundary trace is well defined as a positive, bounded measure $\nu$ on $\partial D$. We consider the corresponding b.v.p., namely, $-L+V u=\tau$ in $D$,$u=\nu$ on $\partial D$. We show that, for $\tau$ and $\nu$ as above, the b.v.p. has a unique solution. Further, under some conditions on the ground state -satisfied for a large family of potentials - we obtain sharp two sided estimates of positive solutions of the b.v.p. Finally we discuss some applications to semilinear problems involving the operator $L_V$.

analysis of PDEsdynamical systemsfunctional analysisoptimization and controlspectral theory

Audience: advanced learners

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