In previous work by Castro, Cossio, and Neuberger [2], it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is less than the first eigenvalue of -Δ with zero Dirichlet boundry condition. One of these solutions changes sign exactly-once and the other two are of one sign. In this paper we show that when this derivative is between the k-th and k+1-st eigenvalues there still exists a solution which changes sign at most k times. In particular, when k=1 the sign-changing exactly-once solution persists although one-sign solutions no longer exist

Castro, Alfonso

Drabek, Pavel

Neuberger, John M

Alfonso Castro

Pavel Drábek

John M. Neuberger

CiteSeerX

A Sign-Changing Solution For A Superlinear Dirichlet Problem, II

In previous work by Castro, Cossio, and Neuberger cite{ccn}, it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is less than the first eigenvalue of $-Delta$ with zero Dirichlet boundry condition. One of these solutions changes sign exactly-once and the other two are of one sign. In this paper we show that when this derivative is between the $k$-th and $k+1$-st eigenvalues there still exists a solution which changes sign at most $k$ times. In particular, when $k=1$ the sign-changing {it exactly-once} solution persists although one-sign solutions no longer exist

Pavel Drabek

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A sign-changing solution for a superlinear Dirichlet problem, II

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Scholarship @ Claremont
All HMC Faculty Publications and Research HMC Faculty Scholarship
2-1-2003
A Sign-Changing Solution for a Superlinear
Dirichlet Problem, II
Alfonso Castro
Harvey Mudd College
Pavel Drabek
University of West Bohemia
John M. Neuberger
Northern Arizona University
This Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion
in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact
scholarship@cuc.claremont.edu.
Recommended Citation
Castro, Alfonso; Drabek, Pavel; and Neuberger, John M., "A Sign-Changing Solution for a Superlinear Dirichlet Problem, II" (2003).
All HMC Faculty Publications and Research. Paper 475.
http://scholarship.claremont.edu/hmc_fac_pub/475
Fifth Mississippi State Conference on Differential Equations and Computational Simulations,
Electronic Journal of Differential Equations, Conference 10, 2003, pp 101–107.
http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
A sign-changing solution for a superlinear
Dirichlet problem, II ∗
Alfonso Castro, Pavel Dra´bek, & John M. Neuberger
Abstract
In previous work by Castro, Cossio, and Neuberger [2], it was shown
that a superlinear Dirichlet problem has at least three nontrivial solutions
when the derivative of the nonlinearity at zero is less than the first eigen-
value of −∆ with zero Dirichlet boundry condition. One of these solutions
changes sign exactly-once and the other two are of one sign. In this paper
we show that when this derivative is between the k-th and k+1-st eigen-
values there still exists a solution which changes sign at most k times. In
particular, when k = 1 the sign-changing exactly-once solution persists
although one-sign solutions no longer exist.
1 Introduction
Let Ω be a smooth bounded region in RN , ∆ the Laplacian operator, and
f ∈ C1(R,R) such that f(0) = 0. In this paper we study the boundary-value
problem
∆u+ f(u) = 0 in Ω
u = 0 in ∂Ω.
(1.1)
We assume that there exist constants A > 0 and p ∈ (1, N+2N−2 ) such that |f ′(u)| ≤
A(|u|p−1 + 1) for all u ∈ R. Hence f is subcritical, i.e., there exists B > 0 such
that |f(u)| ≤ B(|u|p + 1). Also, we assume that there exists m ∈ (0, 1) and
η > 0 such that
muf(u) ≥ 2F (u) (1.2)
for |u| > η, where F (u) = ∫ u
0
f(s) ds. Finally, we make the assumption that f
satisfies
f ′(u) >
f(u)
u
for u 6= 0, and lim
|u|→∞
f(u)
u
=∞ (f is superlinear). (1.3)
∗Mathematics Subject Classifications: 35J20, 35J25, 35J60.
Key words: Dirichlet problem, superlinear, subcritical, sign-changing solution,
deformation lemma.
c©2003 Southwest Texas State University.
Published February 28, 2003.
P. Drabek was partially supported by Ministry of Education of the Czech Republic,
MSM 235200001.
