Multi-bump solutions for nonlinear elliptic equations involving critical Sobolev exponents. (July 2022) - Jin, Sangdon
Abstract: We study the existence of multi-bump solutions of − Δ u − λ u = u p, u > 0 in Ω t, u = 0 on ∂ Ω t, where λ is a real constant, N ≥ 3, p = N + 2 N − 2, and Ω t is a domain that expands as t → ∞ . It is known that the topology of the domain Ω t 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, Ω t b a l l ≡ t q + v | q ∈ M, v ∈ ( T q M ) ⊥, | v | < 1 or Ω t a n n = t q + v | q ∈ M, v ∈ ( T q M ) ⊥, a < | v | < a + 1 , where a > 0 and M is a k -dimensional compact smooth submanifold of R N without boundary ( k = 1, ⋯, N − 1 ) .