On the shape of level sets of least energy solutions to some elliptic problems with nearly critical growth

Abstract

In this paper, we study the shape of level sets of the least energy solutions to some nonlinear elliptic boundary value problems with nearly critical growth and a variable coefficient. Under some assumptions of the coefficient function, we prove the strict starshapedness of the superlevel sets with respect to the maximum points of the solutions, if the domain is bounded and convex in . Moreover the superlevel sets are strictly convex if the boundary of the domain has strictly positive Gauss curvature everywhere.

Keywords:

• Least energy solution
• Critical Sobolev exponent
• Starshapedness
• Gauss curvature

Keywords:

• AMS Subject Classifications:35B33
• 35J60

Acknowledgements

The author wishes to thank the referee for pointing out an error in the first version of this article. Preparation of this work was partially supported by JSPS Grant-in-Aid for Scientific Research, No. 17540186.