J. Neuberger was partially supported by the National Science Foundation DMS-0074326.
101
102 A sign-changing solution EJDE–2003/Conf/10
Let H be the Sobolev space H1,20 (Ω) with inner product 〈u, v〉 =
∫
Ω
∇u · ∇v dζ
(see [1] or [8]). Let 0 < λ1 < λ2 ≤ λ3 ≤ · · · be the eigenvalues of −∆ with
zero Dirichlet boundary condition in Ω. We let {φ1, φ2, . . . } denote a complete
orthonormal set in L2(Ω) of eigenfunctions corresponding to the latter eigenval-
ues.
Our main result is as follows.
Theorem 1 If f ′(0) ∈ [λk, λk+1) then (1.1) has a solution w which changes
sign at most k times, i.e., Ω − w−1{0} consists of at most k + 1 non-empty
connected sets.
Corollary 2 If f ′(0) ∈ [λ1, λ2) then (1.1) has a solution w which changes sign
exactly once.
Our proofs here combine Lyapunov-Schmidt reduction arguments [5], the
mountain pass lemma [11], Sard’s Lemma [12], and the index of critical points
of mountain pass type [6].
To the best of our knowledge, [2] was the first to establish the existence of a
sign-changing solution to (1.1) for a general region in the superlinear case where
f ′(0) < λ1. The proofs in [2] are based on the study of the Nehari manifold
S =
{
u ∈ H : u 6= 0,
∫
Ω
(‖∇u‖2 − uf(u)) dζ = 0}.
Unlike the work in [2], where S is a differentible manifold homeomorphic to the
unit sphere and bounded away from 0, here 0 is a limit point of S. Also S ∪ 0
has a singularity at 0. The intersection of S with planes spanned by {φ1, φk},
k = 2, . . . is a figure eight. The semipostione result in [10] is another example
where a more complicated variational structure is successfully analyzed via our
techniques.
For historical remarks concerning the existence of sign changing solutions to
semilinear elliptic boundary value problems we refer the reader to [4]. See also
[13].
Remark 1 One can easily see that when f ′(0) ≥ λ1 there can be no one signed
solutions. In fact suppose to the contrary that f ′(0) ≥ λ1 and that u is (for
example) a positive solution. Let φ1 be a positive eigenfunction corresponding
to λ1. Then, by multiplying (1.1) by φ1 and integrating we obtain∫
Ω
{∆u+ f(u)}φ1 dζ =
∫
Ω
{u∆φ1 + f(u)φ1} dζ
=
∫
Ω
{f(u)
u
− λ1}uφ1 dζ
=
∫
Ω
{f ′(v)− λ1}uφ1 dζ
>
∫
Ω
{f ′(0)− λ1}uφ1 dζ ≥ 0,
(1.4)
where we have used the mean value theorem to find v ∈ (0, u).
EJDE–2003/Conf/10 A. Castro, P. Dra´bek, & J. M. Neuberger 103
Remark 2 Theorem 1 and Corollary 2 are also valid when the Dirichlet bound-
ary condition in (1.1) is replaced by a homogeneous boundary condition for
which the spectrum of the Laplacian operator consists of isolated eigenvalues of
finite multiplicity converging to ∞. This is the case, for example, of the Neu-
mann boundary condition (∂u/∂η)(x) = 0 for region with Lipschitzian bound-
ary.
2 Preliminary Lemmas
Let k be a positive integer and f ′(0) ∈ (λk, λk+1). We define J : H → R by
J(u) =
∫
Ω
{1
2
|∇u|2 − F (u)} dζ.
By regularity theory for elliptic boundary value problems [8], u is a solution
to (1.1) if and only if u is a critical point of J . Because f is subcritical, J ∈
C2(H,R) (see [11]). The gradient and Hessian of J are given by
J ′(u)(v) = 〈∇J(u), v〉 =
∫
Ω
{∇u · ∇v − f(u)v} dζ, for all v ∈ H, (2.1)
and
〈D2J(u)v, w〉 =
∫
Ω
{∇v · ∇w − f ′(u)vw} dζ, for all u, v, w ∈ H. (2.2)
Let X be the linear subspace generated by {φ1, . . . , φk} and Y the subspace
of H generated by {φk+1, . . . }. By orthogonality properties of eigenfunctions
H = X ⊕ Y . Since (1.3) implies that f ′(t) ≥ f ′(0) > λk, one sees that there
exists m1 > 0 such that
〈D2J(u)x, x〉 ≤ −m1‖x‖2 for all u ∈ H, x ∈ X. (2.3)
Arguing as in Theorem 4 of [5], one sees that there exists a function ψ ∈
C1(Y,X) such that
Jˆ(y) ≡ J(y + ψ(y)) = max
x∈X
J(x+ y) for all y ∈ Y, (2.4)
and
〈∇Jˆ(y), v〉 = 〈∇J(y + ψ(y)), v〉 =
∫
Ω
{∇y · ∇v − f(y + ψ(y))v} dζ, (2.5)
for all y, v ∈ Y . In addition,
ψ(y) is the only critical point of x→ J(x+ y) for all y ∈ Y. (2.6)
Thus
y is a critical point of Jˆ if and only if y + ψ(y) is a critical point of J . (2.7)
104 A sign-changing solution EJDE–2003/Conf/10
Although not obvious (see [5]), Jˆ is of class C2 in spite of only ψ ∈ C1(Y,X).
From (2.5) we see that ∇Jˆ(y) = y +K(y) where K is a compact function of y.
