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 />
The function ˆ<br />
J : X → R defined by ˆ<br />
J(x) = J(x + ψ(x)) is of class C 2 .<br />
An element x0 ∈ X is a critical point of ˆ<br />
J if and only if x0 + ψ(x0) ∈ H is a<br />
critical point of J.<br />
Moreover,<br />
dloc(∇J,x0 + ψ(x0)) = dloc(∇ ˆ<br />
J,x0). (0.7)<br />
m(u0,J) = m(x0, ˆ<br />
J). (0.8)<br />
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 23➲<br />
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