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Abstract

We consider a class of elliptic equation with natural growth. We obtain a region of the natural growth term with precise lower boundary less than zero.
Topological Methods in Nonlinear Analysis
Journal of the Juliusz Schauder Center
Volume 35, 2010, 203–219
AN APPLICATION
OF NONSMOOTH CRITICAL POINT THEORY
Zhouxin Li — Yaotian Shen — Yimin Zhang
Abstract. We consider a class of elliptic equation with natural growth.
We obtain a region of the natural growth term with precise lower boundary
less than zero.
1. Introduction and main results
Let Ω be a bounded domain in RN(N3) with smooth boundary. In this
paper we consider the functional I:W1,p
0(Ω) R, 2 p < N , given by
(1.1) I(u) = Z
j(x, u, u)Z
G(x, u).
Here j(x, s, ξ): Ω ×R×RNis a function which is measurable with respect to x
for all (s, ξ)R×RN, and of class C1with respect to (s, ξ) for almoast every
xΩ, G(x, s) = Rs
0g(x, t)dt, where g(x, s) is a Carath´eodory function.
We are concerned with the existence and nonexistence of nontrivial critical
points of the functional I. Let js(x, s, ξ) and jξ(x, s, ξ ) denote the derivatives of
j(x, s, ξ) with respect to sand ξrespectively, we know that the Euler–Lagrange
2010 Mathematics Subject Classification. 35J45, 35J55.
Key words and phrases. Nonsmooth critical point theory, elliptic equation, natural growth.
This work is supported by National Natural Science Foundation of China under grant
numbers 10771074.
c
2010 Juliusz Schauder Center for Nonlinear Studies
203
204 Z. Li — Y. Shen — Y. Zhang
equation of the functional Iis
(1.2) (div(jξ(x, u, u)) + js(x, u, u) = g(x, u) in Ω,
u= 0 on Ω.
As pointed out by D. Arcoya and L. Boccardo [1] (one can see also [6]) that since
the function j(x, u, u) depends on u, the functional Iis not even Gˆateaux differ-
entiable on W1,p
0(Ω) but only differentiable along directions in W1,p
0(Ω)L(Ω).
For example, when p= 2, if we set j(x, u, u) = u2|∇u|2, then js(x, u, u) =
2u|∇u|2, it is easy to verify that 2u|∇u|2not necessarily belong to W1,p0(Ω),
the topological dual of W1,p
0(Ω). js(x, s, ξ) is called the natural growth term of
Problem (1.2).
The study of Problem (1.2) arise from more concrete case as, for example,
when p= 2,
(1.3)
N
X
ij=1
Dj(aij (x, u)Diu)
+1
2
N
X
ij=1
saij (x, u)DiuDju=g(x, u) in Ω,
u= 0 on Ω.
Existence and multiplicity results for equations like (1.3) have been object of
a very careful analysis since 1994 (see e.g. [1], [3], [4], [6], [13] and references
therein). In these papers, the approaches are variational and the nontrivial crit-
ical points were obtained via the techniques of nonsmooth critical point theory.
In order to get the compactness result, one of the important assumptions in
[1], [3], [4], [6] is
s
N
X
ij=1
saij (x, s)ξiξj0.
This sign condition plays an important part in the proof of the compactness for
a Palais-Smale sequence. But if we consider another assumption, which says,
there exist ν, σ with 2 < σ < 2N /(N2), ν(0, σ 2) such that
s
N
X
ij=1
saij (x, s)ξiξjν
N
X
ij=1
aij (x, s)ξiξj,
we find that the region in which sPN
ij=1 saij (x, s)ξiξjexists will vanish as the
parameter σtends to 2. It is interesting that the author in [13] studied the case
σ= 2, and assume that there exists 0 < α1<1 such that
2α1
N
X
i,j
aij (x, s)ξiξjs
N
X
i,j
saij (x, s)ξiξj0.
