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Table Of Integrals Series And Products

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

/pdf/the-dirichlet-problem-for-the-fractional-laplacian-1w4qdfqmwi.pdf

The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

/pdf/nonlinear-partial-differential-equations-with-applications-204hhsvanv.pdf

Nonlinear partial differential equations with applications

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

Ground state solutions for some indefinite variational problems

We present a unified approach to the method of Nehari manifold for functionals which have a local minimum at 0 and we give several examples where this method is applied to the problem of finding ground states and multiple solutions for nonlinear elliptic boundary value problems. We also consider a recent generalization of this method to problems where 0 is a saddle point of the functional.

The method of Nehari manifold

We investigate the symmetry properties of several radially symmetric minimization problems. The minimizers which we obtain are nodal solutions of superlinear elliptic problems, or eigenfunctions of weighted asymmetric eigenvalue problems, or they lie on the first curve in the Fucik spectrum. In all instances, we prove that the minimizers are foliated Schwarz symmetric. We give examples showing that the minimizers are in general not radially symmetric. The basic tool which we use is polarization, a concept going back to Ahlfors. We develop this method of symmetrization for sign changing functions.

Partial symmetry of least energy nodal solutions to some variational problems

We study the set of solutions of the nonlinear elliptic system (P) { − Δ u + λ 1 u = μ 1 u 3 + β v 2 u in Ω , − Δ v + λ 2 v = μ 2 v 3 + β u 2 v in Ω , u , v > 0 in Ω , u = v = 0 on ∂ Ω , in a smooth bounded domain Ω ⊂ R N , N ⩽ 3 , with coupling parameter β ∈ R  This system arises in the study of Bose–Einstein double condensates We show that the value β = − μ 1 μ 2 is critical for the existence of a priori bounds for solutions of (P)  More precisely, we show that for β > − μ 1 μ 2 , solutions of (P) are a priori bounded In contrast, when λ 1 = λ 2 , μ 1 = μ 2 , (P) admits an unbounded sequence of solutions if β ⩽ − μ 1 μ 2 

/pdf/a-priori-bounds-versus-multiple-existence-of-positive-10fg70bkbe.pdf

A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system

We consider the nonlinear elliptic system 

$$\left \{ \begin{aligned} -&\Delta u +u - u^3 -\beta v^2u = 0\quad \rm{in}\, \mathbb B,\\ -&\Delta v +v - v^3 -\beta u^2v = 0\quad \rm{in}\, \mathbb B,\\ &u,v > 0 \quad \rm{in}\, \mathbb B,\quad u=v=0 \quad \rm{on}\, \partial \mathbb B, \end{aligned} \right.$$

where $N\leqq 3$ and $\mathbb B \subset \mathbb {R}^N$ is the unit ball. We show that, for every $\beta \leqq -1$ and $k \in \mathbb N$, the above problem admits a radially symmetric solution (uβ, vβ) such that uβ − vβ changes sign precisely k times in the radial variable. Furthermore, as $\beta \to -\infty$, after passing to a subsequence, uβ → w+ and vβ → w− uniformly in $\mathbb B$, where w = w+− w− has precisely k nodal domains and is a radially symmetric solution of the scalar equation Δw − w + w3 = 0 in $\mathbb B$, w = 0 on $\partial \mathbb B$. Within a Hartree–Fock approximation, the result provides a theoretical indication of phase separation into many nodal domains for Bose–Einstein double condensates with strong repulsion.

/pdf/radial-solutions-and-phase-separation-in-a-system-of-two-33nz8a9oyl.pdf

Tobias Weth

Papers

Ground state solutions for some indefinite variational problems

The method of Nehari manifold

Partial symmetry of least energy nodal solutions to some variational problems

A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system

Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations