On the Effect of the Domain Shape on the Existence of Large Solutions of Some Superlinear Problems

We discuss the existence, uniqueness and multiplicity of large solutions of the equation u + u p = 0 in " u > 0 in " ; u = 0 on @ " ; where p > 1 and " R 2 is a bounded smooth domain which is obtained by connecting two disconnected domains by a small tunnel. We show that the geometry of the corners dominates the properties of the large solutions (bump-like solutions). When two corners are both open, we prove that there are many solutions with arbitrary number of bumps, while when the two corners are both close, there exists at least one single-bump solution. When one corner is open and another corner is close, we prove that there exists no single-bump solutions.