Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic

Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic

equations near an eigenvalue

Dimitri Mugnai

Dipartimento di Matematica e Informatica Universit` a di Perugia

Via Vanvitelli 1, 06123 Perugia - Italy e-mail: mugnai@dipmat.unipg.it tel: +39 075 5855011, fax: +39 075 5855024

Abstract

We give the precise behaviour of some solutions of a nonlinear el- liptic B.V.P. in a bounded domain when a parameter approaches an eigenvalue of the principal part. If the nonlinearity has some regular- ity and the domain is for example convex, we also prove a nonlinear version of Courant’s Nodal Theorem.

Keywords: eigenvalues, L

− H

estimate, nodal lines, symmetries.

2000AMS Subject Classification: 35B40, 35J65.

1 Introduction

In a celebrated paper, Z. Q. Wang proved that, if Ω is a bounded and smooth domain of R

and g : R −→ R is a C

superlinear and subcritical function such that g(0) = g

(0) = 0, then the nonlinear Dirichlet problem

 −∆u = g(u) in Ω,

u = 0 on ∂Ω, (1.1)

has at least three nontrivial solutions (see [23]). In the same spirit, in [19] it was proved that, if g : Ω × R −→ R is a superlinear and subcritical Carath´ eodory’s function, then there exists δ

> 0 such that ∀ λ ∈ (λ

−δ

, λ

), i ≥ 2, the problem

 −∆u − λu = g(x, u) in Ω,

u = 0 on ∂Ω, (1.2)

∗Supported by the M.I.U.R. project Metodi Variazionali ed Equazioni Differenziali Nonlineari.

has at least three nontrivial solutions (here (λ

)

denotes the sequence of eigenvalues of −∆ on H

(Ω)). In particular in [19] it was proved that two of such solutions have positive energy, approaching 0 as λ → λ

. In this paper we want to prove that such solutions have some finer properties, in- herited by the fact that λ is near an eigenvalue and by the fact that their energy approaches 0. In particular our results are related to the behaviour of the nodal lines of such solutions (see Theorem 2.1) and to some symmetry properties when g and Ω have some symmetries (see Theorem 2.2).

To our knowledge, there are not many results concerning nodal lines of sign-changing solutions, as the ones we are considering. For sure, remarkable results can be found in [4], [6], [7] and [10]. In [6], the authors assume to deal with a function g : Ω × R −→ R of class C

such that

(g

) g(x, 0) = 0 for all x ∈ Ω,

(g

) ∃ p ∈ (2, 2

) such that |g

(x, t)| ≤ C(1 + |t|

) for all x ∈ Ω, t ∈ R;

(g

) g

(x, t) > g(x, t)/t for all x ∈ Ω, t 6= 0;

(g

) ∃ R > 0 and θ > 2 such that 0 < θG(x, t) ≤ tg(x, t) for all x ∈ Ω,

|t| ≥ R, where G(x, t) = R

0

g(x, s) ds.

Under these assumptions, they prove that problem (1.1) has one changing sign solution with exactly two nodal domains and Morse index 2, provided the second eigenvalue of −∆ − g

(x, 0) is strictly positive. Note that in the case of problem (1.2) it means λ < λ

. Under similar assumptions, in [10]

the existence of one solution to problem (1.1) which changes sign exactly once and has connected nodal domains, is proved.

Moreover, in [7] the authors consider a problem similar to (1.2), but with λ < 0; this means that such a problem is substantially similar to problem (1.1), since −∆ − λ induces a positive definite quadratic form whenever λ < λ

. In this case they deal with an autonomous function g = g(t) and they make assumptions analogous to (g

) − (g

) for the autonomous case, with R = 0 in (g

), proving the existence of three nodal solutions, but with the further assumption that Ω is big enough.

Finally, in [4], several existence results are given for problem (1.1) with an autonomous g, according to the assumptions made on g, g

and the growth conditions at infinity (superlinear or asymptotically linear). Concerning the superlinear case, they results can be stated in the following way: if g is of class C

with g(0) = 0, λ

< g

(0) < λ

, i ≥ 1, if sign changing solutions are isolated, then there exists a sign changing solution to problem (1.1).

Therefore, Theorem 2.1 in Section 2 can be considered as a counterpart

of the results stated above. In fact, no regularity assumption is made on

the nonautonomous function g (so no condition is made on its derivative),

no restrictions are made on Ω, a multiplicity result is always given and λ

can be as big as desired, provided it is close to an eigenvalue of −∆. The

novelty of Theorem 2.1 with respect to the results of [19], is that here we give a precise behaviour of the solutions u

for λ ∼ λ

, showing that such solutions change sign and giving some properties of their nodal lines (see Theorems 2.1 and 2.2 and their corollaries).

It is also worth mentioning the problem (in the entire space)

 −∆u + a(x)u = g(x, u) in R

, u ∈ H

(R

),

studied in [5]. Therein, the existence of one sign changing solutions is proved when g is subcritical and superlinear and a satisfies some suitable assump- tions. In addition, if g is odd in u, the existence of a sequence u

of nodal solutions with at most k + 1 nodal regions is proved.

Concerning symmetry properties of solutions of nonlinear subcritical ho- mogeneous boundary problems, depending on the geometry of Ω, there are plenty of results. But in such cases the authors assume to deal with positive solutions in presence of functions g’s of class C

(for example, see [12], [16], [20] and the references quoted therein). For example, in [12] the authors consider the problem







−∆u − λu = g(u) in Ω,

u > 0 in Ω,

u = 0 on ∂Ω,

where Ω is convex in the x

–direction, symmetric w.r.t. the hyperplane x

= 0 and g(0) ≥ 0; using the maximum principle they show that any solution v in H

(Ω) of the linearized problem −∆v − λv = g

(u)v is even–

symmetric in x

. In [20] the author considers the problem

 −∆u = g(x, u) in Ω,

u = 0 on ∂Ω,

where g is even in x

and convex in u, and she shows that any positive solution u is even in x

. Moreover, she also proves that, if Ω is an annulus or a ball, g(|x|, ·) is strictly convex, u is a solution of index 1, then u is axially symmetric. The author underlines the fact that such a result is not easily applicable to changing sign solutions, since changing sign solutions generally have Morse index greater than 1. Regardless of the Morse index, in Theorem 2.2 we will prove such a symmetry result for the solutions under investigation of problem (1.2).

Of course, more general problems can be considered. For example, in [16] it is shown that solutions u of the problem







−∆u = g(u) in Ω, u > 0 in Ω,

u = 0 on ∂Ω,

where Ω is a small perturbation of a symmetric domain for which the Gidas, Ni and Nirenberg symmetry result holds, and g is subcritical or critical, are even–symmetric.

The most natural generalization of the previous problems is obtained in presence of the p–Laplace operator, when the typical problem is the follow- ing:







−∆

u = g(u) in Ω, u > 0 in Ω,

u = 0 on ∂Ω.

Under suitable assumptions on g (typically g of class C

, convex, and sym- metric in the direction ν), symmetry results of the type

Ω symmetric w.r.t. a direction ν =⇒ u is even in the direction ν (1.3) are given, for example, in [13] and [14]. In this sense our results complement the results given in the papers quoted above.

Another aspect of this paper is a contribution to a nonlinear version of Courant’s nodal theorem. More precisely, our result is in the spirit of [11], in which the author proves the following: if Ω is a bounded (possibly non- smooth) domain of R

which is convex and symmetric w.r.t. k orthogonal directions, 1 ≤ k ≤ N , then the nodal lines of the eigenfunctions e

, . . . , e

of the problem

 −∆u = λu in Ω,

u = 0 on ∂Ω,

intersect the boundary. In Theorem 2.1, only assuming Ω smooth enough, we prove that the sign changing solutions found when λ belongs to a suitable left neighborhood of λ

, behave like the eigenfunction e

. Therefore, for example, if i = 2 and Ω is convex, in Corollary 2.1, we prove that the solutions have exactly two nodal domains and the nodal line touches the boundary of Ω.

For completeness, we also recall a version of Courant’s nodal theorem for the p–Laplacian found in [15], and that in the case of sublinear elliptic problems in R

, in [2] the authors prove the existence of radial compactly supported solutions with any given number of nodes.

Finally, in Theorem 2.2, we show a result of symmetry in the spirit of (1.3), showing that both the solutions under considerations, i.e. for λ close to λ

, have the same symmetry as Ω.

Acknowledgments. The author wishes to thank Prof. F. Pacella for suggesting the problem of symmetry for solutions near resonance.

Moreover, the author takes the opportunity to thank the anonymous

referee for the careful reading and for the helpful comments, which improved

the presentation of the paper.

2 Assumptions and main results

Throughout this paper, we will consider a bounded domain Ω ⊂ R

which is so smooth that the classical regularity results for solutions of elliptic equa- tions can be applied. To this purpose it is enough to assume, for example, that ∂Ω is of class C

.

Moreover, we will assume some usual conditions for a standard superlin- ear and subcritical nonlinearity:

(g

) g : Ω × R −→ R is a Carath´eodory’s function;

(g

) there exist constants a > 0, s > 1 (and s <

if N ≥ 3) such that

∀ t ∈ R and for a.e. x ∈ Ω

|g(x, t)| ≤ a|t|

; (g

) ∀ t 6= 0 and for a.e. x in Ω

0 < µG(x, t) ≤ g(x, t)t, where µ = s + 1 and G(x, t) = R

0

g(x, σ) dσ;

(g

) there exists c

> 0 such that G(x, t) ≥ c

|t|

∀ t ∈ R and for a.e. x ∈ Ω.

Such assumptions are quite natural and common when one studies non- linear subcritical problems (see [1], [21], [17], the papers quoted above,. . . ).

Remark 2.1. In order to get existence of nodal solutions, in the papers quoted in the Introduction, the authors always assume to deal with a regular function g (at least of class C

), usually even independent of x. Here we only assume g to be continuous in the second variable.

Remark 2.2. We remark that, usually, in (g

) one requires only µ > 2. But by (g

) and (g

), one immediately gets µ ≤ s + 1 (and so s > 1). Therefore a stronger assumption is made on µ; however such an assumption is satisfied whenever g(x, t) “behaves” like |t|

t.

Remark 2.3. If g is continuous on Ω × R, condition (g

) comes by integra- tion of (g

), with the constant c

replaced by a function c

(x).

From now on, we will denote by e

any nontrivial element of E

with ke

k

= 1, where E

denotes the eigenspace of −∆ on H

(Ω) associated to the eigenvalue λ

. Since Ω is of class C

, it is a classical fact that e

is smooth.

We are now able to state the first result of this paper.

Theorem 2.1. Under hypotheses (g

) − (g

), for any i ≥ 2 there exists

τ

> 0 such that for any λ ∈ (λ

− τ

, λ

), problem (1.2) has at least two

distinct continuous nontrivial sign–changing solutions u

and u

such that,

for any k = 1, 2,

(a) u

→ 0 in H

(Ω);

(b) for any sequence (µ

)

⊂ (λ

− τ

, λ

) converging to λ

, the sequence



ku^k_µnk



n

is compact in H

(Ω) and the set of its limits points is con- tained in the set {e ∈ E

: kek =

i

};

(c) ∃ C = C(i) > 0 such that ku

k

≤ Cku

k.

Here kuk denotes the norm of u in H

(Ω), i.e. the L

–norm of Du.

Remark 2.4. A straightforward corollary of (a) and (b) of Theorem 2.1 is that u

→ 0 uniformly in Ω as λ → λ

for any k = 1, 2.

A remarkable consequence of the previous theorem is the following non- linear version of Courant’s Nodal Theorem.

Corollary 2.1. Assume that Ω ⊂ R

, N ≥ 2, is such that the Courant’s nodal Theorem holds, that g : Ω×R −→ R is of class C

and that t

(x, t) >

tg(x, t) for all (x, t) ∈ Ω × R. Moreover, assume that λ

is simple and that (g

) − (g

) hold. Then the solutions u

’s found in Theorem 2.1 for i = 2 have exactly two nodal domains and the nodal line of each u

intersects ∂Ω.

Remark 2.5. According to Theorem 1.1 in [11], the Courant’s nodal The- orem holds holds, for example, if Ω ⊂ R

is convex and symmetric w.r.t. k orthogonal directions, 1 ≤ k ≤ N .

Proof of Corollary 2.1. Since each eigenvalue of −∆ has finite multiplicity and λ

is simple, we have λ

< λ < λ

< λ

. Since each u

is found by a linking structure involving the decomposition H

(Ω) = Span(e

) ⊕ Span(e

)⊕Span(e

, . . .) (see [19]), it is a classical fact that i(u

) ≤ 2 (see [8]) and #{nodal regions} ≤ i(u

), where i(u

) denotes the Morse index of u

(it is enough to adapt the proof of [8] under the assumptions on g

, similarly to [3]). We remark that, since u

∈ C(Ω), no further growth assumption on g

are needed. Moreover, since each u

changes sign, for

λk

→ t

e

(t

= 1/λ

), we get that each u

has exactly two nodal domains. Let us set Ω

= {x ∈ Ω : t

e

(x) > 0}.

