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2020 | Book

Linking Systems Read chapter Read first chapter

Critical Point Theory

Sandwich and Linking Systems

Author: Martin Schechter

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This monograph collects cutting-edge results and techniques for solving nonlinear partial differential equations using critical points. Including many of the author’s own contributions, a range of proofs are conveniently collected here, Because the material is approached with rigor, this book will serve as an invaluable resource for exploring recent developments in this active area of research, as well as the numerous ways in which critical point theory can be applied.
Different methods for finding critical points are presented in the first six chapters. The specific situations in which these methods are applicable is explained in detail. Focus then shifts toward the book’s main subject: applications to problems in mathematics and physics. These include topics such as Schrödinger equations, Hamiltonian systems, elliptic systems, nonlinear wave equations, nonlinear optics, semilinear PDEs, boundary value problems, and equations with multiple solutions. Readers will find this collection of applications convenient and thorough, with detailed proofs appearing throughout.
Critical Point Theory will be ideal for graduate students and researchers interested in solving differential equations, and for those studying variational methods. An understanding of fundamental mathematical analysis is assumed. In particular, the basic properties of Hilbert and Banach spaces are used.

Table of Contents

Frontmatter
Chapter 1. Linking Systems
Abstract
Many problems arising in science and engineering call for the solving of the Euler equations of functionals, i.e., equations of the form
$$\displaystyle G'(u)=0, $$
where G(u) is a C 1 functional (usually representing the energy) arising from the given data. As an illustration, the equation
$$\displaystyle -\Delta u(x)=f(x,u(x)) $$
is the Euler equation of the functional
$$\displaystyle G(u)=\frac {1}{2}\|\nabla u\|{ }^2-\int F(x,u(x)) \,dx $$
on an appropriate space, where
$$\displaystyle F(x,t) = \int ^{t}_{0}f(x,s)\,ds, $$
and the norm is that of L 2. The solving of the Euler equations is tantamount to finding critical points of the corresponding functional. The history of this approach goes back to the calculus of variations. Then the desire was to find extrema of certain expressions G (functionals). Following the approach of calculus, one tried to find all critical points of G, substitute them back in G, and see which one gives the required extremum. This worked fairly well in one dimension where G′(u) = 0 is an ordinary differential equation. However, in higher dimensions, it turned out that it was easier to find the extrema of G than solve G′(u) = 0. This led to the approach of solving equations of the form G′(u) = 0 by finding extrema of G.
Martin Schechter
Chapter 2. Sandwich Systems
Abstract
As we saw in Chap. 1, an important requirement for a linking pair is that they must separate the functional, i.e., they must satisfy
$$\displaystyle a_0 := \sup _{A} G \le b_0 := \inf _{B}G. $$
If a linking pair does not separate the functional, nothing can be said concerning a potential critical point. This raises the questions, “Is there anything one can do if one cannot find linking sets that separate the functional?” “Are there sets that can lead to critical sequences even though they do not separate the functional?” Fortunately, there are.
Martin Schechter
Chapter 3. Linking Sandwich Sets
Abstract
The theory presented in Chap. 2 gives precise criteria for the existence of sandwich pairs, but it does not provide a list of such pairs. There is good reason for this. The criterion
$$\displaystyle \sigma (1)\, A \cap B \ne \phi , \quad \forall \sigma \in \Sigma _Q $$
is very difficult to verify in practice, while the corresponding statement for linking pairs is easier. We were able to provide a reasonable list of linking sets at the end of Chap. 1, but we have not yet been able to do so for sandwich sets. In this chapter we shall focus our attention on this matter. It turns out that we can obtain a lot of help from the theory of linking sets.
Martin Schechter
Chapter 4. The Monotonicity Trick
Abstract
The use of linking or sandwich pairs cannot produce critical points by themselves. The most they can produce are sequences satisfying
