Registered users receive a variety of benefits including the ability to customize email alerts, create favorite journals list, and save searches.
Please note that a Project Euclid web account does not automatically grant access to full-text content. An institutional or society member subscription is required to view non-Open Access content.
Contact firstname.lastname@example.org with any questions.
Let be a domain in , , . Let and let be a uniformly parabolic operator , , whose coefficients, depending on , are periodic in and satisfy some regularity assumptions. Let be the matrix whose entry is and let be the unit exterior normal to . Let be a -periodic function (that may change sign) defined on whose restriction to belongs to for some large enough . In this paper, we give necessary and sufficient conditions on for the existence of principal eigenvalues for the periodic parabolic Steklov problem on , on , , on . Uniqueness and simplicity of the positive principal eigenvalue is proved and a related maximum principle is given.
The higher order quasilinear elliptic equation subject to Dirichlet boundary conditions may have unique and regular positive solution. If the domain is a ball, we obtain a priori estimate to the radial solutions via blowup. Extensions to systems and general domains are also presented. The basic ingredients are the maximum principle, Moser iterative scheme, an eigenvalue problem, a priori estimates by rescalings, sub/supersolutions, and Krasnosel′skiĭ fixed point theorem.