Multi-bump solutions for nonlinear elliptic equations involving critical Sobolev exponents

Abstract

We study the existence of multi-bump solutions of −Δu−λu=up,u>0inΩt,u=0on∂Ωt,where λ is a real constant, N≥3, p=N+2N−2 , and Ωt is a domain that expands as t→∞. It is known that the topology of the domain Ωt affects the existence and non-existence of solutions. In this article, we prove the existence of multi-bump solutions for more general domains; for example, Ωtball≡{tq+v|q∈M,v∈(TqM)⊥,|v|<1} or Ωtann={tq+v|q∈M,v∈(TqM)⊥,a<|v|<a+1}, where a>0 and M is a k-dimensional compact smooth submanifold of RN without boundary (k=1,⋯,N−1).