Download Contributions to Nonlinear Analysis: A Tribute to D.G. de by Emerson A. M. Abreu, Paulo Cesar Carrião (auth.), Thierry PDF

By Emerson A. M. Abreu, Paulo Cesar Carrião (auth.), Thierry Cazenave, David Costa, Orlando Lopes, Raúl Manásevich, Paul Rabinowitz, Bernhard Ruf, Carlos Tomei (eds.)

This quantity represents a extensive survey of present examine within the fields of nonlinear research and nonlinear differential equations. it really is devoted to Djairo G. de Figueiredo at the social gathering of his 70th birthday. the gathering of 34 study and survey articles displays the big variety of pursuits of Djairo de Figueiredo, together with:
- a variety of forms of nonlinear partial differential equations and platforms, specifically equations of elliptic, parabolic, hyperbolic and combined type;
- equations of Schrödinger, Maxwell, Navier-Stokes, Bernoulli-Euler, Seiberg-Witten, Caffarelli-Kohn-Nirenberg;
- lifestyles, strong point and multiplicity of solutions;
- severe Sobolev progress and hooked up phenomena;
- qualitative homes, regularity and form of solutions;
- inequalities, a-priori estimates and asymptotic behavior;
- a number of purposes to types equivalent to asymptotic membranes, nonlinear plates and inhomogeneous fluids.
The contributions of such a lot of uncommon mathematicians to this quantity record the significance and lasting effect of the mathematical learn of Djairo de Figueiredo. The e-book therefore is a resource of recent principles and effects and may attract graduate scholars and mathematicians attracted to nonlinear problems.

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Extra resources for Contributions to Nonlinear Analysis: A Tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday

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Q(x) + 1 Hereafter, let us denote by I∞ , the Euler–Lagrange functional related to the problem −Δm u + um−1 = λus in IRN , (P∞ ) u ≥ 0, u = 0 and u ∈ W 1,m (IRN ), which is given by I∞ (v) = |v|m |∇v|m + dx − λ m m IRN ∗ ∗ where N < m, s ∈ (m − 1, m − 1) and m = Nm N −m . 1 Iλ satisfies the Mountain Pass Geometry. Proof. 1) and Ω 1 |v|q(x)+1 q− +1 dx ≤ − |v|q(x)+1 ≤C v q(x) + 1 q +1 q− +1 . 2), there exists ρ > 0 such that Iλ (v) ≥ ρ for v = r. 5) it follows Iλ (tφ) → −∞ as t → +∞. 6) it follows the Mountain Pass Geometry.

All the finite energy solutions of the system (43), (44) possess nonnegative energy if and only if assumption W + is satisfied. Moreover if W + is satisfied, the bond energy (42) of any finite energy solution (A, ϕ) of (43), (44) is not positive. Proof. The only if part has already been proved. The if part follows immediately from Prop. 2. By (42) the bond energy is 1 ρ(σ)ϕ + W (σ) dx, 2 Eb = − 2 σ = |A| − ϕ2 , ρ(σ) = W (σ) ϕ. Now, by (53), we have W (σ) dx ≥ 0. Moreover, by (44) we easily derive ρ(σ)ϕdx = W (σ)ϕ2 dx = Then we conclude that Eb is not positive.

We divide this section in subsections considering different situations involving the geometry of the function q(x) at infinity. 1. First Case: Equality at Infinity In this subsection, we show our first result considering the case where the function q is equal to a constant at infinity. The next proposition establishes an important inequality involving the minimax level c∞ of I∞ . 1 Let {un } be a (P S)d sequence for Iλ converging weakly to 0 in W 1,p(x) (IRN ). e in IRN and q(x) ≡ s, for all |x| ≥ R.

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