Also since ∇Jˆ is a variational vector field of class C1 we have dimker(∇Jˆ)′(y)
= dimker(D2Jˆ(y)) = codim(D2(Jˆ(y))(Y )) = codim(∇J)′(y)(Y ) for all y ∈ Y .
Thus we may apply Sard’s lemma [12] to conlcude the following lemma.
Lemma 1 There exists {qn} ⊂ Y with qn ↓ 0 as n→∞ such that if ∇Jˆ(u) =
qn then the Hessian D2Jˆ(u) is invertible.
Proof This proof follows immediately from the fact that the regular values of
∇Jˆ is the complement of a set of first category. In partiuclar it is dense. ♦
Let qn be as in the previous lemma. We define Jn : Y → R by Jn(y) =
Jˆ(y)−〈qn, y〉. We note that D2Jn(y) = D2Jˆ(y) for each y ∈ Y . Also from (12)
of [5]
〈D2Jˆ(y)h, h〉 = 〈D2J(y + ψ(y))(h+ ψ′(y)h), h+ ψ′(y)h〉. (2.8)
Lemma 2 The functional Jn has a critical point yn such that the Morse index
of D2J(yn + ψ(yn)) is less than or equal to k + 1.
proof In order to establish the existence of the critical point un we prove that
Jn satisifies the hypotheses of the Mountain Pass Lemma [11]. Since f ′(0) <
λk+1, we have 〈D2J(0)y, y〉 = J ′′(0)(y, y) ≥ (1 − f ′(0)/λk+1)
∫
Ω
|∇y|2 dζ > 0
for y ∈ Y . Thus 0 is a strict local minimum of J restricted to Y . Thus there
exist δ, η > 0 such that J(y) ≥ η for all ||y|| = δ. This and (2.4) imply that for
‖y‖ = δ we have
Jˆ(y) ≥ J(y) ≥ η > 0 . (2.9)
Now taking n sufficiently large so that ‖qn‖ ≤ η/(2δ) we have
Jn(y) ≥ η − ‖qn‖δ > η/2 > 0 , (2.10)
for ‖y‖ = δ.
Next we note that, since 0 is a critical point of J , ψ(0) = 0. Hence Jn(0) = 0.
Additionally, since we are assuming f to be superlinear, there exit numbers
m1 > λk+1 and m2 such that
2F (t) ≥ m1t2 +m2 (2.11)
for all t ∈ R. Hence
Jn(tφk+1) = Jˆ(tφk+1)− 〈qn, tφk+1〉 ≤ J(tφk+1)− 〈qn, tφk+1〉
≤ (t‖φk+1‖2 −m1
∫
Ω
(tφk+1)2 dζ −m2|Ω|)/2 + t‖qn‖‖φk+1‖
→ −∞ as t→ +∞,
(2.12)
where we have used that ‖φk+1‖2 = λk+1
∫
Ω
φ2k+1 dζ. For future reference we
note that, without loss of generality, we may assume that ‖qn‖ ≤ 1 for all n.
EJDE–2003/Conf/10 A. Castro, P. Dra´bek, & J. M. Neuberger 105
Thus (2.12) implies that
Jn(tφk+1) ≤ (λk+1 −m1)(−m2|Ω|)− λk+1‖φk+1‖2(λk+1 −m1) ≡ K˜ for all n, t ≥ 0.
(2.13)
From (2.12), for each n, there exists a real number tn > 0 with ‖tnφk+1‖ ≥ 2δ
such that Jˆn(tnφk+1) < 0.
Next we show that Jn satisfies the Palais-Smale condition. Suppose that
{yj} is a sequence so that {Jn(yj)} is bounded, say |Jn(yj)| ≤ M for all j
and ∇Jn(yj) → 0 as j → ∞. For ease of notation, let u = yj + ψ(yj) and
T = m2 uf(u)− F (u). Then
M +
m
2
||u|| ≥ Jn(yj)− m2 [〈∇Jn(yj), yj〉+ 〈∇J(yj + ψ(yj)), ψ(yj)〉]
= (
1
2
− m
2
)||u||2 +
∫
Ω
T dζ − (1− m
2
)〈qn, u〉
≥ (1
4
− m
4
)(||u||2 − ‖qn‖2) +M1|Ω|,
(2.14)
where M1 ∈ R is a lower bound for T (see (1.2)). The latter inequality implies
that {yj + ψ(yj)} is bounded. Hence without loss of generality we may assume
that {yj} converges weakly to y¯ ∈ Y and that ψ(yj) converges to x¯ ∈ X. Since
the imbedding of H in Lp+1(Ω) is compact, we may assume that f(yj + ψ(yj))
converges strongly in L1(Ω). Since ∇Jn(yj) = yj + K(yj) + qn → 0 with K
compact, we see that {yj} has a convergent subsequence. This proves that Jn
satisfies the Palais-Smale condition.
Now by the Mountain Pass Lemma there exists yn ∈ Y such that ∇Jn(yn) =
0 and
Jn(yn) = inf
σ∈Σ
[
max
t∈[0,1]
Jn(σ(t))
]
, (2.15)
where Σ = {σ : [0, 1]→ Y ;σ is continuous, σ(0) = 0 and σ(1) = tnφk+1}. Thus
from (2.10), (2.13) and (2.15) we have
η/2 ≤ Jn(yn) ≤ K˜. (2.16)
In addition, since D2Jn(yn) is invertible (see Lemma 1), by Theorem 2 of [6],
we may assume that the Morse index of D2Jn(yn) is 1. Since D2J(yn + ψ(yn))
is negative definite in a subspace of dimension k, namely X, then we conclude
that the Morse index of D2J(yn + ψ(yn)) is at most k + 1, and this concludes
the proof. ♦
3 Proof of Main Theorem
First we consider the case f ′(0) > λk. Let {yn} be as in Lemma 2. By (2.16)
one sees that {yn+ψ(yn)} is bounded. Thus, without loss of generality, we may
assume that {yn} converges weakly to y¯ ∈ Y and {ψ(yn)} converges to x¯ ∈ X.
106 A sign-changing solution EJDE–2003/Conf/10
From (2.4) we have 0 = ∇Jn(yn) = yn +K(yn)− qn where K is a compact op-
erator. Since, in addition {qn} converges to 0, actually {yn} converges strongly
