Multiple Solutions for a Nonlinear Dirichlet Problem via - Mathematics

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Sketch of the Proof of Theorem A: We finally prove (c). Let us suppose m(u5,J) = k. Writing u5 as u5 = x5 + ψ(x5), Lemma 1 asserts that m(x5, J) ˆ = k and so x5 is a local maximum of J. ˆ Thus, x4 and x5 are points of minima of − ˆ J. The Mountain Pass Theorem implies that there exists a critical point x6 of the mountain pass type for −J. ˆ MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 44➲ ▲ ▲ ➲ ❏ ✘
Sketch of the Proof of Theorem A: We finally prove (c). Let us suppose m(u5,J) = k. Writing u5 as u5 = x5 + ψ(x5), Lemma 1 asserts that m(x5, J) ˆ = k and so x5 is a local maximum of J. ˆ Thus, x4 and x5 are points of minima of − ˆ J. The Mountain Pass Theorem implies that there exists a critical point x6 of the mountain pass type for −J. ˆ Because of nondegeneracy, m(x6,− ˆ J) = 1 (see [Hofer]). Hence, m(x6, ˆ J) = k − 1. Again, by Lemma 1 u6 = x6 + ψ(x6) is a critical point of J whose Morse index is k − 1. Finally, using a degree counting, as before, we get u7. In this way we got (c), and our proof is complete. � MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 45➲ ▲ ▲ ➲ ❏ ✘
Page 1 and 2: MULTIPLE SOLUTIONS FOR A NONLINEAR
Page 3 and 4: In this lecture we study the nonlin
Page 5 and 6: Let λ1,λ2,λ3,... be the sequence
Page 7 and 8: Throughout this paper we assume f s
Page 9 and 10: Theorem A. If f satisfies (f1), (f2
Page 11 and 12: Theorem A. If f satisfies (f1), (f2
Page 13 and 14: REMARK: We prove the existence of u
Page 15 and 16: REMARK: We prove the existence of u
Page 17 and 18: Sketch of the Proof of Theorem A: B
Page 19 and 20: Sketch of the Proof of Theorem A: L
Page 21 and 22: Sketch of the Proof of Theorem A: T
Page 33 and 34: Sketch of the Proof of Theorem A: O
Page 37 and 38: Sketch of the Proof of Theorem A: S
Page 39 and 40: Sketch of the Proof of Theorem A: C
Page 41 and 42: Sketch of the Proof of Theorem A: N
Page 43: Sketch of the Proof of Theorem A: W
Page 47 and 48: References Solutions of a Nonlinear
Page 49: Fin Copyright c○ 2008 Jorge Cossi

Multiple Solutions for a Nonlinear Dirichlet Problem via - Mathematics

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