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

This quantity is a suite of articles provided on the Workshop for Nonlinear research held in João Pessoa, Brazil, in September 2012. The effect of Bernhard Ruf, to whom this quantity is devoted at the celebration of his sixtieth birthday, is perceptible during the assortment by means of the alternative of subject matters and strategies. the numerous participants reflect on glossy subject matters within the calculus of adaptations, topological tools and regularity research, including novel functions of partial differential equations. based on the culture of the workshop, emphasis is given to elliptic operators inserted in numerous contexts, either theoretical and utilized. issues comprise semi-linear and entirely nonlinear equations and structures with varied nonlinearities, at sub- and supercritical exponents, with spectral interactions of Ambrosetti-Prodi style. additionally taken care of are analytic points in addition to functions akin to diffusion difficulties in mathematical genetics and finance and evolution equations with regards to electromechanical devices.

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5) R and R g(x, un )v → R where u ∈ K is the weak limit of (un ) in H 1 (R). 1. The weak limit u is null in Oc , that is, u(t) = 0 ∀t ∈ Oc . Hence, u ∈ H01 (O). In fact, for each m ∈ N, we deﬁne Δm = t ∈ R; V (t) > 1 m . It is immediate to see that ∞ P = {t ∈ R; V (t) > 0} = Δm . 1. The last inequality, together with Fatou’s Lemma, lead to |u|2 = 0 ∀m ∈ N. e. in P . Now, the claim follows using the continuity of u. 7), R |un |2 + R (1 + λn V )|un |2 ≤ R (1 + λn V )un u + R un u − R g(x, un )(u − un ).

J0 |EG ) by J1 (resp. J0 ). In the following, we denote by WG1,2 (Ω) = W 1,2 (Ω) ∩ Fix(G). L. 2] (see also [6] for related results). Since EG = W01,2 (Ω) ∩ Fix(G) → WG1,2 (Ω), it follows that EG → → Lt (Ω) for 2 < t < 2∗ . 1. 3. 1, it follows that dϕ1 is compact from EG in EG . 46 S. Barile and A. Salvatore Proof. , E → L t0 −1 (Ω). , 2 2∗ < t0 < , p−1 p−1 ⎪ 2∗ ⎪ ⎩ < t0 < 2, 2∗ − 1 is solvable because, as 2 < p < 2∗ , max 2 2∗ , ∗ p−1 2 −1 < min 2∗ , 2 . 20]). 4. 7) does not hold for t = 2. 3. 9]) to the functional J0 .

4) d≥ 1 un + 2 2 λ − 1 θ f (un )un − Ω 1 2k (1 + λV (x))|un |2 + on (1). O. A. 5), d≥ 1 1 1 − − 2 θ 2k un + 2 λ − zn un+ + on (1). θ Since k > θ−2 and zn → 0 in Eλ , the last inequality implies that (un + ) is bounded in Eλ . Therefore, (un ) is bounded in Eλ . Now, we will show that (un ) has a subsequence that converges strongly in Eλ . Since (un − ) converges to 0 in Eλ , without loss of generality, we will assume that R un ≥ 0 for all n ∈ N. We begin by ﬁxing R > 0 so large in order that Ω ⊂ − R 2, 2 1 and a function η ∈ C (R, R) satisfying • • • • 0 ≤ η(t) ≤ 1, ∀t ∈ R; η(t) = 0, |t| ≤ R2 ; η(t) = 1, |t| ≥ R; |η (t)| ≤ C R , ∀t ∈ R.