Boundedness of Stable Solutions to Semilinear Elliptic Equations: A Survey

Abstract

This article is a survey on boundedness results for stable solutions to semilinear elliptic problems.For these solutions, we present the currently known L∞${L^{\infty}}$ estimates that hold for all nonlinearities.Such estimates are known to hold up to dimension 4.They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only beenestablished in the radial case.

Keywords

• stable solutions
• semilinear elliptic equations
• boundedness of solutions
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