Multiple Solutions for a Nonlinear Dirichlet Problem via - Mathematics

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Theorem A. If f satisfies (f1), (f2), (f3), k ≥ 4 is an even number, and all the solutions of (0.1) are nondegenerate then problem (0.1) has at least five solutions: u1 = 0, u2 > 0 in Ω, u3 < 0 in Ω, u4, and u5. Moreover, (a) m(u4,J) = k. (b) u5 changes sign, m(u5,J) is even and m(u5,J) ≤ k. (c) If m(u5,J) = k, there are two additional solutions u6 and u7 of (0.1). Moreover m(u6,J) = k − 1. (d) If u4 is of one sign, there are also two additional solutions u6 and u7 of (0.1). MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 12➲ ▲ ▲ ➲ ❏ ✘
REMARK: We prove the existence of u2 and u3 using mountain pass arguments of the Ambrosetti-Rabinowitz type. MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 13➲ ▲ ▲ ➲ ❏ ✘
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: Theorem A. If f satisfies (f1), (f2
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 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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