An Application of Nonsmooth Critical Point Theory 205
Under this condition and other certain hypotheses, the author in [13] proved
that there exists at least one weak solution of problem (1.3).
As to the general case j(x, s, ξ), problem (1.1) was also studied by many
authors, see for example [11] for p= 2 and [1], [15], [16] for 1 p < N. In
these papers, the assumption on the natural growth term is sjs(x, s, ξ)0. As
mentioned above, this sign condition plays an important part in the proof of
the compactness for a Palais–Smale sequence. On the other hand, the technique
in [11], [15], [16] is variational via the nonsmooth critical point theory based
on the notion of weak slope proposed by J. N. Corvellec, M. Degiovanni and
M. Marzocchi in [7], [8], which is different to [1].
Let a0, 2 p < N and
j(x, s, ξ) = 1
p1 + 1
1 + |s|a|ξ|p,
then by direct computation, one gets
sjs(x, s, ξ) = 1
p
a|s|a
(1 + |s|a)2|ξ|p0.
The equality holds if and only if s= 0 or ξ= 0. Thus, it remains an in-
teresting question whether problem (1.1) has a nontrivial critical point when
sjs(x, s, ξ)<0.
In this paper we discuss the general case of problem (1.1) for j(x, s, ξ) and
2p<N. Motivated by [11], [13], we study the existence of nontrivial critical
points for Problem (1.1) when the sign condition is dropped (see condition (j3)
in this section).
A crucial step in proving our main result is to show the compactness of the
Palais–Smale sequence of the functional Iwhen the sign condition is dropped.
Under certain hypotheses on the functions jand g, we get the desire result.
We use the nonsmooth critical point theory in [7], [8] to prove the existence of
one or infinitely many nontrivial critical points of I. Moreover, we use the Po-
hozaev identity in [12] to prove that the corresponding Euler–Lagrange equation
of problem (1.1), i.e. (1.2), has no weak solution in C2(Ω) C1(Ω) when Ω is
star-shaped and condition (j3) fails (see (j6) in this section).
In this paper, we give the following assumptions on the functions jand g.
The function j(x, s, ξ): Ω ×R×RNis measurable with respect to xfor all
(s, ξ)R×RN, and of class C1with respect to (s, ξ) for almost every xΩ.
We also assume that there exist α, β with βα > 0 and γ > 0 such that for
almost every xΩ and every (s, ξ)R×RN
α|ξ|pjξ(x, s, ξ)ξβ|ξ|p,(j1)
|js(x, s, ξ)| γ|ξ|p.(j2)
206 Z. Li — Y. Shen — Y. Zhang
Regarding the function g(x, s), we assume that gis a Carath´eodory function
and that there exist p < q < p:= N p/(Np) and a(x)L(Ω), b > 0 such
that
(g1)|g(x, s)| ≤ a(x) + b|s|q1.
We also assume that there exist p < σ < p, and a0(x)L1(Ω), b0(x)Lm(Ω)
with m=p/(pr),1< r < p such that
(g2)σG(x, s)sg(x, s) + a0(x) + b0(x)|s|r.
Let µ= (1 σ
p), we assume that there exist ν > 0, R > 0 such that, for
almost every xΩ and every (s, ξ)R×RNwith |s|> R,
µjξ(x, s, ξ)ξjs(x, s, ξ )s,(j3)
jξ(x, s, ξ)ξpj(x, s, ξ),(j4)
and
(j5)σj(x, s, ξ)jξ(x, s, ξ )ξjs(x, s, ξ)sν|ξ|p,
where σis given by (g2).
We will prove different existence of critical points for the functional Iin
dependence on different growth rate of the function g(x, s). First we study the
nonsymmetric case. In this case, we assume that for almost every xΩ,
(g3) lim sup
|s|→0
g(x, s)
|s|p1< αλ1βλ1<lim inf
|s|→∞
g(x, s)
|s|p2s,
where λ1is the first eigenvalue of the p-Laplacian operator p.
Then we have the following result.
