Badiale M., Serra E. Semilinear Elliptic Equations for Beginners: Existence Results via the Variational Approach

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1.7 Some Spectral Properties of Elliptic Operators 31

and I :H

) →R deﬁned by

I(u)=



|∇u|

dx +



dx −



hu dx =

u

−



hu dx.

This procedure is called the Dirichlet principle.

We now give a ﬁrst example of a nonlinear problem which can easily solved with

the same techniques.

Theorem 1.6.6 Let  ⊂ R

, with N ≥ 3, be open and bounded. Assume that

q ∈L

∞

() satisﬁes q ≥0 a.e. in , and let f : R → R be a continuous function

such that (1.9) holds. If

f(t)t ≤0 and (f (t) −f (s))(t −s) ≤0 ∀t,s ∈R, (1.24)

then, for every h ∈L

(), the problem



−u +q(x)u =f(u)+h(x) in ,

u =0 on ∂

(1.25)

has exactly one solution.

Proof Let F(t)=



f(s)ds and deﬁne a differentiable functional J :H

() →R

J(u)=



|∇u|

dx +



q(x)u

dx −



F(u)dx −



hu dx.

Critical points of J are weak solutions of (1.25). We have, for all u, v ∈H

(),



(u) −J



(v))(u −v) =u −v

−



(f (u) −f (v)(u −v)dx ≥u −v

so that J is strictly convex. Furthermore it is easy to see that F(t)≤ 0 for all t , and

hence

J(u)≥

u

−



F(u)dx −|h|

|u|

≥

u

−Cu,

by the Poincaré and Sobolev inequalities. This shows that J is also coercive. As

the functional is coercive, convex and continuous, it has a global minimum point,

which is a critical point. Since J is strictly convex, this is the only critical point, by

Proposition 1.5.9. 

As obvious examples of functions f satisfying the assumption of the previous

theorem, one can take any f(t)=−|t|

p−2

t with p ∈(1, 2

∗

],orf(t)=−arctan t.

1.7 Some Spectral Properties of Elliptic Operators

We now recall some well-known spectral properties of elliptic operators; for the sake

of simplicity and in view of later applications we state the results for an operator

of the form − + q(x), the prototype of all (uniformly) elliptic operators. The

32 1 Introduction and Basic Results

results can be extended with minor modiﬁcations to cases where e.g. the Laplacian

is replaced by the operator −div(A(x)∇), under suitable natural assumptions on the

matrix A.

Let  be a bounded open subset of R

, and let q ∈L

∞

().

Deﬁnition 1.7.1 We say that λ ∈ R is an eigenvalue of the operator − + q(x)

under Dirichlet boundary conditions if there exists ϕ ∈H

() \{0} such that



−ϕ +q(x)ϕ =λϕ in ,

ϕ =0on∂.

(1.26)

The function ϕ is called an eigenfunction associated to λ.

Remark 1.7.2 Problem (1.26) is to be interpreted in the weak sense, namely ϕ solves

it provided



∇ϕ ·∇vdx+



q(x)ϕvdx =λ



ϕv dx for all v ∈H

().

If q ≡0, the corresponding numbers λ are simply called the eigenvalues of the

Laplacian (often even omitting “under Dirichlet boundary conditions”).

The next theorem collects the basic properties of eigenvalues and eigenfunctions;

for its proof, see [18]. Since we present the theory for the Dirichlet problem only,

when we talk about eigenvalues we always mean that Dirichlet boundary conditions

are imposed.

Theorem 1.7.3 Let  ⊂R

be bounded and open and let q ∈L

∞

(). Then there

exist sequences {λ

}

⊂R and {ϕ

}

⊂H

() such that

1. each λ

is an eigenvalue of − +q(x) and each ϕ

is an eigenfunction corre-

sponding to λ

;

2. λ

→+∞as k →∞;

3. {ϕ

}

is an orthonormal basis of L

();

4. for every k, ϕ

∈L

∞

() and ϕ

(x) =0 a.e. in .

Remark 1.7.4 It is customary to list the eigenvalues in an inﬁnite increasing se-

quence λ

≤λ

≤···; thus eigenvalues are repeated according to their (ﬁnite) mul-

tiplicity. Two eigenfunctions ϕ

and ϕ

are always different for i = j (they are or-

thogonal in L

), even if they correspond to the same eigenvalue.

Notation For a ﬁxed operator on a ﬁxed domain, eigenvalues are denoted by λ

If one wants to stress the dependence of λ

on the operator or on the domain one

writes λ

(− +q) or λ

().