to y¯. Also since Jn(yn) ≥ η/2 > 0, we see that Jˆ(y¯) ≥ η/2 > 0 and y¯ 6= 0. Now
for v ∈ X one has
〈x¯, v〉 −
∫
Ω
{f(y¯ + x¯)v} dζ
= lim
n→∞
[〈ψ(yn), v〉 − ∫
Ω
{vf(yn + ψ(yn))} dζ − 〈qn, ψ(yn)〉
]
= 0.
(3.1)
Hence x¯ = ψ(y¯) and ∇J(x¯ + y¯) = 0. Thus x¯ + y¯ 6= 0 is a solution to (1.1).
Let us see that x¯ + y¯ has at most k + 1 nodal regions. If not, by defining vj ,
j = 1, . . . , k+2, as x¯+ y¯ onWj and as zero on Ω¯−Wj , then from (1.3), (2.1) and
(2.2), we see that 〈D2J(x¯+ y¯)vj , vj〉 < 0. Since the v′js are mutually orthogonal
then we have thatD2J(x¯+y¯) is negative definite on a k+2-dimensional subspace.
By continuity then D2J(yn + ψ(yn)) is negative definite on the same k + 2-
dimensional subspace. This contradicts that the Morse index ofD2J(yn+ψ(yn))
is less than or equal to k+1. This contradiction proves that x¯+ y¯ is a solution
to (1.1) having at most k + 1 nodal regions.
Finally we consider the case f ′(0) = λk. Let {j} be a sequence of positive
numbers converging to 0. Without loss of generality we may assume that j <
(λk+1−λk)/2 for all positive integers j. By our previous arguments, there exists
a sequence {uj = x¯j + y¯j} of functions in H that satisfy
∆uj + juj + f(uj) = 0 in Ω
uj = 0 in ∂Ω.
(3.2)
In addition, each xj + yj is the limit a of a sequence {xn,j + yn,j} with each
xn,j + yn,j satisfying (2.16). Hence, by continuity η/2 ≤ J(x¯j + y¯j) ≤ K˜ and∫
Ω
{∇v · ∇v − (f ′(uj) + j)v2} dζ
defines a quadratic form of Morse index at most k + 1. Arguing as above one
sees that x¯j + y¯j has a convergent subsequence with limit x¯+ y¯. The function
x¯+ y¯ is a solution to (1.1) and, as in the case f ′(0) > λk, it has at most k + 1
nodal regions. This concludes the proof of our main theorem; the corollary is
obvious.
References
[1] R. Adams, Sobolev Spaces, New York: Academic Press (1975).
[2] A. Castro, J. Cossio and J. M. Neuberger, A Sign-Changing Solution for
a Superlinear Dirichlet Problem, Rocky Mountain J. of Math., 27, No. 4
(1997), pp. 1041-1053.
EJDE–2003/Conf/10 A. Castro, P. Dra´bek, & J. M. Neuberger 107
[3] A. Castro, J. Cossio and J. M. Neuberger, On Multiple Solutions of a Non-
linear Dirichlet Problem, Nonlinear Analysis TMA, 30, No. 6 (1997), pp.
3657-3662.
[4] A. Castro, J. Cossio and J.. M. Neuberger, A Minmax Principle, Index of the
Critical Point, and Existence of Sign-Changing Solutions to Elliptic Bound-
ary Value Problems, Electronic J. of Diff. Eq., Vol. 1998 (1998), No. 2, pp.
1-18.
[5] A. Castro, and A. C. Lazer, Critical Point Theory and the Number of So-
lutions of a Nonlinear Dirichlet Problem, Annli di Mat. Pura ed Applicata
(IV), Vol. CXX (1979), pp. 113-137.
[6] H. Hofer, The Topological Degree at a Critical Point of Mountain Pass Type,
Proceedings of Symposia in Pure Mathematics, 45 Part I (1986).
[7] D. Kinderlehrer and G. Stampacchia, Introduction to Variational Inequalities
and Their Applications, New York: Academic Press (1979).
[8] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Sec-
ond Order, Berlin, New York: Springer-Verlag (1983).
[9] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory,
Cambridge Tracts in Mathematics, Cambridge University Press (1993).
[10] John M. Neuberger, A Sign-Changing Solution for a Superlinear Dirichlet
Problem with a Reaction Term Nonzero at Zero, Nonlinear Analysis, 33
(1998), pp. 427-441.
[11] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applica-
tions to Differential Equations, Regional Conference Series in Mathematics,
65, Providence,R.I.: AMS (1986).
[12] S. Smale, An Infinite Dimensional Version of Sard’s Theorem, Amer. J.
Math., 87 (1965), pp. 861-866.
[13] Z. Q. Wang, On a Superlinear Elliptic Equation, Ann. Inst. H. Poincare
Analyse Non Lineaire 8 (1991) 43-57.
Alfonso Castro
Division of Mathematics and Statistics, University of Texas at San Antonio,
San Antonio, TX 78249-0664, USA
E-mail: acastro@utsa.edu
Pavel Drabek
Department of Mathematics, University of West Bohemia,
306 14 Pilsen, Czech Republic.
E-mail address: pdrabek@kma.zcu.cz
John M. Neuberger
Department of Mathematics, Northern Arizona University,
Flagstaff, AZ 86011-5717 USA.
E-mail address: John.Neuberger@nau.edu


