Multiple Solutions for a Nonlinear Dirichlet Problem via - Mathematics

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Sketch of the Proof of Theorem A: Consequently, there must exist another critical point u5 of J whose local degree is Thus, m(u5,J) is even. Since 〈D 2 J(u)y,y〉 ≥ (1 − γ 1 = (−1) m(u5,J) . λk+1 )�y� 2 ; ∀u ∈ H ∀y ∈ Y. (0.13) it follows that m(u5,J) ≤ k. Observe that u5 is a sign-changing solution of (0.1). This prove Theorem A part (b). MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 40➲ ▲ ▲ ➲ ❏ ✘
Sketch of the Proof of Theorem A: Now suppose u4 is of one sign. Without loss of generality we can assume u4 ∈ S2. Let P be a subregion of S2 such that P ∩ K = {u2,u4}. From (0.3), (0.4), (0.12) and the excision property of Leray-Schauder degree it follows that −1 = d(∇J,S2,0) = dloc(∇J,u2) + dloc(∇J,u4) + d(∇J,S2 \ P,0) = −1 + (−1) k + d(∇J,S2 \ P,0). Hence, the existence property of the Leray-Schauder degree implies there must be another signed solution of (0.1). MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 41➲ ▲ ▲ ➲ ❏ ✘
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: Sketch of the Proof of Theorem A: C
Page 43 and 44: Sketch of the Proof of Theorem A: W
Page 45 and 46: 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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