Remark 1.7.5 When λ

(− +q) > 0 (this happens certainly if q(x) ≥0a.e.in,

by the characterization of λ

in Theorem 1.7.6 below), it is easy to see that the

quantity

(u|v) =



∇u ·∇vdx+



q(x)uv dx

1.7 Some Spectral Properties of Elliptic Operators 33

deﬁnes a scalar product on H

() that induces a norm ·equivalent to the usual

one. One checks immediately that the set {

√

}

is an orthonormal basis of H

()

with respect to the scalar product deﬁned above.

For future reference we notice that we can write

u =

∞



k=1

in L

() and |u|

∞



k=1

with α





uϕ

dx,or

u =

∞



k=1

√

in H

() and u

∞



k=1

with β

= (u|

√

). The numbers α

and β

are the Fourier coefﬁcients of u in

() and in H

() respectively. Of course, β

√

Eigenvalues and eigenfunctions play a very relevant role in nonlinear elliptic

problems. For this reason we add some more information.

The most “important” eigenvalue is the smallest, λ

, also called the ﬁrst eigen-

value or the principal eigenvalue. It enjoys a number of special features, some of

which we list in the next result.

Theorem 1.7.6 (Variational characterization of the ﬁrst eigenvalue) Let  be

an open and bounded subset of R

and let q ∈ L

∞

(). Deﬁne a functional

Q :H

() \{0}→ as

Q(u) =





|∇u|

dx +





q(x)u





This functional is called the Rayleigh quotient. Then

1. min

u∈H

()\{0}

Q(u) =λ

;

2. Q(u) =λ

if and only if u is a (weak) solution of



−u +q(x)u =λ

u in ,

u =0 on ∂;

(1.27)

3. every non identically zero solution of (1.27) has constant sign in  (in particular,

is a.e. different from zero in );

4. the set of solutions of (1.27) has dimension one; one says that λ

is simple.

The fact that the ﬁrst eigenvalue is simple is normally expressed by writing the

sequence of eigenvalues as λ

<λ

≤ λ

≤···, with the strict inequality between

the ﬁrst two terms. Since the eigenfunctions associated to λ

are all of the form

34 1 Introduction and Basic Results

θϕ

,forθ ∈ R and for some ϕ

∈H

(), it is customary to say that ϕ

is the ﬁrst

eigenfunction, in the sense that there exists a unique ϕ

∈H

() such that

⎧

⎪

⎨

⎪

⎩

−ϕ

+q(x)ϕ

=λ

in ,

=0on∂,

> 0in,

normalized.

(1.28)

The normalization can be expressed by ϕ

=1, or |ϕ

= 1 or by other similar

statements, depending on the convenience of the moment.

Remark 1.7.7 All eigenfunctions but ϕ

change sign. Indeed, since they form an

orthonormal basis of L

(), they satisfy in particular





dx =0fork =1. As

is positive, ϕ

must change sign.

The role of eigenvalues is of fundamental importance for the solvability of linear

problems. We give the main result, that can be seen as a direct application of the

Fredholm alternative in Functional Analysis (see [11, 18]).

Theorem 1.7.8 Let  be a bounded open subset of R

. Assume that q ∈ L

∞

()

and that h ∈L

(). Consider the problem



−u +q(x)u =λu +h in ,

u =0 on ∂.

(1.29)

Then

1. Problem (1.29) has a unique (weak) solution for every h ∈L

() if and only if

λ is not an eigenvalue of − +q(x).

2. If λ is an eigenvalue, then problem (1.29) has a solution if and only if h is or-

thogonal in L

() to ker(− +q(x) −λ).

In this case, the general solution of (1.29) is of the form u =u

+ϕ, where u

is a

ﬁxed solution of (1.29) and ϕ is any function in ker(− +q(x) −λ).

As an application of the spectral properties discussed so far, we state a very

useful property that allows one to prove easily that solutions of certain equations are

nonnegative. This is quite important in many problems where by physical reasons,

for example, one is only interested in positive solutions.

We are talking about a version of the maximum principle, a fundamental tool in

the study of differential equations. For a complete discussion of this principle, we

refer the reader to [18]or[41].

Proposition 1.7.9 Let  ⊂ R

be open and bounded, and let q ∈L

∞

(). Assume

that u ∈H

() satisﬁes

−u +q(x)u ≥0 in 

1.8 Exercises 35

in the weak sense, that is,



∇u ·∇ϕdx+



q(x)uϕ dx ≥0 ∀ϕ ∈H

(), ϕ(x) ≥ 0 a.e. in .(1.30)

If λ

(− +q) > 0, then u(x) ≥0 almost everywhere in .