English

A Sign-Changing Solution for a Superlinear Dirichlet Problem, II

Scholarship@Claremont

Claremont CollegesScholarship @ ClaremontAll HMC Faculty Publications and Research HMC Faculty Scholarship2-1-2003A Sign-Changing Solution for a SuperlinearDirichlet Problem, IIAlfonso CastroHarvey Mudd CollegePavel DrabekUniversity of West BohemiaJohn M. NeubergerNorthern Arizona UniversityThis Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusionin All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contactscholarship@cuc.claremont.edu.Recommended CitationCastro, Alfonso; Drabek, Pavel; and Neuberger, John M., "A Sign-Changing Solution for a Superlinear Dirichlet Problem, II" (2003).All HMC Faculty Publications and Research. Paper 475.http://scholarship.claremont.edu/hmc_fac_pub/475Fifth Mississippi State Conference on Differential Equations and Computational Simulations,Electronic Journal of Differential Equations, Conference 10, 2003, pp 101–107.http://ejde.math.swt.edu or http://ejde.math.unt.eduftp ejde.math.swt.edu (login: ftp)A sign-changing solution for a superlinearDirichlet problem, II ∗Alfonso Castro, Pavel Dra´bek, & John M. NeubergerAbstractIn previous work by Castro, Cossio, and Neuberger [2], it was shownthat a superlinear Dirichlet problem has at least three nontrivial solutionswhen the derivative of the nonlinearity at zero is less than the first eigen-value of −∆ with zero Dirichlet boundry condition. One of these solutionschanges sign exactly-once and the other two are of one sign. In this paperwe show that when this derivative is between the k-th and k+1-st eigen-values there still exists a solution which changes sign at most k times. Inparticular, when k = 1 the sign-changing exactly-once solution persistsalthough one-sign solutions no longer exist.1 IntroductionLet Ω be a smooth bounded region in RN , ∆ the Laplacian operator, andf ∈ C1(R,R) such that f(0) = 0. In this paper we study the boundary-valueproblem∆u+ f(u) = 0 in Ωu = 0 in ∂Ω.(1.1)We assume that there exist constants A > 0 and p ∈ (1, N+2N−2 ) such that |f ′(u)| ≤A(|u|p−1 + 1) for all u ∈ R. Hence f is subcritical, i.e., there exists B > 0 suchthat |f(u)| ≤ B(|u|p + 1). Also, we assume that there exists m ∈ (0, 1) andη > 0 such thatmuf(u) ≥ 2F (u) (1.2)for |u| > η, where F (u) = ∫ u0f(s) ds. Finally, we make the assumption that fsatisfiesf ′(u) >f(u)ufor u 6= 0, and lim|u|→∞f(u)u=∞ (f is superlinear). (1.3)∗Mathematics Subject Classifications: 35J20, 35J25, 35J60.Key words: Dirichlet problem, superlinear, subcritical, sign-changing solution,deformation lemma.c©2003 Southwest Texas State University.Published February 28, 2003.P. Drabek was partially supported by Ministry of Education of the Czech Republic,MSM 235200001.J. Neuberger was partially supported by the National Science Foundation DMS-0074326.101102 A sign-changing solution EJDE–2003/Conf/10Let H be the Sobolev space H1,20 (Ω) with inner product 〈u, v〉 =∫Ω∇u · ∇v dζ(see [1] or [8]). Let 0 < λ1 < λ2 ≤ λ3 ≤ · · · be the eigenvalues of −∆ withzero Dirichlet boundary condition in Ω. We let {φ1, φ2, . . . } denote a completeorthonormal set in L2(Ω) of eigenfunctions corresponding to the latter eigenval-ues.Our main result is as follows.Theorem 1 If f ′(0) ∈ [λk, λk+1) then (1.1) has a solution w which changessign at most k times, i.e., Ω − w−1{0} consists of at most k + 1 non-emptyconnected sets.Corollary 2 If f ′(0) ∈ [λ1, λ2) then (1.1) has a solution w which changes signexactly once.Our proofs here combine Lyapunov-Schmidt reduction arguments [5], themountain pass lemma [11], Sard’s Lemma [12], and the index of critical pointsof mountain pass type [6].To the best of our knowledge, [2] was the first to establish the existence of asign-changing solution to (1.1) for