Let us prove that the nodal line Γ

of each u

cannot be closed. In fact, there exists ε > 0 such that ∀ λ ∈ (λ

− τ

, λ

) and ∀ x ∈ Ω

with d(x, Γ) ≥ ε it holds u

(x) ≥ 0 and ∀ x ∈ Ω \ Ω

such that d(x, Γ) ≥ ε it holds u

(x) ≤ 0.

Otherwise we could find, for example, a point x

and then, by continuity of u

, an open set A

contained in Ω

, where u

is negative and with x

converging to a point x

∈ ∂Ω. Of course

(x

) ≥ 0.

Assume that

λk

→ t

e

in W

(Ω) for some q. By the trace theorem there exists a universal constant T > 0 such that

D u

ku

k − t

De

≤ T

D u

ku

k − t

De

≤ T

u

ku

k − t

e

→ 0 as λ → λ

. Therefore we can assume that D

λk

→ t

De

a.e. on ∂Ω. In particular we can choose x

∈ A

converging to a point x

∈ ∂Ω where D

λk

converges to t

De

. Then

(x

) →

(x

) and a contradiction arises, since by classical results

< 0.

Finally, let us show that

λk

→ t

e

in W

(Ω), where q is chosen so large that W

(Ω) ,→ L

(Ω) (see the following Section for more details).

First of all, we get that

λk

* t

e

in W

(Ω) as λ → λ

. Indeed the function v

:=

λk

− t

e solves the problem

( −∆v

= λv

+ (λ − λ

)t

e

+

λk

in Ω,

u = 0 on ∂Ω.

By by the regularity inequality of Calder´ on–Zygmund (see [22, Theorem B.2]) and by assumption (g

) we get

kv

k

≤ C 

λ

kv

k

+ a ku

k

ku

k + (λ

− λ)kt

e

k



(2.4) for some universal constant C > 0. By (c) of Theorem 2.1 ku

k

≤ c

ku

k

for some c

> 0 independent of λ and also kv

k

is bounded; therefore also kv

k

is bounded. Thus we can assume that

λk

* t

e

in W

(Ω). But then v

→ 0 in W

(Ω), so that (2.4) implies that v

→ 0 in W

(Ω).

Concerning the symmetry result, we will specialize the class of nonlin- earities, assuming

(g g

) g : Ω × R −→ R is of class C

and there exist b ∈ L

(Ω) and p ≥ 1 such that

|g

(x, t)| ≤ b(x)|t|

∀ t ∈ R and for a.e. x ∈ Ω.

Here q ≥ 1 if N < 3 and q = N/2 if N ≥ 3.

Theorem 2.2. Let Ω be a bounded domain of R

which is convex and sym- metric w.r.t. two orthogonal directions, say the x

–directions, i = 1, 2. As- sume g (g

)–(g

)–(g

)–(g

) and g(x

, −x

, −s) = −g(x, s) for any s ∈ R and for a.e. x ∈ Ω. Then there exists τ

≤ τ

such that ∀ λ ∈ (λ

− τ

, λ

), any solution u

of problem (1.2) found in Theorem 2.1 satisfies u

(x) =

−u

(x

, −x

).

3 Proof of Theorem 2.1

Let us introduce the C

functional f

: H

(Ω) −→ R defined by f

(u) = 1

2 Z

Ω

|Du|

dx − λ 2

Z

Ω

u

dx − Z

Ω

G(x, u) dx, whose critical points are the solutions of problem (1.2).

Let us denote by u

one of the two families of solutions to (1.2) found in [19] with the property that f

(u

) → 0 as λ → λ

, i ≥ 2.

The proof of Theorem 2.1 will be given in several steps.

First step: u

→ 0 in H

(Ω).

Proof. Since u

solves (1.2), we have Z

Ω

Du

· Dv dx − λ Z

Ω

u

v dx − Z

Ω

g(x, u

)v dx = 0 ∀ v ∈ H

(Ω), (3.5) and in particular

Z

Ω

|Du

|

dx − λ Z

Ω

u

dx − Z

Ω

g(x, u

)u

dx = 0. (3.6) In this way

f

(u) = Z

Ω

 1

2 g(x, u

)u

− G(x, u

)



dx ≥  µ

2 − 1  Z

Ω

G(x, u

) dx > 0.

