On least Energy Solutions to A Semilinear Elliptic Equation in A Strip

الأصل · EN

We consider the following semilinear elliptic equation on a strip: {arrayl Δu-u + uᵖ=0 in ⁿ⁻¹ × (0, L), u>0, ∂ u/∂ ν=0 on ∂ (ⁿ⁻¹ × (0, L)) array. where 1< p≤ N+2/N-2. When 1<p <N+2/N-2, it is shown that there exists a unique L* >0 such that for L ≤ L*, the least energy solution is trivial, i.e., doesn't depend on xₙ, and for L >L*, the least energy solution is nontrivial. When N ≥ 4, p=N+2/N-2, it is shown that there are two numbers L*<L** such that the least energy solution is trivial when L ≤ L*, the least energy solution is nontrivial when L ∈ (L*, L**], and the least energy solution does not exist when L >L**. A connection with Delaunay surfaces in CMC theory is also made.