Theorem 1.1. Assume conditions (j1)–(j5)and (g1)–(g3)hold, then there
exists a nontrivial critical point uW1,p
0(Ω) of problem (1.1).
Next we study the symmetric case. In this case, we assume that for almost
every xΩ, j(x, s, ξ) = j(x, s, ξ), g(x, s) = g(x, s) and
(g4) lim
|s|→∞
g(x, s)
|s|p2s=.
Then we have the following result.
Theorem 1.2. Assume conditions (j1)–(j5)and (g1)–(g2),(g4)hold, then
there exist a sequence {un}of nontrivial critical points of problem (1.1) in
W1,p
0(Ω) such that I(un)→ ∞ as n→ ∞.
To get the nonexistence result, we assume, besides (j1)–(j2), the following
(g1)0and (g2)0for simplicity.
An Application of Nonsmooth Critical Point Theory 207
We assume that g(x, s)g(s) and there exist p < q, σ < pand b > 0 such
that
(g1)0|g(s)| ≤ b|s|q1,
and
(g2)00< σG(x, s)sg(x, s).
For example, g(x, s) = |s|q2swith p < q < pand σ=q.
Then we have the following result.
Theorem 1.3. Let RN(N3) be bounded and star-shaped, assume
that jdoes not depend on xand that (j1)–(j2)and (g1)0–(g2)0hold. Moreover,
assume that for every (s, ξ)R×RN
(j6)sjs(s, ξ)<µjξ(s, ξ )ξ,
where µ= (1 σ/p), then (1.2) has no nontrivial solution in C2(Ω) C1(Ω).
The paper is arranged as follows. In Section 2, we set the abstract frame-
work and specify its connections with our problem. In Section 3, we study the
compactness of the Palais–Smale sequence. In Section 4, we prove the existence
and nonexistence of nontrivial critical points.
Throughout this paper we denote by k·k,k·kqand k·k1,p0the standard
norms of W1,p
0(Ω), Lq(Ω) and W1,p0(Ω), respectively. ” (“*”) indicates the
strong (weak) convergence in the corresponding function space.
2. Mathematical background
In this section we give some definitions and abstract critical point theories
(for the proof, see [7], [8]) will be used in this paper. These definitions and
theories also have been used in [6], [11], [15], [16].
Definition 2.1. Let Xbe a complete metric space endowed with the metric
d,f:XRbe a continuous function, and uX. We denote by |df |(u) the
supremum of the real numbers σin [0,) such that there exist δ > 0 and
a continuous map
H:B(u;δ)×[0, δ]X,
such that for every vin B(u;δ) and for every tin [0, δ] it results
d(H(v, t), v)t,(2.1)
f(H(v, t)) f(v)σt.(2.2)
where B(u;δ) is the open ball of center uXand of radius δ. The extended
real number |df|(u) is called the weak slope of fat u.
If Xis a Finsler manifold of class C1, it turns out that |df|(u) = kf0(u)k.
208 Z. Li — Y. Shen — Y. Zhang
Definition 2.2. Let Xbe a complete metric space, f:XRbe a con-
tinuous function. A point uXis a critical point of fif |df |(u) = 0. We say
that cRis a critical value of fif there exists a critical point uXof fwith
f(u) = c.
Definition 2.3. Let Xbe a complete metric space, f:XRbe a contin-
uous function and cR. We say that fsatisfies the Palais–Smale condition at
level c((PS)cin short), if every sequence {un}in Xsuch that |df |(un)0 and
f(un)cadmits a subsequence {unk}converging in X.
Theorem 2.4. Let Xbe a Banach space endowed with the norm k·k and
f:XRa continuous function. First, suppose that there exist wX,η > f(0)
and r > 0such that
f(u)> η, for all uX, kuk=r,(2.3)
f(w)< η, kwk> r.(2.4)
We set Γ = {γ: [0,1] X, is continuous and γ(0) = 0,γ(1) = w}. Finally,
suppose that fsatisfies (PS)ccondition at the level
c= inf
γΓsup
t[0,1]
f(γ(t)) <,
Then, there exists a nontrivial critical point uof fsuch that f(u) = c.