But f

(u

) → 0 as λ → λ

, and so R G(x, u

) → 0 as λ → λ

, which implies u

−→ 0 in L

(Ω)

by (g

). Therefore, also u

→ 0 strongly in H

(Ω). In fact f

(u

) = 1

2 Z

Ω

|Du

|

dx − 1 2 λ

Z

Ω

u

dx − Z

Ω

G(x, u

)u

dx, and passing to the limit we get

ku

k → 0. (3.7)

We can now assume that there is u ∈ H

(Ω) such that u

/ku

k * u in H

(Ω).

Second step: u solves −∆u = λ

u in H

(Ω).

Proof. If v ∈ H

(Ω), (3.5) gives Z

Ω

Du

ku

k · Dv dx − λ Z

Ω

u

ku

k v dx = Z

Ω

g(x, u

)

ku

k v dx. (3.8) The growth assumption (g

), H¨ older and Sobolev inequalities imply

    Z

Ω

g(x, u

) ku

k v dx

≤ a Z

Ω

|u

|

ku

k |v| dx ≤ cku

k

kvk

L

2∗

2∗−s

, where c > 0 is independent of u

.

In this way, passing to the limit in (3.8), Z

Ω

Du · Dv dx = λ

Z

Ω

uv dx ∀ v ∈ H

(Ω), (3.9) i.e. u solves −∆u = λ

u in H

(Ω), and so there exists t

∈ R such that u = t

e

, where e

∈ E

(see Theorem 2.1).

Third step: u

/ku

k → t

e

strongly in H

(Ω), as λ → λ

, where t

=

i

. Proof. Equation (3.6) and (g

) imply that

1 = lim

λ→λi

Z

Ω

|Du

|

ku

k

dx = λ

Z

Ω

u

dx.

In fact,     Z

Ω

g(x, u

) ku

k u

dx

≤ a Z

Ω

|u

|

ku

k dx ≤ c

ku

k

by H¨ older and Sobolev inequalities, c

= c

(Ω, N ) > 0 being independent of u

.

But equation (3.9) implies that λ

Z

Ω

u

dx = Z

Ω

|Du|

dx, so that kuk = 1 and then the convergence is strong.

The last equation gives the possible values of t

. Now, let us prove the following preliminary result.

Proposition 3.1. Assume N ≤ 5 and one of the following hypotheses:

• s < 4 if N = 3;

• s < 2 if N = 4;

• s <

if N = 5.

Then u

∈ C(Ω) and there exist τ

> 0 and C = C(Ω, N, i) > 0 such that

∀ λ ∈ (λ

− τ

, λ

)

ku

k

≤ Cku

k.

Proof. Since Ω is of class C

, it is a classical fact (see for example [22, Lemma B.3 and below]) that any solution u of (1.2) belongs to C(Ω). In order to get the L

− H

estimate, let us note that u ∈ W

(Ω), provided λu + g(x, u) ∈ L

(Ω), and there exists C > 0 such that

kuk

≤ Ckλu + g(x, u)k

. (3.10) Of course (g

) implies that λu + g(·, u) ∈ L

(Ω) if λ|u| + a|u|

∈ L

(Ω).

Moreover

kλu + a|u|

k

≤ λkuk

+ k|u|

k

= λkuk

+ kuk

and by Sobolev inequality

≤ γ 

kuk + kuk

 ,

provided qs ≤ 2

, and γ > 0 is a constant depending on Ω, N , q and i.

Since ku

k → 0 as λ → λ

, there exists τ

> 0 such that ku

k ≤ 1

∀ λ ∈ (λ

− τ

, λ

), so that fo such λ’s ku

k

≤ ku

k. Therefore, (3.10) and the estimate above imply the existence of a constant C

= C

(Ω, N, q, i) > 0 such that

ku

k

≤ Cku

k ∀ λ ∈ (λ

− τ

, λ

).

If 1

q − 2 N < 0, Morrey’s Embedding Theorem guarantees that

ku

k

≤ C

ku

k

, where C

is a universal constant.

Therefore we require q >

in order to get ku

k

≤ C

ku

k for some C

> 0 depending on Ω, N, q and i.

Of course, if N = 1 or N = 2, it is clear that any q large enough can be found.

If N = 3 one has to choose

< q ≤

, which is solvable only if s < 4.

If N = 4 one has to choose 2 < q ≤

, which is solvable only if s < 2.

If N = 5 one has to choose

< q ≤

, which is solvable only if s <

.

If N ≥ 6 it is clear that the system

< q ≤

cannot be solved.

However, some refinements of the considerations above can be done in- voking Moser’s iteration technique (see [18]) and its development due to Brezis and Kato (see [9]). Therefore we now generalize Proposition 3.1 to get the following general L

− H

estimate, after remarking again that u

∈ C(Ω). In this way Theorem 2.1 will be completely proved.

