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

This quantity is a set 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 during the assortment by way of the alternative of subject matters and methods. the various individuals think about sleek issues within the calculus of diversifications, topological equipment and regularity research, including novel functions of partial differential equations. according to the culture of the workshop, emphasis is given to elliptic operators inserted in numerous contexts, either theoretical and utilized. subject matters comprise semi-linear and entirely nonlinear equations and platforms with various nonlinearities, at sub- and supercritical exponents, with spectral interactions of Ambrosetti-Prodi style. additionally taken care of are analytic features in addition to functions equivalent to diffusion difficulties in mathematical genetics and finance and evolution equations concerning 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**

**Example text**

1. 8) and λn R for some subsequence. e. O . Proof. From now on, we will prove the lemma only for the sequence (un ), because the same arguments can be applied to (wn ). 6), that is, R |un |2 + R (1 + λn V )|un |2 ≤ R (1 + λn V )un u + R un u − R g(x, un )(u − un ). O. A. Corrˆea Using the fact that V (t)u(t) = 0 for all t ∈ R, it follows that R |un |2 + R (1 + λn V )|un |2 ≤ R un u + R un u − R g(x, un )(u − un ). 2, un → u in H 1 (R) for some subsequence. Hence, lim inf n→+∞ R (|un |2 + |un |2 ) = lim n→+∞ R (un u + un u) = R R (|u |2 + |u|2 ), (|u |2 + |u|2 ), and lim n→+∞ R g(x, un )(u − un ) = 0.

O . Proof. From now on, we will prove the lemma only for the sequence (un ), because the same arguments can be applied to (wn ). 6), that is, R |un |2 + R (1 + λn V )|un |2 ≤ R (1 + λn V )un u + R un u − R g(x, un )(u − un ). O. A. Corrˆea Using the fact that V (t)u(t) = 0 for all t ∈ R, it follows that R |un |2 + R (1 + λn V )|un |2 ≤ R un u + R un u − R g(x, un )(u − un ). 2, un → u in H 1 (R) for some subsequence. Hence, lim inf n→+∞ R (|un |2 + |un |2 ) = lim n→+∞ R (un u + un u) = R R (|u |2 + |u|2 ), (|u |2 + |u|2 ), and lim n→+∞ R g(x, un )(u − un ) = 0.

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, . . , N − 1, respectively the polar moment of inertia are given by N −1 Iyi = RN −1 yi2 E(w, y) dy, I0 = N −1 Iyi = i=1 RN −1 i=1 yi2 E(w, y) dy, respectively. Now, using the fact of E(w, y) is a symmetric function, we conclude that Iy1 = · · · = IyN −1 , which implies that I0 = (N − 1)Iy1 .