Theorem 2.5. Let Xbe a Banach space, f:XRa continuous even
functional. Assume that there exists a strictly increasing sequence {Wk}of finite
dimensional subspaces of Xwith the fol lowing properties:
(a) there exist ρ > 0,η > f(0) and a subspace VXof finite codimension
such that
f(u)η, for all uV, kuk=ρ;
(b) there exists a sequence {Rk}in (ρ, )such that
f(u)f(0),for all uWk,kuk ≥ Rk;
(c) fsatisfies (PS)ccondition for any cη.
Then there exists a sequence {uk}of critical points of fwith
lim
k→∞ f(uk) = .
Definition 2.6. A sequence {un} ⊂ W1,p
0(Ω) is a Concrete–Palais–Smale
sequence at level c((CPS)cin short) if there exists ynW1,p0(Ω) with yn0
such that
(2.5) I(un)c,
An Application of Nonsmooth Critical Point Theory 209
(2.6) hI0(un), ϕi=Z
[jξ(x, un,un)ϕ+js(x, un,un)ϕ]Z
g(x, un)ϕ
=hyn, ϕi,
for all ϕC
0(Ω). Moreover, we say that Isatisfies the (CPS)ccondition if
every (CPS)csequence is strongly compact in W1,p
0(Ω).
The next result connects the previous notions with abstract critical point
theory.
Theorem 2.7. The functional I:W1,p
0(Ω) Ris continuous and
|dI|(u)sup
ϕC
0(Ω),kϕk=1 Z
[jξ(x, u, u)ϕ+js(x, u, u)ϕg(x, u)ϕ],
for every uW1,p
0(Ω). In particular, if |dI|(u)<, then we have
|dI|(u)≥ k − div(jξ(x, u, u)) + js(x, u, u)g(x, u)k1,p0.
Proof. See [5, Theorem 2.1.3].
3. Compactness results
In this section, we prove that the functional Isatisfies the (CPS)ccondition
in W1,p
0(Ω), so does the (PS)ccondition. Indeed, if {un} ⊂ W1,p
0(Ω) is a (PS)c
sequence of I, then by Theorem 2.7, it is also a (CPS)csequence. Then, if I
satisfies the (CPS)ccondition, we can deduce that {un}admits a convergent
subsequence.
Proposition 3.1. Assume (j1)–(j5)hold, let uW1,p
0(Ω) and assume that
there exists a wW1,p0(Ω) such that for every vW1,p
0L(Ω)
(3.1) Z
jξ(x, u, u)v+Z
js(x, u, u)v=hw, vi.
Then jξ(x, u, u)u, js(x, u, u)uL1(Ω) and
(3.2) Z
jξ(x, u, u)u+Z
js(x, u, u)u=hw, ui.
Proof. Let kR+be fixed, we define the following cutoff functions:
(3.3) Tk(u) = (uif |u| ≤ k,
sgnu·kif |u|> k, Gk(u) = uTk(u).
Then for every vW1,p
0(Ω), we have Tk(v)W1,p
0L(Ω). Thus, we can take
Tk(u) as a test function in (3.1) and get
(3.4) Z
jξ(x, u, u)Tk(u) + Z
js(x, u, u)Tk(u) = hw, Tk(u)i.
210 Z. Li — Y. Shen — Y. Zhang
Since from (j1), we can deduce that
j(x, s, ξ) = Z1
0
jξ(x, s, tξ)ξ dt Z1
0
α||pt1dt =α
p|ξ|p,
j(x, s, ξ) = Z1
0
jξ(x, s, tξ)ξ dt Z1
0
β||pt1dt =β
p|ξ|p.
This means that
(3.5) α
p|ξ|pj(x, s, ξ)β
p|ξ|p
for almost every xΩ and every (s, ξ)R×RN. Let Rbe given by (j3), we
denote
(3.6) AR:= {xΩ : |u|> R}, BR:= Ω \AR.