Proposition 3.2. Under the assumptions of Theorem 2.1, there exist τ

> 0 and C = C(Ω, N, i) > 0 such that ∀ λ ∈ (λ

− τ

, λ

), u

∈ C(Ω) and

ku

k

≤ Cku

k.

Proof. Let us rewrite the equation −∆u − λu = g(x, u) as

−∆u = α(x)(1 + |u|), (3.11)

where

α(x) := λu(x) + g(x, u(x)) 1 + |u(x)| .

By standard regularity results, any solution in H

(Ω) of (3.11) belongs to L

(Ω) for any q ∈ [1, ∞). Such a result is based on the classical iteration technique

if u ∈ L

(Ω) ⇒ u ∈ L

(Ω), where

q

+ 1 = (q

+ 1) N

N − 2 , j ≥ 1 and q

= 0, (3.12) which provides the following estimate for any q

< ∞ (for example, see [22, Lemma B.3]):

Z

Ω

|D|u|

|

dx ≤ C

(1 + c

)

1

2

−

2 j

4(qj+1)²

, (3.13)

where

C

= max n

kαk

|Ω|

, 3kuk

o

(3.14) and c

≥ 0 is such that

Z

{|α|>c∗}

|α|

dx ≤ 1 24 .

More precisely, one proves that |u|

∈ H

(Ω) ,→ L

(Ω), and then u ∈ L

N −2

(Ω) ∀ j. This implies that u ∈ L

(Ω) ∀ q < ∞. In this way, by the Calder´ on–Zygmund inequality (see [22, Theorem B.2]), u ∈ W

(Ω)

∀ q < ∞ and

kuk

≤ C(q)kλu + g(·, u)k

. (3.15) If q is sufficiently large (2q > N ), W

(Ω) ,→ C(Ω) and there exists ˜ C

> 0 such that

kvk

≤ ˜ C

kvk

∀ v ∈ W

(Ω). (3.16)

Thus, by (3.15) and (3.16), there exists C

(q) > 0 such that

kuk

≤ C

(q)kλu + g(·, u)k

. ≤ C

(q) λ

kuk

+ akuk

. (3.17) If qs ≤ 2

, then kuk

, kuk

≤ Ckuk, where C is independent of u. But kuk → 0 as λ → λ

, so that there exists τ

> 0 such that kuk

≤ kuk for any λ ∈ (λ

− τ

, λ

) and from (3.17) the Proposition is proved (this is essentially the case of Proposition 3.1).

If qs > 2

, first we observe that kuk

≤ C(Ω)kuk

, C(Ω) being indepen- dent of u, so that it will be enough to estimate kuk

in terms of kuk.

First of all, note that if u ∈ H

(Ω) ∩ L

(Ω) and r > 2

, then Z

Ω

|u|

dx ≤ kuk

kuk

≤ γkuk

kuk

by Sobolev Theorem, so that

kuk

≤ C

kuk

2∗

∞ r

kuk

, (3.18)

where C

> 0 is independent of u.

Therefore, (3.17) and (3.18) imply kuk

≤ C

(kuk

+ kuk

) ≤ C

(kuk

2∗

∞qs

kuk

+ kuk

“ 1−^2∗_qs

”

∞

kuk

), (3.19) where C

, C

> 0 are independent of u, but depend on i.

If one proves that

ku

k

→ 0 as λ → λ

, (3.20) then

ku

k

“ 1−^2∗_qs

”

∞

≤ ku

k

2∗

∞qs

.

Moreover, ku

k → 0, and then ku

k

≤ ku

k

for every λ in a suitable left neighborhood of λ

, say (λ

− τ

, λ

).

In this way (3.19) implies ku

k

≤ C

(ku

k

2∗

∞ qs

ku

k

+ ku

k

2∗

∞ qs

ku

k

), that is

ku

k

2∗

∞qs

≤ C

ku

k

for some C

> 0 independent of u

, so that the thesis follows.

Therefore, in order to conclude, we only need to prove (3.20). Its proof

consists in a standard bootstrap argument based on inequality (3.13). This

leads to an estimate which is less interesting than the one stated in the

Proposition, but is enough to prove the main result. In fact, such a bootstrap argument gives an inequality of the form

ku

k

≤ Cku

k

(3.21)

for some C > 0 and δ ∈ (0, 1) independent of u and for any λ ∈ (λ

− τ

, λ

), and this is enough to prove (3.20) thanks to (3.7). The proof of (3.21) is classical and we only sketch it for the sake of completeness.

