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Linking Methods in Critical Point Theory

verfasst von: Martin Schechter

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

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As is well known, The Great Divide (a.k.a. The Continental Divide) is formed by the Rocky Mountains stretching from north to south across North America. It creates a virtual "stone wall" so high that wind, rain, snow, etc. cannot cross it. This keeps the weather distinct on both sides. Since railroad trains cannot climb steep grades and tunnels through these mountains are almost formidable, the Canadian Pacific Railroad searched for a mountain pass providing the lowest grade for its tracks. Employees discovered a suitable mountain pass, called the Kicking Horse Pass, el. 5404 ft., near Banff, Alberta. (One can speculate as to the reason for the name.) This pass is also used by the Trans-Canada Highway. At the highest point of the pass the railroad tracks are horizontal with mountains rising on both sides. A mountain stream divides into two branches, one flowing into the Atlantic Ocean and the other into the Pacific. One can literally stand (as the author did) with one foot in the Atlantic Ocean and the other in the Pacific. The author has observed many mountain passes in the Rocky Mountains and Alps. What connections do mountain passes have with nonlinear partial dif ferential equations? To find out, read on ...

Inhaltsverzeichnis

Frontmatter

Chapter 1. Critical Point Theory

Abstract

Many nonlinear problems can be reduced to the form Many nonlinear problems can be reduced to the form

$$G'(u) = 0,$$

(1.1.1)

where G is a C¹-functional on a Banach space E. In this case the problems can be attacked by specialized, important techniques which can produce results where other methods fail. The history of this approach can be traced back to the calculus of variations in which equations of the form (1.1.1) are the Euler-Lagrange equations of the functional G. The original method was to find maxima or minima of G by solving (1.1.1) and then show that some of the solutions are extrema. This approach worked well for one dimensional problems. In this case it is easier to solve (1.1.1) than it is to find a maximum or minimum of G. However, in higher dimensions it was realized quite early that it is easier to find maxima and minima of G than it is to solve (1.1.1). Consequently, the tables were turned, and critical point theory was devoted to finding extrema of G. This approach is called the direct method in the calculus of variations. If an extremum point of G can be identified, it will automatically be a solution of (1.1.1).

Martin Schechter

Chapter 2. Linking

Abstract

It is interesting that the concept of linking is of importance in critical point theory. To the average person two objects are said to be linked if they cannot be pulled apart. This is basically the idea we shall use in finding critical points. Let E be a Banach space. We introduce the set Ф of mappings Γ(t) ∈ C(E x [0, 1], E) with the following properties:

for each t ∈ [0, 1), Γ(t) is a homeomorphism of E onto itself and Γ(t)^-1 is continuous on E x [0, 1)

Γ(0) = I

for each Γ(t) ∈ Ф there is a u₀ ∈ E such that Γ(1)u = u₀ for all u ∈ E and Γ(t)u → u₀ as t → 1 uniformly on bounded subsets of E.

Martin Schechter

Chapter 3. Semilinear Boundary Value Problems

Abstract

Many elliptic semilinear problems can be described in the following way. Let Ω be a domain in Rⁿ and let A be a selfadjoint operator on L²(Ω). We assume that A ≥ λ₀ > 0 and that

$$C_0^\infty (\Omega ) \subset D: = D({A^{1/2}}) \subset {H^{m,2}}(\Omega )$$

(3.1.1)

for some m > 0, where C ₀ ^∞ (Ω) denotes the set of test functions in Ω (i.e., infinitely differentiable functions with compact supports in Ω) and H^m,2(Ω) denotes the Sobolev space described in Appendix I to this chapter. If m is an integer, the norm in H^m,2(Ω) is given by

$${\left\| u \right\|_{m,2}}: = {\left( {\sum\limits_{\left| \mu \right|m} {{{\left\| {{D^u}u} \right\|}^2}} } \right)^{1/2}}.$$

(3.1.2)

Martin Schechter

Chapter 4. Alternative Methods

Abstract

In Chapter II we showed how to construct critical sequences, i.e., sequences that satisfy

$$G\left( {u_k } \right) \to c, - \infty < c \leqslant \infty ,G'\left( {u_k } \right)/\left( {\left\| {u_k } \right\| + 1} \right)^\beta \to 0$$

(4.1.1)

for some β ≥ 0, where G is a C¹-functional on a Banach space E (cf. Section 2.7). For our applications, (4.1.1) leads to a critical point provided the sequence is bounded (cf. Theorem 3.4.1). In the present chapter we shall show that, by fine tuning our arguments, we can obtain an alternative of the form: Either

(a)

there exists a Palais-Smale sequence, i.e., a sequence satisfying

$$G\left( {u_k } \right) \to c, - \infty < c < \infty ,G'\left( {u_k } \right) \to 0,$$

(4.1.2)

(b)

there is a sequence satisfying

$$\begin{gathered}G\left( {u_k } \right) \to c, - \infty < c \leqslant \infty ,\rho _k = \left\| {u_k } \right\| \to \infty \hfill \\G\left( {u_k } \right)/\rho _{\rho _k }^{\beta + 1} \to 0,G'\left( {u_k } \right)/\rho _{\rho _k }^\beta \to 0. \hfill \\\end{gathered}$$

(4.1.3)

Martin Schechter

Chapter 5. Bounded Saddle Point Methods

Abstract

Although a critical sequence does not necessarily lead to a critical point, we saw in Chapter III (cf. Theorem 3.4.1) that in some applications, bounded critical sequences do indeed lead to critical points. Thus one might ask if there are criteria that can be imposed which will produce bounded critical sequences. We study this question in Section 5.2. There we require the two linking sets A, B to be contained in a ball of radius R and impose a boundary condition on the sphere comprising the boundary of the ball to prevent deformations of the sets from exiting the ball. We then show that this indeed produces a bounded Palais-Smale sequence. However, the boundary condition is an additional restriction which asserts itself in the applications. As we shall see in Section 5.8, the restriction is not as severe as those used to cause a Palais-Smale sequence to be bounded. Consequently, the boundary condition pays for itself in applications.