Example 1.7.10 A typical use of the preceding proposition goes as follows. Suppose

that the function f :  × R → R is continuous and satisﬁes f(x,t)≥ 0fort ≥ 0

and f(x,0) =0 (for every x ∈). We are interested in nonnegative solutions of



−u +q(x)u =f(x,u) in ,

u =0on∂.

(1.31)

We deﬁne

g(x, t) =



f(x,t) if t ≥0,

0ift<0.

Of course g is continuous. Then we solve (if possible) Problem (1.31) with f re-

placed by g. We obtain in this way a function u ∈H

() that satisﬁes

−u +q(x)u =g(x,u) ≥0

at least in the weak sense. If λ

(− +q) > 0, then by Proposition 1.7.9 we obtain

that u(x) ≥0a.e.in. But then, for almost every x in , g(x,u(x)) =f (x,u(x)),

so that actually u (is nonnegative) and solves (1.31).

Remark 1.7.11 Under regularity conditions, it is possible to prove that nonnegative

solutions are actually everywhere positive in . This is the content of the strong

maximum principle, which is beyond the aims of this book. We refer the interested

reader to [18, 41].

1.8 Exercises

1. Let  ⊂R

be open and bounded. Consider the problem



−u =1in,

u =0on∂.

(1.32)

(a) Deﬁne the notion of weak solution for this problem, and ﬁnd a functional

on H

() whose critical points correspond to weak solutions of (1.32).

(b) Show that (1.32) has exactly one solution.

the equation in polar coordinates and look for a radial solution.)

2. Let  ⊂R

, with N ≥3 be open and bounded. Take q ∈L

∞

() and let A(x)

be an N ×N matrix with continuous entries (on

).

For p ∈(2, 2

∗

], consider the problem



−div(A(x)∇u) +q(x)u =|u|

p−2

u in ,

u =0on∂.

(1.33)

36 1 Introduction and Basic Results

Deﬁne the notion of weak solution for this problem, and ﬁnd a functional on

() whose critical points correspond to weak solutions of (1.33).

3. Let f : R → R be a continuous function satisfying (1.10) and (1.24). Assume

that a ∈L

∞

) is such that

a(x) ≥c>0

for a.e. x ∈R

. Prove that for every h ∈L

), the problem



−u +u =a(x)f (u) +h(x),

u ∈H

)

has exactly one solution.

4. Let  be an open subset of R

, N ≥3, and take a function q ∈L

N/2

(). Prove

that the function a :H

() ×H

() →R deﬁned by

a(u,v) =



quv dx

is bilinear and continuous. Deduce that the functional J : H

() → R deﬁned

J(u)=



is differentiable and compute J



5. (Best constant in the Poincaré inequality.) Let  ⊂ R

be open and bounded.

Prove that



dx ≤



|∇u|

dx for all u ∈H

()

and the constant 1/λ

is the best (i.e., smallest) possible.

6. (Dependence of λ

on the domain.) For every  ⊂R

open and bounded, de-

note by λ

() the ﬁrst eigenvalue of the Laplacian on H

(). Prove that λ

()

is strictly decreasing with respect to inclusion, in the sense that

 ⊂



implies λ

() > λ

(



7. (Dependence of λ

on q.) Let  be open and bounded. Prove that

(a) λ

(− +q(x)) is a nondecreasing function of q, in the sense that

p,q ∈L

∞

() and p(x) ≤q(x) a.e. imply λ

(− +p) ≤ λ

(− +q);

(b) λ

is a continuous function of q, in the sense that

|λ

(− +q) −λ

(− +p)|≤q −p

∞

()

8. Let  be open and bounded. For h ∈L

() and λ ∈R, deﬁne I :H

() →R

I(u)=



|∇u|

dx −



dx −



hu dx.

Prove that I is coercive on H

() if and only if λ<λ

1.9 Bibliographical Notes 37

9. More generally, assume that  and h are in the previous exercise, let q ∈

∞

(), and deﬁne J :H

() →R by

J(u)=



|∇u|

dx +



q(x)u

dx −



hu dx.

(a) Prove that if λ

(− +q(x)) > 0, then the expression

(u, v) =



∇u∇vdx+



q(x)uv dx

deﬁnes on H

() a scalar product that induces a norm equivalent to the

standard one.

(b) Prove that J is coercive on H

() if and only if λ

(− +q(x)) > 0.

10. Prove Proposition 1.7.9.(Hint:test(1.30) with u

−

and use the variational char-

acterization of λ

1.9 Bibliographical Notes

• Section 1.1: A very gentle introduction to the Calculus of Variations is the book

by Hildebrandt and Tromba [25]. For a history of the early Calculus of Variations,

see Goldstine [23].

