Contributions to Nonlinear Analysis: A Tribute to D.G. de by Thierry Cazenave, David Costa, Orlando Lopes, Raúl

By Thierry Cazenave, David Costa, Orlando Lopes, Raúl Manásevich, Paul Rabinowitz, Bernhard Ruf, Carlos Tomei

This quantity includes learn and survey articles within the fields of nonlinear research and nonlinear differential equations, written by means of the world over finished researchers. The articles mirror the state-of-the-art in those vital fields of present learn. the quantity is devoted to D.G. de Figueiredo, thereby honoring the $64000 contributions and lasting impression of a distinct mathematician.

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Extra info for Contributions to Nonlinear Analysis: A Tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday (Progress in Nonlinear Differential Equations and Their Applications)

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These invariants of the motion are analyzed in [3]. Here we shall only write the expressions of the energy and of the momentum. • Energy: E (A, ϕ) = ∂A ∂t 1 2 2 2 2 2 − |∇ϕ| + |∇ × A| − W |A| − ϕ2 dx. (39) • Momentum: 3 P (A, ϕ) = i=1 ∂Ai ∂ϕ + ∂t ∂xi ∇Ai dx. (40) Now let us consider a constant of the motion, the charge, which is not related to the invariance of (25) under the action of the Poincar`e group. The charge is defined as C (A, ϕ) = ρ (A, ϕ) dx = 2 |A| − ϕ2 ϕdx. W (41) The charge C = C (A, ϕ) is constant along the solutions A, ϕ of (26), (27).

Souto Proof. ∇φ + |un |p(x)−2 un φ dx = λ IRN |un |q(x)−1 un φdx + on (1). ∇φ + |u|p(x)−2 uφ dx = λ IRN |u|q(x)−1 uφdx. 1, the function u can not be zero. 2. 2 Let {un } be a (P S)c sequence for Iλ converging weakly to 0 in W 1,p(x) (IRN ). Assume the limit lim q(x) = s. (Q2 ) |x|→∞ Then c ≥ c∞ , where c∞ denotes the minimax level of the Euler–Lagrange functional I∞ . Proof. un → 0. un = |∇un |p(x) − |∇un |m dx + {|x|≤R} |un |p(x) − |un |m dx +λ IRN |un |s+1 − |un |q(x)+1) dx. 5) are on (1). 6) where on (1) = |x|≤R |un |s+1 − |un |q(x)+1 dx.

50 V. Bienci and D. Fortunato Moreover, let F and F be the sets of the fixed points for the actions Tg and Tg respectively, namely F = u : R3 → R3 : Tg u = u for all g ∈ O(3) F = w : R3 → R : Tg w = w for all g ∈ O(3) . And finally we set V = u ∈ D(R3 , R3 ) ∩ F : ∇ · u = 0 W = Dp,q (R3 , R) ∩ F . It can be shown that for all g ∈ O(3) , u : R3 → R3 and w : R3 → R Tg (∇ · u) = ∇ · (Tg u) ∇(Tg w) = Tg (∇w) where ∇· denotes the divergence operator. So (u ∈ F and w ∈ F ) =⇒ (u + ∇w ∈ F ) . (66) The following compactness result can be proved by adapting to our case a well known radial lemma ([4]).

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