$$\displaystyle G(u_k)\to a,\quad (1+\|u_k\|)G'(u_k)\to 0. $$
If such a sequence has a convergent subsequence, we obtain a critical point. Lacking such information, we cannot eliminate the possibility that
$$\displaystyle \|u_k\| \to \infty , $$
which destroys any hope of obtaining a critical point from this sequence. On the other hand, knowing that the sequence is bounded does not guarantee a critical point either. But there is a difference. In many applications, knowing that a sequence satisfying (4.1) is bounded allows one to obtain a convergent subsequence. This is just what is needed. For such applications it would be very helpful if we could obtain a bounded sequence satisfying (4.1). This leads to the question: Is there anything we can do to obtain such a sequence?
Martin Schechter
Chapter 5. Infinite Dimensional Linking
Abstract
Let N be a closed, separable subspace of a Hilbert space E. We can define a new norm |v|w satisfying |v|w ≤∥v∥, ∀v ∈ N and such that the topology induced by this norm is equivalent to the weak topology of N on bounded subsets of N. This can be done as follows: Let {e k} be an orthonormal basis for N. Define
$$\displaystyle (u,v)_w=\sum _{k=1}^{\infty }\frac {(u,e_k)(v, e_k)}{2^k}, \quad u,v\in N. $$
Martin Schechter
Chapter 6. Differential Equations
Abstract
We shall make use of various extensions of Picard’s theorem in a Banach space. Some are well known.
Martin Schechter
Chapter 7. Schrödinger Equations
Abstract
We now consider some applications of the materials presented in Chaps. 1–6. We wish to show how powerful these methods are in obtaining results better than those given by other methods. In Chaps. 7–9 we deal with some problems involving Schrödinger equations.
Martin Schechter
Chapter 8. Zero in the Spectrum
Abstract
In the previous chapter we noted that in our study of semilinear elliptic partial differential equations of the form
$$\displaystyle \mathcal A u = f(x,u), u \in D $$
in unbounded domains, we required that the resolvent set of \(\mathcal A\) not be empty. For convenience, we assumed that it contain 0. This allowed us to choose an interval \((a,b) \subset \rho (\mathcal A),\) where a < 0 < b. We let \( D = D(|\mathcal A|{ }^{(1/2)}).\) With the scalar product \((u,v)_D= (|\mathcal A|{ }^{(1/2)}u,|\mathcal A|{ }^{(1/2)}v),\) it became a Hilbert space. We let
$$\displaystyle N= E(-\infty , a], \quad M = E[b, \infty ) $$
be the negative and positive invariant subspaces of \(\mathcal A.\) Then
$$\displaystyle (\mathcal A v,v) \le a \|v\|{ }^2, \quad v \in D \cap N, $$
and
$$\displaystyle (\mathcal A w,w) \ge b\|w\|{ }^2, \quad w \in D \cap M. $$
The hypotheses of our theorems depended on a and b using the fact that 0 was embedded in \(\rho (\mathcal A).\) The purpose of the present chapter is to study the situation when 0 is a boundary point of \(\rho (\mathcal A)\) and the arguments do not work.
Martin Schechter
Chapter 9. Global Solutions
Abstract
In the previous two chapters, we studied the semilinear Schrödinger equation
$$\displaystyle - \Delta u + V(x) u = f(x,u), \quad u \in H^1({\mathbb R^n}), $$
where V (x) is a given potential. We needed the linear operator − Δu + V (x)u to have a nonempty resolvent. To achieve this, we assumed that V (x) was periodic in x. This forced us to assume the same for f(x, u), and we had to deal with several restrictions in our methods. In this chapter we study the equation without making any periodicity assumptions on the potential or on the nonlinear term. But we must be assured that the linear operator has nonempty resolvent. To accomplish this, we make assumptions on V (x) which guarantee that the essential spectrum of − Δu + V (x)u is the same as that of − Δu. In other words, our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in [0, ∞) being the absolutely continuous part of the spectrum. There may be no negative eigenvalues, a finite number of negative eigenvalues, or an infinite number of negative eigenvalues. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.
Martin Schechter
Chapter 10. Second Order Hamiltonian Systems
Abstract
We consider the system
$$\displaystyle -\ddot x(t)=\; B(t)x(t) +\nabla _xV(t,x(t)), $$
where
$$\displaystyle x(t)=(x_1(t),\cdots ,x_n(t)) $$
is a map from I = [0, T] to \(\mathbb R^n\) such that each component x j(t) is a periodic function in H 1 with period T, and the function V (t, x) = V (t, x 1, ⋯ , x n) is continuous from \(\mathbb R^{n+1}\) to \(\mathbb R\) with
