Download Analysis and Topology in Nonlinear Differential Equations: A by Djairo G de Figueiredo, João Marcos do Ó, Carlos Tomei PDF

By Djairo G de Figueiredo, João Marcos do Ó, Carlos Tomei

This quantity is a suite of articles awarded on the Workshop for Nonlinear research held in João Pessoa, Brazil, in September 2012. The impact of Bernhard Ruf, to whom this quantity is devoted at the get together of his sixtieth birthday, is perceptible through the assortment by way of the alternative of topics and strategies. the various members give some thought to sleek subject matters within the calculus of diversifications, topological tools and regularity research, including novel purposes of partial differential equations. based on the culture of the workshop, emphasis is given to elliptic operators inserted in several contexts, either theoretical and utilized. subject matters contain semi-linear and completely nonlinear equations and structures with various nonlinearities, at sub- and supercritical exponents, with spectral interactions of Ambrosetti-Prodi style. additionally handled are analytic features in addition to functions comparable to diffusion difficulties in mathematical genetics and finance and evolution equations regarding electromechanical devices.

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Additional resources for Analysis and Topology in Nonlinear Differential Equations: A Tribute to Bernhard Ruf on the Occasion of his 60th Birthday

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On the other hand, by a rotation of coordinates we may assume that the x1 , . . , xN −1 axes lie along principal directions corresponding to λ1 , . . , λN −1 at z. So, the Hessian matrix can be described as ⎤ ⎡ 0 λ1 0 · · · ⎢ 0 λ2 · · · 0 ⎥ ⎥ ⎢ D2 G(x) = ⎢ . ⎥. .. .. ⎦ ⎣ .. . 0 0 ··· λN −1 Thus, at z = (0, 0) we have N −1 D2 G(0)y, y E(w, y) dy RN −1 = i=1 RN −1 λi yi2 E(w, y) dy. ´ and E. Medeiros E. M. do O 20 By the definition of the mass moment of inertia we have that the moment of inertia about the yi -axis, i = 1, .

From this, Iλ (tϕ+ ) → −∞ as t → +∞, and thus, setting e = tϕ+ for t large enough, we derive that e > r and Iλ (e) < Iλ (uλ ). 2]. 1. 1, Problem (PA ) has a positive solution at the mountain pass level for all λ > 0, that is, there is wλ ∈ K verifying Iλ (wλ ) = cλ and Iλ (wλ )(v − wλ ) ≥ 0 ∀v ∈ K, where cλ is the mountain pass level of Iλ . Multiplicity of Positive Solutions 33 Proof. 1 with the Mountain Pass Theorem, we have that the mountain pass level cλ associated with Iλ is a critical value, hence there is wλ ∈ K such that Iλ (wλ ) = cλ and Iλ (wλ )(v − wλ ) ≥ 0 ∀v ∈ K.

1) Ω Let us point out that the norm in W 1,2 (Ω) given by |∇u| dx + |u| dx 2 Ω 2 1 2 Ω is equivalent to the norm · . Indeed, as Ω is a subset of RN which lies between two hyperplanes, by Poincar´e inequality a constant k > 0 exists such that |∇u|2 dx + Ω |u|2 dx ≤ k Ω |∇u|2 dx Ω (see [1, p. 159]). Hence, · is equivalent to the classical norm in W01,2 (Ω). For this reason, even if we have a problem with “zero mass” we don’t use the space D1,2 (Ω) but we study (Pf ) in W01,2 (Ω) as in the “positive mass case”, thus simplifying the argument in [3].

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