Summability properties of solutions of second-order elliptic equations
The authors consider the Dirichlet problem \[ \sum^{n}_{i,j=1}(\partial /\partial x_ i)(a_{ij}\partial u/\partial x_ j)+V(x)u+\lambda u=f,\quad u|_{\partial \Omega}=0 \] where the function V satisfies the following condition: If for any \(\epsilon >0\) there exists c(\(\epsilon)\) such that \[ -\int_{\Omega}V^ 2_{\phi} dx\leq \epsilon \int_{\Omega}| {\bar \nabla}\phi |^ 2 dx+c(\epsilon)\int_{\Omega}\phi^ 2 dx \] for each \(\phi\) belonging to \(C_ 0^{\infty}(\Omega).\) Under some additional conditions on V the authors obtain two estimates of the solution to the problem. In particular, the authors prove the estimate of the weak solution \[ \int_{\Omega}| x|^{-2} | u(x)|^ q dx\leq K \exp (qj/(j+q))\int^{1/2}_{j/(q+2j)}t^{-2} \ln [t^{-1}c(t)]dt. \] If u is a weak solution to the Dirichlet problem for an equation \(\Delta u=f\) in \(L^ 2\) then they prove an estimate \(\int_{\Omega} | x|^{-2} (e^{\beta | u|}- 1)dx<\infty,\) \(\forall \beta <(n-2)^ 2\pi^{n/2}/(\Gamma (1+n/2)\| f\|_{n/2},\omega)^ 3,\quad f\in L_{\omega}^{n/2}.\)
scientific article; zbMATH DE number 4071442
scientific article