$$\displaystyle \nabla _xV(t,x)=(\partial V/\partial x_1,\cdots ,\partial V/ \partial x_n) \in C(\mathbb R^{n+1},\mathbb R^n). $$
For each \(x \in \mathbb R^n,\) the function V (t, x) is periodic in t with period T.
Martin Schechter
Chapter 11. Core Functions
Abstract
Let \(\mathcal A\) be a self-adjoint operator on L 2( Ω), where Ω is a bounded domain in \(\mathbb R^n.\) Let f(x, t) be a Carathéodory function on \( \Omega \times \mathbb R.\) A well-known semilinear problem is to solve
$$\displaystyle \mathcal A u = f(x,u), \quad u \in D(\mathcal A). $$
In particular, one searches for properties of f(x, t) which guarantee the existence of solutions. This is not a trivial situation; there does not appear to be a criterion which tells us whether or not the problem is solvable.
Martin Schechter
Chapter 12. Custom Monotonicity Methods
Abstract
Consider the problem
$$\displaystyle - \Delta u=f(x,u), \; x \in \Omega \,; \quad u=0 \;\;\mathrm {on}\; \; \partial \Omega , $$
where \(\Omega \subset \mathbb R^n\) is a bounded domain whose boundary is a smooth manifold, and f(x, t) is a continuous function on \(\bar \Omega \times \mathbb R.\) The following theorem will be a corollary of the results of this chapter.
Martin Schechter
Chapter 13. Elliptic Systems
Abstract
In this chapter we show how monotonicity methods combined with infinite dimensional sandwich pairs can be used to solve very general systems of equations whether or not they are semibounded.
Martin Schechter
Chapter 14. Flows and Critical Points
Abstract
In this chapter we study equations of the form
$$\displaystyle \begin{aligned} \left\{\begin{aligned} - {\Delta_{\mathit p}} u & = f(x,u)\;\quad && \;\text{in }\; \Omega\\ {} u & = 0 && \text{on }\; \partial{\Omega} \end{aligned}\right \}, \end{aligned}$$
where Ω is a bounded domain in \(\mathbb R^n,\, n \ge 1\), \({\Delta _{\mathit p}} u = \operatorname {\mathrm {div}} \big (|\nabla u|{ }^{p-2}\, \nabla u\big )\) is the p-Laplacian of u, 1 < p < ∞, and f is a Carathéodory function on \(\Omega \times \mathbb R\) with subcritical growth. We show that sandwich pairs can be used in solving such problems.
Martin Schechter
Chapter 15. The Semilinear Wave Equation
Abstract
In this chapter we study periodic solutions of the Dirichlet problem for the semilinear wave equation:
$$\displaystyle \square u-\mu u:= u_{tt}- u_{rr} -\mu u = p(t,r,u),\quad t \in \mathbb R, \quad 0 <r < R, $$
$$\displaystyle u(t,R) = u(t,0) = 0, \quad t\in {\mathbb R}, $$
$$\displaystyle u(t+T,r) = u(t,r),\quad t\in {\mathbb R},\quad 0 \le r \le R. $$
Martin Schechter
Chapter 16. Nonlinear Optics
Abstract
Light waves propagating in a photo refractive crystal are governed by a nonlinear Schrödinger equation of the form
$$\displaystyle i\frac {\partial u}{\partial z} +D \Delta u=g(x, |u|{ }^2)u, $$
where D > 0 is the beam diffraction coefficient and the functions are periodic with respect to the variables \(x=(x_1, x_2) \in \Omega \subset \mathbb R^2.\) Here,
$$\displaystyle g(x, |u|{ }^2)=\frac {K}{ 1+V(x)+|u|{ }^2}, $$
where V (x) is a continuous, nonnegative function periodic in \(\overline \Omega .\) Steady state solutions satisfy the following equation over a periodic domain \(\Omega \subset \mathbb R^2:\)
$$\displaystyle \Delta u= \frac {Pu}{1+V(x)+|u|{ }^2} + \lambda u, $$
where P, λ are parameters. The solutions u are to be periodic in Ω with the same periods as those of Ω. This equation has the trivial solution u = 0.
Martin Schechter
Chapter 17. Radially Symmetric Wave Equations
Abstract
In this chapter we study periodic solutions of the Dirichlet problem for the semilinear wave equation
$$\displaystyle \square u:= u_{tt}-\Delta u = f(t,x,u),\quad t\in {\mathbb R},\quad x\in \mathcal B_R, $$
$$\displaystyle u(t,x) = 0,\quad t\in {\mathbb R},\quad x\in \partial \mathcal B_R, $$
$$\displaystyle u(t+T,x) = u(t,x),\quad t\in {\mathbb R},\quad x\in \mathcal B_R, $$
where
$$\displaystyle \mathcal B_R = \{x\in {\mathbb R}^n:|x|<R\}. $$
Martin Schechter
Chapter 18. Multiple Solutions
Abstract
A typical characteristic of nonlinear problems is the fact that many times there are multiple solutions. Usually it is difficult to obtain any solutions, let alone more than one. In general, it is much harder to determine when there are even two.
Martin Schechter
Backmatter
Metadata
Title
Critical Point Theory
Author
Martin Schechter
Copyright Year
2020
Electronic ISBN
978-3-030-45603-0
Print ISBN
978-3-030-45602-3
DOI
https://doi.org/10.1007/978-3-030-45603-0