a general region in the superlinear case wheref ′(0) < λ1. The proofs in [2] are based on the study of the Nehari manifoldS ={u ∈ H : u 6= 0,∫Ω(‖∇u‖2 − uf(u)) dζ = 0}.Unlike the work in [2], where S is a differentible manifold homeomorphic to theunit sphere and bounded away from 0, here 0 is a limit point of S. Also S ∪ 0has a singularity at 0. The intersection of S with planes spanned by {φ1, φk},k = 2, . . . is a figure eight. The semipostione result in [10] is another examplewhere a more complicated variational structure is successfully analyzed via ourtechniques.For historical remarks concerning the existence of sign changing solutions tosemilinear elliptic boundary value problems we refer the reader to [4]. See also[13].Remark 1 One can easily see that when f ′(0) ≥ λ1 there can be no one signedsolutions. In fact suppose to the contrary that f ′(0) ≥ λ1 and that u is (forexample) a positive solution. Let φ1 be a positive eigenfunction correspondingto λ1. Then, by multiplying (1.1) by φ1 and integrating we obtain∫Ω{∆u+ f(u)}φ1 dζ =∫Ω{u∆φ1 + f(u)φ1} dζ=∫Ω{f(u)u− λ1}uφ1 dζ=∫Ω{f ′(v)− λ1}uφ1 dζ>∫Ω{f ′(0)− λ1}uφ1 dζ ≥ 0,(1.4)where we have used the mean value theorem to find v ∈ (0, u).EJDE–2003/Conf/10 A. Castro, P. Dra´bek, & J. M. Neuberger 103Remark 2 Theorem 1 and Corollary 2 are also valid when the Dirichlet bound-ary condition in (1.1) is replaced by a homogeneous boundary condition forwhich the spectrum of the Laplacian operator consists of isolated eigenvalues offinite multiplicity converging to ∞. This is the case, for example, of the Neu-mann boundary condition (∂u/∂η)(x) = 0 for region with Lipschitzian bound-ary.2 Preliminary LemmasLet k be a positive integer and f ′(0) ∈ (λk, λk+1). We define J : H → R byJ(u) =∫Ω{12|∇u|2 − F (u)} dζ.By regularity theory for elliptic boundary value problems [8], u is a solutionto (1.1) if and only if u is a critical point of J . Because f is subcritical, J ∈C2(H,R) (see [11]). The gradient and Hessian of J are given byJ ′(u)(v) = 〈∇J(u), v〉 =∫Ω{∇u · ∇v − f(u)v} dζ, for all v ∈ H, (2.1)and〈D2J(u)v, w〉 =∫Ω{∇v · ∇w − f ′(u)vw} dζ, for all u, v, w ∈ H. (2.2)Let X be the linear subspace generated by {φ1, . . . , φk} and Y the subspaceof H generated by {φk+1, . . . }. By orthogonality properties of eigenfunctionsH = X ⊕ Y . Since (1.3) implies that f ′(t) ≥ f ′(0) > λk, one sees that thereexists m1 > 0 such that〈D2J(u)x, x〉 ≤ −m1‖x‖2 for all u ∈ H, x ∈ X. (2.3)Arguing as in Theorem 4 of [5], one sees that there exists a function ψ ∈C1(Y,X) such thatJˆ(y) ≡ J(y + ψ(y)) = maxx∈XJ(x+ y) for all y ∈ Y, (2.4)and〈∇Jˆ(y), v〉 = 〈∇J(y + ψ(y)), v〉 =∫Ω{∇y · ∇v − f(y + ψ(y))v} dζ, (2.5)for all y, v ∈ Y . In addition,ψ(y) is the only critical point of x→ J(x+ y) for all y ∈ Y. (2.6)Thusy is a critical point of Jˆ if and only if y + ψ(y) is a critical point of J . (2.7)104 A sign-changing solution EJDE–2003/Conf/10Although not obvious (see [5]), Jˆ is of class C2 in spite of only ψ ∈ C1(Y,X).From (2.5) we see that ∇Jˆ(y) = y +K(y) where K is a compact function of y.Also since ∇Jˆ is a variational vector field of class C1 we have dimker(∇Jˆ)′(y)= dimker(D2Jˆ(y)) = codim(D2(Jˆ(y))(Y )) = codim(∇J)′(y)(Y ) for all y ∈ Y .Thus we may apply Sard’s lemma [12] to conlcude the following lemma.Lemma 1 There exists {qn} ⊂ Y with qn ↓ 0 as n→∞ such that if ∇Jˆ(u) =qn then the Hessian D2Jˆ(u) is invertible.Proof This proof follows immediately from the fact that the regular values of∇Jˆ is the complement of a set of first category. In partiuclar it is dense. ♦Let qn be as in the previous lemma. We define Jn : Y → R by Jn(y) =Jˆ(y)−〈qn, y〉. We note that D2Jn(y) = D2Jˆ(y) for each y ∈ Y . Also from (12)of [5]〈D2Jˆ(y)h, h〉 = 〈D2J(y + ψ(y))(h+ ψ′(y)h), h+ ψ′(y)h〉. (2.8)Lemma 2 The functional Jn has