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 />
We finally prove (c). Let us suppose m(u5,J) = k. Writing u5 as<br />
u5 = x5 + ψ(x5), Lemma 1 asserts that m(x5, J) ˆ = k and so x5 is a local<br />
maximum of J. ˆ<br />
Thus, x4 and x5 are points of minima of − ˆ<br />
J. The Mountain Pass<br />
Theorem implies that there exists a critical point x6 of the mountain pass<br />
type <strong>for</strong> −J. ˆ<br />
Because of nondegeneracy, m(x6,− ˆ<br />
J) = 1 (see [Hofer]). Hence,<br />
m(x6, ˆ<br />
J) = k − 1. Again, by Lemma 1 u6 = x6 + ψ(x6) is a critical point of<br />
J whose Morse index is k − 1. Finally, using a degree counting, as<br />
be<strong>for</strong>e, we get u7. In this way we got (c), and our proof is complete. �<br />
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 45➲<br />
▲ ▲<br />
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