Multiple Solutions for a Nonlinear Dirichlet Problem via - Mathematics
Multiple Solutions for a Nonlinear Dirichlet Problem via - Mathematics
Multiple Solutions for a Nonlinear Dirichlet Problem via - Mathematics
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Sketch of the Proof of Theorem A:<br />
Lemma 1. (the Lyapunov-Schmidt reduction method)<br />
Let H be a real separable Hilbert space. Let X and Y be closed<br />
subspaces of H such that H = X ⊕Y . Let J : H → R be a functional of<br />
class C 2 . If there are m > 0 and α > 1 such that<br />
〈∇J(x + y) − ∇J(x + y1),y − y1〉 ≥ m�y − y1� α<br />
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 19➲<br />
<strong>for</strong> all x ∈ X, y, y1 ∈ Y<br />
▲ ▲<br />
(0.6)<br />
➲ ❏ ✘