a critical point yn such that the Morse indexof D2J(yn + ψ(yn)) is less than or equal to k + 1.proof In order to establish the existence of the critical point un we prove thatJn satisifies the hypotheses of the Mountain Pass Lemma [11]. Since f ′(0) <λk+1, we have 〈D2J(0)y, y〉 = J ′′(0)(y, y) ≥ (1 − f ′(0)/λk+1)∫Ω|∇y|2 dζ > 0for y ∈ Y . Thus 0 is a strict local minimum of J restricted to Y . Thus thereexist δ, η > 0 such that J(y) ≥ η for all ||y|| = δ. This and (2.4) imply that for‖y‖ = δ we haveJˆ(y) ≥ J(y) ≥ η > 0 . (2.9)Now taking n sufficiently large so that ‖qn‖ ≤ η/(2δ) we haveJn(y) ≥ η − ‖qn‖δ > η/2 > 0 , (2.10)for ‖y‖ = δ.Next we note that, since 0 is a critical point of J , ψ(0) = 0. Hence Jn(0) = 0.Additionally, since we are assuming f to be superlinear, there exit numbersm1 > λk+1 and m2 such that2F (t) ≥ m1t2 +m2 (2.11)for all t ∈ R. HenceJn(tφk+1) = Jˆ(tφk+1)− 〈qn, tφk+1〉 ≤ J(tφk+1)− 〈qn, tφk+1〉≤ (t‖φk+1‖2 −m1∫Ω(tφk+1)2 dζ −m2|Ω|)/2 + t‖qn‖‖φk+1‖→ −∞ as t→ +∞,(2.12)where we have used that ‖φk+1‖2 = λk+1∫Ωφ2k+1 dζ. For future reference wenote that, without loss of generality, we may assume that ‖qn‖ ≤ 1 for all n.EJDE–2003/Conf/10 A. Castro, P. Dra´bek, & J. M. Neuberger 105Thus (2.12) implies thatJn(tφk+1) ≤ (λk+1 −m1)(−m2|Ω|)− λk+1‖φk+1‖2(λk+1 −m1) ≡ K˜ for all n, t ≥ 0.(2.13)From (2.12), for each n, there exists a real number tn > 0 with ‖tnφk+1‖ ≥ 2δsuch that Jˆn(tnφk+1) < 0.Next we show that Jn satisfies the Palais-Smale condition. Suppose that{yj} is a sequence so that {Jn(yj)} is bounded, say |Jn(yj)| ≤ M for all jand ∇Jn(yj) → 0 as j → ∞. For ease of notation, let u = yj + ψ(yj) andT = m2 uf(u)− F (u). ThenM +m2||u|| ≥ Jn(yj)− m2 [〈∇Jn(yj), yj〉+ 〈∇J(yj + ψ(yj)), ψ(yj)〉]= (12− m2)||u||2 +∫ΩT dζ − (1− m2)〈qn, u〉≥ (14− m4)(||u||2 − ‖qn‖2) +M1|Ω|,(2.14)where M1 ∈ R is a lower bound for T (see (1.2)). The latter inequality impliesthat {yj + ψ(yj)} is bounded. Hence without loss of generality we may assumethat {yj} converges weakly to y¯ ∈ Y and that ψ(yj) converges to x¯ ∈ X. Sincethe imbedding of H in Lp+1(Ω) is compact, we may assume that f(yj + ψ(yj))converges strongly in L1(Ω). Since ∇Jn(yj) = yj + K(yj) + qn → 0 with Kcompact, we see that {yj} has a convergent subsequence. This proves that Jnsatisfies the Palais-Smale condition.Now by the Mountain Pass Lemma there exists yn ∈ Y such that ∇Jn(yn) =0 andJn(yn) = infσ∈Σ[maxt∈[0,1]Jn(σ(t))], (2.15)where Σ = {σ : [0, 1]→ Y ;σ is continuous, σ(0) = 0 and σ(1) = tnφk+1}. Thusfrom (2.10), (2.13) and (2.15) we haveη/2 ≤ Jn(yn) ≤ K˜. (2.16)In addition, since D2Jn(yn) is invertible (see Lemma 1), by Theorem 2 of [6],we may assume that the Morse index of D2Jn(yn) is 1. Since D2J(yn + ψ(yn))is negative definite in a subspace of dimension k, namely X, then we concludethat the Morse index of D2J(yn + ψ(yn)) is at most k + 1, and this concludesthe proof. ♦3 Proof of Main TheoremFirst we consider the case f ′(0) > λk. Let {yn} be as in Lemma 2. By (2.16)one sees that {yn+ψ(yn)} is bounded. Thus, without loss of generality, we mayassume that {yn} converges weakly to y¯ ∈ Y and {ψ(yn)} converges to x¯ ∈ X.106 A sign-changing solution EJDE–2003/Conf/10From (2.4) we have 0 = ∇Jn(yn) = yn +K(yn)− qn where K is a compact op-erator. Since, in addition {qn} converges to 0, actually {yn} converges stronglyto y¯. Also since Jn(yn) ≥ η/2 > 0, we see that Jˆ(y¯) ≥ η/2 > 0 and y¯ 6= 0. Nowfor v ∈ X one has〈x¯, v〉 −∫Ω{f(y¯ + x¯)v} dζ= limn→∞[〈ψ(yn), v〉 − ∫Ω{vf(yn + ψ(yn))} dζ − 〈qn, ψ(yn)〉]= 0.