Then by (j2), we have
(3.7) |js(x, u, u)u| ≤ |∇u|ponBR.
Denote
+:= {xΩ : 0 js(x, u, u)u},(3.8)
:= {xΩ : µjξ(x, u, u)ujs(x, u, u)u0}.(3.9)
Then, by (j5) and (3.5), we have
(3.10) |js(x, u, u)u| ≤ σj(x, u, u)σβ
p|∇u|pon AR+,
by (j3) and (j1), we have
(3.11) |js(x, u, u)u| ≤ µjξ(x, u, u)uµβ|∇u|pon AR.
Combining (3.7), (3.10) and (3.11), we have
|js(x, u, u)u| ≤ +σ
p+µβ|∇u|p.
This means that js(x, u, u)uL1(Ω). Thus, we can use Lebesgue Dominated
Convergence Theorem to pass the limit in (3.4) and get (3.2).
Proposition 3.2. Assume (j1)–(j5)and (g1)–(g2)hold, then every Concre-
te–Palais–Smale sequence {un}is bounded in W1,p
0(Ω).
Proof. Assume that {un} ⊂ W1,p
0(Ω) such that (2.5) and (2.6) hold. Let
us fix ε > 0 and consider the function ϑε:RRdefined by
ϑε(s) =
0 for 0 sR,
(1 + ε)(sR) for RsRε,
sfor Rεs,
ϑε(s) for s0,
An Application of Nonsmooth Critical Point Theory 211
where Ris given in (j3) and Rε= (1 + ε)R/ε. Then, for every uW1,p
0(Ω), it
results
(3.12) |∇ϑε(u)| ≤ (1 + ε)|∇u|.
Moreover, ϑε(u) has the same sign of u. By Proposition 3.1, we can take unas
test functions in (2.6) and get
(3.13) ZAR,n
[jξ(x, un,un)ϑε(un) + js(x, un,un)ϑε(un)]
=ZAR,n
g(x, un)un+ZAR,n
g(x, un)(ϑε(un)un) + hyn, ϑε(un)i.
where AR,n is defined as in (3.6). Since ϑε(un) has the same sign of un, by (j5)
and (3.12), we can deduce that
ZAR,n+
n
[σj(x, un,un)jξ(x, un,un)ϑε(un)js(x, un,un)ϑε(un)]
ZAR,n+
n
[σj(x, un,un)(1 + ε)jξ(x, un,un)unjs(x, un,un)un]
νZAR,n+
n
|∇un|pεβ Z
|∇un|p,
and by (j4), (3.5),
ZAR,n
n
[σj(x, un,un)jξ(x, un,un)ϑε(un)js(x, un,un)ϑε(un)]
ZAR,n
n
[σj(x, un,un)(1 + ε)jξ(x, un,un)un]
σp
pαZAR,n
n
|∇un|pεβ Z
|∇un|p,
where Ω+
nand Ω
nare defined as in (3.8) and (3.9). Thus we get
(3.14) min ν, σp
pαZAR,n
|∇un|p2εβ Z
|∇un|p
ZAR,n
[σj(x, un,un)jξ(x, un,un)ϑε(un)js(x, un,un)ϑε(un)].
On the other hand, by (3.5), we have
(3.15) α
pZBR,n
|∇un|pZBR,n
j(x, un,un),
where BR,n is defined as in (3.6). Now let
ν0:= min ν, σp
pα, σ
pα>0
212 Z. Li — Y. Shen — Y. Zhang
and compute σI(un)− hyn, ϑε(un)i. By (3.13)–(3.15), we can deduce that
ν0Z
|∇un|p2εβ Z
|∇un|pσZ
j(x, un,un)(3.16)
ZAR,n
[jξ(x, un,un)ϑε(un) + js(x, un,un)ϑε(un)]
=σI(un) + Z
G(x, un)− hyn, ϑε(un)i − Z
g(x, un)ϑε(un)
σI(un) + ZBR,n
G(x, un)
+ZAR,n
[σG(x, un)g(x, un)un]
+hyn, ϑε(un)i| +ZAR,n
g(x, un)(ϑε(un)un)
.
