Multiple Solutions for a Nonlinear Dirichlet Problem via - Mathematics

Sketch of the Proof of Theorem A:
The function ˆ
J : X → R defined by ˆ
J(x) = J(x + ψ(x)) is of class C 2 , and
x0 ∈ X is a critical point of ˆ
J if and only if x0 + ψ(x0) ∈ H is a critical
point of J.
Because f ′ (∞) ∈ (λk,λk+1) it follows that
ˆ
J(x) → −∞ as �x� → ∞ for x ∈ X. (0.11)
Because dimX < ∞, there exists x4 ∈ X such that
J(x4) ˆ = máxJ.
ˆ
X
Hence, u4 = x4 + ψ(x4) is a critical point of J.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 31➲
▲ ▲
➲ ❏ ✘