From (3.17) it is enough to estimate ku

k

and ku

k

in terms of ku

k.

To this aim, take q so large that qs > N , so that W

(Ω) ,→ C(Ω) and there exists c = c(N, q, s) > 0 such that

kvk

≤ ckDvk

for every v ∈ W

(Ω). (3.22) Let j ∈ N be such that qs ≤

, where q

is as in (3.12) and (3.13).

In this way, we will apply (3.22) to the function |u|

, and from (3.13) we get

kuk

≤ c





C

(1 + c

)

1

2

−

4(qj+1)²





1 2qj +2

for some c

= c

(N, q, s) > 0. Thus, to get (3.21), in view of (3.14) it is enough to estimate kαk

2

and kuk

in terms of powers of kuk.

First observe that kαk

2

≤ λ

u 1 + |u|

2

+ ak|u|

k

2

. (3.23)

Since s < 2

− 1, we get (s − 1)

< 2

and by Sobolev inequality k|u|

k

2

≤ γkuk

(3.24)

for some universal constant γ.

As for      

u 1+|u|

2

, if N ≤ 6 we have Z

Ω

 |u|

1 + |u|



dx ≤ Z

Ω

|u|

dx,

and N/2 ≤ 2

, so that Sobolev inequality gives      

u 1+|u|

2

≤ γ

kuk. In the case N > 6, note that for any ε ∈ (0, 1), it holds

1 + |u| ≥ 1

ε

(1 − ε)

|u|

= C

|u|

, where we have set

C

= ε

(1 − ε)

.

Since N ≥ 7, one can take ε ∈



1 − 4

N − 2 , 1 − 2 N



, (3.25)

so that

Z

Ω

 |u|

1 + |u|



dx ≤ C

Z

Ω

|u|

dx, By (3.25),

1 ≤ (1 − ε)N/2 ≤ 2

, and by Sobolev inequality

Z

Ω

 |u|

1 + |u|



dx ≤ ¯ Ckuk

, (3.26) for some ¯ C > 0 independent of u.

Finally, (3.23), (3.24) and (3.26) imply the existence of a constant M = M (Ω, λ

) > 0 independent of u such that

kαk

≤ M (kuk

+ kuk + kuk

).

Concerning the norm kuk

appearing in (3.14), if 2q

+ 2 ≤ 2

it can be estimated directly by kuk. If 2q

+ 2 > 2

, we can rewrite the analogous of (3.13) for q

+ 1, that is

Z

Ω

|D|u|

|

dx ≤ C

(1 + c

)

1

2

−

4(qj−1+1)²

, (3.27)

so that by Sobolev inequality and (3.12), we can estimate kuk

in terms of the left hand side of (3.27). If q

+ 1 ≤ 2

we conclude. Otherwise, after a finite number of steps we are reduced to estimate u

(Ω) in terms of powers of kuk for some k ∈ N and the usual bootstrap argument applies.

Finally, since ku

k ≤ 1 ∀ λ ∈ (λ

− τ

, λ

), (3.21) follows.

4 Proof of Theorem 2.2

In this Section, we will always assume assumptions g (g

), (g

), (g

) and (g

).

Lemma 4.1. Assume that Ω is symmetric w.r.t. the j–th axis, 1 ≤ j ≤ N , that g(x

, . . . , −x

, . . . , x

, −s) = −g(x, s), and that u solves (1.2). Then also u(x) := −u(x

, . . . , −x

, . . . , x

) solves (1.2).

Note that such a symmetry on g leads to the existence of infinitely many

solutions to problem (1.2) (see, for example, [21]).

Proof. Let ϕ ∈ H

(Ω). Then Z

Ω

Du · Dϕ dx − Z

Ω

h

λu − g(x, u) i

ϕ dx (change x

7→ −x

)

= Z

Ω

Du · Dϕ(x

, . . . , −x

, . . . , x

) dx

− Z

Ω

[λu − g(x, u)]ϕ(x

, . . . , −x

, . . . , x

) dx = 0, since u is a solution of (1.2).

Now set

U

(x) := u

(x) − u

(x).

We will prove Theorem 2.2 by proving the following final

Lemma 4.2. Under the assumptions of Lemma 4.1, if every element in E

(see Section 2) has only one nodal line, which is not closed, then there exists τ

∈ (0, τ

] such that U

≡ 0 for any λ ∈ (λ

− τ

, λ

).