Note that
(3.17) ZAR,n
g(x, un)(ϑε(un)un)
=Z{xΩ:R<|un|<Rε}
g(x, un)(ϑε(un)un)
C(R, ε),
and by (g1), we have
(3.18) ZBR,n
G(x, un)
ZBR,n
(|a(x)||un|+b|un|q)C(R).
Now choose ε=ν0/4β, combine (3.16)–(3.18), by (g2) and Sobolev inequality,
we get
(3.19) ν0
2kunkpC+1 + ν0
4βkynk1,p0kunk+kb0kmkunkr,
where C=C(R, ε, c), cis given by (2.5). Note that 1 < r < p and kynk1,p00,
(3.19) yields the conclusion.
Proposition 3.3. Assume (j1)–(j5)and (g1)–(g2)hold, then every bounded
Concrete–Palais–Smale sequence {un}converges strongly to uW1,p
0(Ω).
Proof. By assumptions, there exists a uin W1,p
0(Ω) such that up to a sub-
sequence, unconverges weakly to uin W1,p
0(Ω), unconverges strongly to uin
Lq(Ω), 1 < q < pand unconverges to ualmost everywhere in Ω.
For kR+be fixed, without lost of generality, we assume that k > R, where
Ris given by (j3), we denote
Ak,n ={xΩ : |un|> k}, Bk,n = \Ak,n,
A+
k,n ={xΩ : un> k}, A
k,n ={xΩ : un<k}.
Let Tkand Gkas defined in (3.3). Without lost of generality, we assume that
|∇Gk(un)| 6= 0 and |∇Tk(un)| 6= 0 for every n. We divide the proof in two steps.
An Application of Nonsmooth Critical Point Theory 213
Step 1. We prove that for any ε > 0, there exists k > 0 large enough such
that limn→∞ kGk(un)k ≤ ε. Because Gk(un) = unin Ak,n and Gk(un) = 0
in Bk,n, Proposition 3.1 implies that we can take ϕ=Gk(un) as test functions
in (2.6) and get
(3.20) ZAk,n
jξ(x, un,un)Gk(un) + ZAk,n
js(x, un,un)Gk(un)
=ZAk,n
g(x, un)Gk(un) + hyn, Gk(un)i.
By (j3), (j1) and note that Gk(un) has the same sign of un, we have
(3.21) ZA+
k,n
js(x, un,un)Gk(un) = ZA+
k,n
js(x, un,un)(unk)
=ZA+
k,n+
n1k
unjs(x, un,un)un
+ZA+
k,n
n1k
unjs(x, un,un)un
ZA+
k,n
n
js(x, un,un)un
µZA+
k,n
n
jξ(x, un,un)un≥ −µZA+
k,n
jξ(x, un,un)un,
where Ω+
nand Ω
nare defined as in (3.8) and (3.9). Analogously,
(3.22) ZA
k,n
js(x, un,un)Gk(un)≥ −µZA
k,n
jξ(x, un,un)un,
Therefore, combining (3.21) and (3.22), we get
(3.23) ZAk,n
js(x, un,un)Gk(un)≥ −µZAk,n
jξ(x, un,un)un.
Note that un=Gk(un) in Ak,n, from (3.20), (3.23) and according to (j1),
we get
α(1 µ)kGk(un)k2ZAk,n
|g(x, un)||Gk(un)|dx +kynk1kGk(un)k.
Since g(x, un)g(x, u) and yn0 in W1,p0(Ω), respectively, we get the
conclusion.