(3.1)Hence x¯ = ψ(y¯) and ∇J(x¯ + y¯) = 0. Thus x¯ + y¯ 6= 0 is a solution to (1.1).Let us see that x¯ + y¯ has at most k + 1 nodal regions. If not, by defining vj ,j = 1, . . . , k+2, as x¯+ y¯ onWj and as zero on Ω¯−Wj , then from (1.3), (2.1) and(2.2), we see that 〈D2J(x¯+ y¯)vj , vj〉 < 0. Since the v′js are mutually orthogonalthen we have thatD2J(x¯+y¯) is negative definite on a k+2-dimensional subspace.By continuity then D2J(yn + ψ(yn)) is negative definite on the same k + 2-dimensional subspace. This contradicts that the Morse index ofD2J(yn+ψ(yn))is less than or equal to k+1. This contradiction proves that x¯+ y¯ is a solutionto (1.1) having at most k + 1 nodal regions.Finally we consider the case f ′(0) = λk. Let {j} be a sequence of positivenumbers converging to 0. Without loss of generality we may assume that j <(λk+1−λk)/2 for all positive integers j. By our previous arguments, there existsa sequence {uj = x¯j + y¯j} of functions in H that satisfy∆uj + juj + f(uj) = 0 in Ωuj = 0 in ∂Ω.(3.2)In addition, each xj + yj is the limit a of a sequence {xn,j + yn,j} with eachxn,j + yn,j satisfying (2.16). Hence, by continuity η/2 ≤ J(x¯j + y¯j) ≤ K˜ and∫Ω{∇v · ∇v − (f ′(uj) + j)v2} dζdefines a quadratic form of Morse index at most k + 1. Arguing as above onesees that x¯j + y¯j has a convergent subsequence with limit x¯+ y¯. The functionx¯+ y¯ is a solution to (1.1) and, as in the case f ′(0) > λk, it has at most k + 1nodal regions. This concludes the proof of our main theorem; the corollary isobvious.References[1] R. Adams, Sobolev Spaces, New York: Academic Press (1975).[2] A. Castro, J. Cossio and J. M. Neuberger, A Sign-Changing Solution fora Superlinear Dirichlet Problem, Rocky Mountain J. of Math., 27, No. 4(1997), pp. 1041-1053.EJDE–2003/Conf/10 A. Castro, P. Dra´bek, & J. M. Neuberger 107[3] A. Castro, J. Cossio and J. M. Neuberger, On Multiple Solutions of a Non-linear Dirichlet Problem, Nonlinear Analysis TMA, 30, No. 6 (1997), pp.3657-3662.[4] A. Castro, J. Cossio and J.. M. Neuberger, A Minmax Principle, Index of theCritical Point, and Existence of Sign-Changing Solutions to Elliptic Bound-ary Value Problems, Electronic J. of Diff. Eq., Vol. 1998 (1998), No. 2, pp.1-18.[5] A. Castro, and A. C. Lazer, Critical Point Theory and the Number of So-lutions of a Nonlinear Dirichlet Problem, Annli di Mat. Pura ed Applicata(IV), Vol. CXX (1979), pp. 113-137.[6] H. Hofer, The Topological Degree at a Critical Point of Mountain Pass Type,Proceedings of Symposia in Pure Mathematics, 45 Part I (1986).[7] D. Kinderlehrer and G. Stampacchia, Introduction to Variational Inequalitiesand Their Applications, New York: Academic Press (1979).[8] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Sec-ond Order, Berlin, New York: Springer-Verlag (1983).[9] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory,Cambridge Tracts in Mathematics, Cambridge University Press (1993).[10] John M. Neuberger, A Sign-Changing Solution for a Superlinear DirichletProblem with a Reaction Term Nonzero at Zero, Nonlinear Analysis, 33(1998), pp. 427-441.[11] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applica-tions to Differential Equations, Regional Conference Series in Mathematics,65, Providence,R.I.: AMS (1986).[12] S. Smale, An Infinite Dimensional Version of Sard’s Theorem, Amer. J.Math., 87 (1965), pp. 861-866.[13] Z. Q. Wang, On a Superlinear Elliptic Equation, Ann. Inst. H. PoincareAnalyse Non Lineaire 8 (1991) 43-57.Alfonso CastroDivision of Mathematics and Statistics, University of Texas at San Antonio,San Antonio, TX 78249-0664, USAE-mail: acastro@utsa.eduPavel DrabekDepartment of Mathematics, University of West Bohemia,306 14 Pilsen, Czech Republic.E-mail address: pdrabek@kma.zcu.czJohn M. NeubergerDepartment of Mathematics, Northern Arizona University,Flagstaff, AZ 86011-5717 USA.E-mail address: John.Neuberger@nau.edu

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A Sign-Changing Solution for a Superlinear Dirichlet Problem, II

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