Remark 4.1. The assumption ”every element in E

has only one nodal line, which is not closed” is of course satisfied when i = 2 and the Courant Nodal Theorem holds.

Proof of Lemma 4.2. Assume by contradiction that U

6≡ 0. Of course U

→ 0 in H

(Ω) and uniformly as a consequence of (3.7) and of Proposition 3.2.

Now, we prove that, along sequences, U

/kU

k → ±

λi

e

. Indeed we can assume that U

/kU

k * U in H

(Ω) and a.e. in Ω as λ → λ

. By Lemma 4.1, U

solves

 −∆U

− λU

= g(x, u

) − g(x, u

) in Ω

U

= 0 on ∂Ω.

Thus, if ϕ ∈ H

(Ω), Z

Ω

DU

kU

k · Dϕ dx − λ Z

Ω

U

kU

k ϕ dx = Z

Ω

g(x, u

) − g(x, u

)

kU

k ϕ dx. (4.28) By the Mean Value Theorem, there exists v

with values between u

and u

such that

|g(x, u

) − g(x, u

)| ≤ |g

(x, v

)||U

|.

By assumption g (g

),

|g

(x, v

)||U

| ≤ b(x)|v

|

|U

| ≤ 2

b(x)(ku

k

+ ku

k

)|U

|.

Therefore, (4.28) implies that U solves −∆U = λ

U in H

(Ω). Indeed, 

   Z

Ω

g(x, u

) − g(x, u

) kU

k φ dx

≤ 2

(ku

k

+ ku

k

) Z

Ω

b(x) |U

|

kU

k |φ| dx

≤ 2

(ku

k

+ ku

k

) kU

k

kU

k kbk

kφk

,

and since u

→ 0 uniformly, the assertion is proved, for kU

k

≤ γkU

k by Sobolev inequality (we wrote the estimates in the case N ≥ 3, the case N < 3 being analogous). Thus there exists t ∈ R such that U = te

. But from (4.28) we also get

Z

Ω

|DU

|

kU

k

dx − λ Z

Ω

U

kU

k

dx = Z

Ω

g(x, u

) − g(x, u

) kU

k

U

dx.

Reasoning as above, 

   Z

Ω

g(x, u

) − g(x, u

) kU

k

U

dx

≤ 2

(ku

k

+ ku

k

) kU

k

kU

k

kbk

. Thus, passing to the limit,

1 = λ

Z

Ω

U

dx and therefore

Z

Ω

|DU |

dx = λ

Z

Ω

U

dx = 1, so that U

/kU

k → U strongly in H

(Ω) and t = ±

λi

. Thus there exists e ∈ E

such that

U

kU

k −→ 1

√ λ

e in H

(Ω) and a.e. in Ω. (4.29) Since e changes sign and its nodal line is not closed, and since Ω is symmetric w.r.t. every x

−axes, there exists x = (x

, . . . , x

) ∈ Ω and j ∈ {1, . . . , N } such that e(x) > 0 and e(¯ x) < 0, where we set ¯ x = (x

, . . . , −x

, . . . , x

). Moreover, by (4.29), we can also suppose that

U

(x) kU

k → 1

√ λ

e(x) > 0 (4.30)

and U

(¯ x)

kU

k → 1

√ λ

e(¯ x) < 0. (4.31)

If λ is sufficiently close to λ

, by (4.30) we would have

u

(x) > u

(x) = −u

(x

, . . . , −x

, . . . , x

) (4.32) and by (4.31)

u

(x

, . . . , −x

, . . . , x

) < u

(x) = −u

(x). (4.33) Therefore, (4.32) and (4.33) give u

(x) > u

(x) and a contradiction arises.

Then U

≡ 0.

We are now able to conclude.

Proof of Theorem 2.2. By assumption g(x

, −x

, −s) = −g(x, s), so that Lemma 4.1 implies that both u

(x

, x

) and −u

(x

, −x

) solve problem (1.2). By Theorem 1.2 in [11], since Ω ⊂ R

is convex and symmetric w.r.t.

x

and x

, any second eigenfunction of −∆ in H

(Ω) changes sign exactly once and the eigenspace associated to λ

is spanned by eigenfunctions each of which is odd in one variable and even in the other one; thus any linear combination of them has its nodal line intersecting ∂Ω. Then, by Lemma 4.2 applied for i = N = 2, if λ ∈ (λ

− τ

, λ

) we get u

(x

, x

) = −u

(x

, −x

).

2

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