Martin Schechter

Chapter 6. Estimates on Subspaces

Abstract

As a consequence of Theorem 2.7.2, if E = M ⊕ N is a decomposition of a Banach space into closed subspaces such that M ≠ E, N ≠ E, dim N < ∞ and

$$\mathop {\sup }\limits_N G < \infty ,\mathop {\inf }\limits_M G > - \infty ,$$

(6.1.1)

then there is a sequence {u_k} ⊂ E such that

$$G\left( {u_k } \right) \to c,G'\left( {u_k } \right) \to 0.$$

(6.1.2)

Martin Schechter

Chapter 7. The Fučík Spectrum

Abstract

In Section 3.5 and again in Section 6.3, we mentioned the situation in which

$$f\left( {x,t} \right)/t \to {\alpha _ \pm }\left( x \right){\text{ }}as{\text{ }}t \to \pm \infty .$$

(7.1.1)

Martin Schechter

Chapter 8. Resonance

Abstract

In Section 3.6 we briefly discussed the situation in which

$$f\left( {x,t} \right)/t \to \alpha _ \pm \left( x \right){\text{ }}as{\text{ }}t \to \pm \infty .$$

(8.1.1)

Martin Schechter

Chapter 9. Boundary Conditions

Abstract

In Section 3.4 we saw that there was a distinct advantage to obtaining a bounded Palais-Smale sequence. A method of obtaining such sequences was presented in Section 5.2. In keeping with the mathematical principle “There is no free lunch,” we had to impose a boundary condition on a sphere of radius R in order to produce a bounded Palais-Smale sequence. This boundary condition causes additional restrictions in applications. However, in general, these additional restrictions are more than offset by the requirements that the Palais-Smale conditions be satisfied. In this chapter we shall present some applications in which the boundary condition “pays for itself” in that it imposes no additional restriction and the usual theory does not work without it. In the first application we can apply Theorem 5.2.1 directly. This will be presented in the next section. For the second we shall need a theorem which can be called the direct opposite of Theorem 5.2.1 in that it produces a Palais-Smale sequence completely outside a ball of radius R. This will be presented in Section 9.3. The application will be given in Section 9.4.

Martin Schechter

Chapter 10. Multiple Solutions

Abstract

There are various tools that one can use in critical point theory to show that G’(u) = 0 has multiple solutions. One of them is to solve

$$G\left( u \right) = c,G'\left( u \right) = 0$$

(10.1.1)

for different values of c. We have used this device several times, especially when we verified that a solution was not trivial. In this chapter we shall use another method which helps us locate a region in Hilbert space where a particular solution is stuated. If we find a solution in another region we are sure that we have another solution. This process can be repeated. We present the theory in the next section and give applications in Section 10.3.

Martin Schechter

Chapter 11. Nonlinear Eigenvalues

Abstract

In Chapter II we obtained critical sequences for the solution of

$$G'\left( u \right) = 0$$

(11.1.1)

by finding linking sets which separate the values of G or sequences of such sets. In Chapter V we applied this technique to show that one obtains solutions of (11.1.1) inside a ball or solutions of (11.1.2)

$$G'\left( u \right) = \beta u$$

(11.1.2)

on the boundary. As a result, one obtains either solutions of (11.1.1) or a rich family of eigenfunctions satisfying (11.1.2).

Martin Schechter

Chapter 12. Strong Resonance

Abstract

As we saw in Chapter VIII, the equation

$$Au = f\left( {x,u} \right)$$

(12.1.1)

has asymptotic resonance at infinity if

$$f\left( {x,t} \right)/t \to {\lambda _\ell }as\left| t \right| \to \infty,$$

(12.1.2)

where λ_ℓ is an eigenvalue of A. However, there are various degrees of resonance. If we write

$$f\left( {x,t} \right) = \lambda _\ell t + p\left( {x,t} \right),$$

(12.1.3)

then (12.1.2) will be true as long as

$$p\left( {x,t} \right)/t \to 0{\text{ }}as{\text{ }}\left| t \right| \to \infty .$$

(12.1.4)

Martin Schechter

Chapter 13. Notes, Remarks and References

Abstract

Here we include some observations pertaining to various sections.

Martin Schechter

Backmatter

Titel: Linking Methods in Critical Point Theory
verfasst von: Martin Schechter
Verlag: Birkhäuser Boston
Electronic ISBN: 978-1-4612-1596-7
Print ISBN: 978-1-4612-7210-6
DOI: https://doi.org/10.1007/978-1-4612-1596-7

Springer Professional

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Critical Point Theory

Chapter 2. Linking

Chapter 3. Semilinear Boundary Value Problems

Chapter 4. Alternative Methods

Chapter 5. Bounded Saddle Point Methods

Chapter 6. Estimates on Subspaces

Chapter 7. The Fučík Spectrum

Chapter 8. Resonance

Chapter 9. Boundary Conditions

Chapter 10. Multiple Solutions

Chapter 11. Nonlinear Eigenvalues

Chapter 12. Strong Resonance

Chapter 13. Notes, Remarks and References

Backmatter