• Section 1.2: Proofs, details, and general theory can be found in almost any

book in Functional Analysis. A good reference is Brezis [11], which is PDE-

oriented. Topics more strictly related to differential equations can also be found

in Evans [18].

• Section 1.3: The reader who wishes to see a more systematic treatment of differ-

ential calculus in normed space is referred to Ambrosetti and Prodi [4].

• Section 1.4: For a more complete discussion on weak solutions, and particularly

for the extension of the deﬁnitions in this section to nonhomogeneous Dirichlet

problems and Neumann problems, see Chap. IX of [11].

• Section 1.5: The development of the results and techniques outlined in this sec-

tion, namely minimization of functionals under coercivity, convexity or semi-

continuity properties is the Calculus of Variations, and particularly the so-called

direct methods in the Calculus of Variations. This is a very wide topic that has

reached a high degree of complexity, matched by outstanding success in solving

huge classes of problems taken from everywhere in science. The readers inter-

ested in these methods can consult Giusti [21], especially the ﬁrst chapter.

• Section 1.7: The missing proofs in this section can be found in [11, 18].

Chapter 2

Minimization Techniques: Compact Problems

Throughout this chapter we show how techniques based on minimization arguments

can be used to establish existence results for various types of problems.

Our aim is not to describe the most general results, but to give a series of ex-

amples, and to show how simple techniques can be reﬁned to treat more complex

cases.

2.1 Coercive Problems

We begin with the following problem, that will provide our guideline through the

whole chapter. We want to ﬁnd a (weak) solution to



−u +q(x)u =f(u)+h(x) in ,

u =0on∂.

(2.1)

In this section, the general framework is speciﬁed by the assumptions

)⊂R

is bounded and open, q ∈L

∞

() and q(x) ≥0a.e.in.

)h∈L

().

We equip H

() with the scalar product

(u|v) =



∇u ·∇vdx+



q(x)uv dx, (2.2)

and we denote by ·the induced norm, equivalent to the standard one.

Remark 2.1.1 In assumption (h

), the requirement q(x) ≥ 0 a.e. is used only to

obtain λ

(− + q(x)) > 0, which guarantees that (2.2) is indeed a scalar product

and that the induced norm is equivalent to the standard norm of H

(),seeRe-

mark 1.7.5 and Exercise 9 in Chap. 1. Therefore in all the results of this chapter, and

similarly in all the subsequent chapters, the assumption q ≥ 0 could be replaced be

the “abstract” condition

(− +q(x)) > 0, (2.3)

M. Badiale, E. Serra, Semilinear Elliptic Equations for Beginners, Universitext,

40 2 Minimization Techniques: Compact Problems

and everything would work perfectly well with no changes in the proofs. The point

is, precisely, that (2.3) is abstract, and nobody knows for which general q’s it is sat-

isﬁed. We prefer, in this book, to assume an explicit sign condition on q, rather than

an indirect one on λ

. The reader should however keep in mind this clariﬁcation.

We begin by assuming the following hypothesis on the nonlinearity f .

)f:R →R is continuous and bounded.

Setting F(t)=



f(s)ds, the computations carried out in Example 1.3.20 show

that the functional I :H

() →R deﬁned by

I(u)=



|∇u|

dx +



q(x)u

dx −



F(u)dx −



hu dx

u

−



F(u)dx −



hu dx

is differentiable on H

(). Its critical points are the weak solutions of (2.1).

Note that unless F is concave, which we do not assume, the functional I needs

not be convex.

Theorem 2.1.2 Under the assumptions (h

)–(h

), Problem (2.1) admits at least

one solution.

Remark 2.1.3 The leading idea of the proof is that since f is bounded, the term





F(u)dx should grow at most linearly with respect to u, as well as the last

term. If this is true, the functional I can be seen as an “at most linear” perturbation

of the quadratic term u

. This suggests the existence of a global minimum. Let us

see how all this really works.

Proof We break it into two steps. We make repeated use of Hölder and Sobolev

inequalities.

Step 1. The functional I is coercive. Note ﬁrst that since f is bounded, then

|F(t)|≤M|t|

for some M>0 and all t ∈R. Hence





F(u)dx



≤M



|u|dx ≤Cu,

where the last inequality comes from the continuity of the embedding of H

()

into L

(). This conﬁrms the idea of the linear growth as in the preceding remark.

Thus

I(u)=

u

−



F(u)dx −



hu dx ≥

u

−Cu−|h|

|u|

≥

u

−Cu,

which shows that I is coercive.