Step 2. We prove that for a fixed klarge enough, kTk(un)Tk(u)k → 0 as
ntends to infinity. Let vn=Tk(un)Tk(u) and ϕ(t) = teηt2. Since ϕ(vn)
214 Z. Li — Y. Shen — Y. Zhang
W1,p
0(Ω) L(Ω), we can take ϕ=ϕ(vn) as test functions in (2.6) and get
(3.24) Z
g(x, un)ϕ(vn)dx +hyn, ϕ(vn)i
=Z
ϕ0(vn)jξ(x, un,un)vn+Z
ϕ(vn)js(x, un,un) := I + II.
Firstly, by the definitions of Tkand Gk,
(3.25) I = ZAk,n
ϕ0(vn)jξ(x, un,Gk(un))vn
+ZBk,n
ϕ0(vn)jξ(x, un,Tk(un))vn:= III + IV.
According to (j1) and by H¨older inequality, we get
|III|=ZAk,n
ϕ0(vn)jξ(x, un,Gk(un))Gk(un)Gk(un)vn|∇Gk(un)|2
(3.26)
βϕ0(2k)ZAk,n
|∇Gk(un)|p1|∇vn|
βϕ0(2k)kGk(un)kp1k∇vnkLp(Ak,n )εn.
Here and in the following, we use εnto denote a quantity which tends to zero as
ntends to infinity. Since un* u in W1,p
0(Ω), thus vn*0 in W1,p
0(Ω), by (j1),
we get
(3.27) IV αZBk ,n
ϕ0(vn)|∇Tk(un)|p2Tk(un)vn.
Recall that for p2 and all x, y RN[10, Lemma 4.1, p. 5709],
(3.28) (|x|p2x− |y|p2y, x y)Cp|xy|p,
where Cp>0 is a constant. Note that
ZBk,n
ϕ0(vn)|∇Tk(u)|p2(Tk(u))vn0.
Let x=Tk(un), y =Tk(u) in (3.28), by direct computation, we get
(3.29) IV C1ZBk,n
ϕ0(vn)|∇vn|pεn,
where C1>0 is a constant. Combining (3.26) and (3.29), we get
(3.30) I C1ZBk,n
ϕ0(vn)|∇vn|pεn.
An Application of Nonsmooth Critical Point Theory 215
Secondly, we consider II in (3.24). By (j2), we have
(3.31) II γZAk,n
|ϕ(vn)||∇Gk(un)|p+ZBk,n
|ϕ(vn)||∇Tk(un)|p
γZAk,n
|ϕ(vn)||∇Gk(un)|p+ 2p1γZBk,n
|ϕ(vn)||∇Tk(u)|p
+ 2p1γZBk,n
|ϕ(vn)||∇vn|p:= V + VI + VII.
Since un* u in W1,p
0(Ω), thus vn*0 in W1,p
0(Ω), we have
(3.32) VI = 2p1γZBk,n
|ϕ(vn)||∇Tk(u)|p0.
Moreover, by Step 1 and note that |ϕ(vn)| ≤ ϕ(2k), we have
(3.33) V = γZAk,n
|ϕ(vn)||∇Gk(un)|pεn.
Combining (3.31)–(3.33), we get
(3.34) II C2ZBk ,n
|ϕ(vn)||∇vn|p+εn,
where C2>0 is a constant. According to Lemma 1.2 in [5], for a, b > 0, we have
0(t)b|ϕ(t)| ≥ a/ for every tRwith η > (b/2a)2. Taking a=C1,b=C2
in ϕ, and combining (3.24), (3.30) and (3.34), we get
(3.35) C1
2ZBk,n
|∇vn|pZ
g(x, un)ϕ(vn)dx +εn0.
On the other hand, since vn=Tk(un)Tk(u) = signun·kTk(u) in Ak,n, we
have
(3.36) ZAk,n
|∇vn|p=ZAk,n
|∇Tk(u)|p0.
Thus (3.35) and (3.36) imply that kTk(un)Tk(u)k → 0.
Finally, since for any fixed kR+,
kunuk ≤ kTk(un)Tk(u)k+kGk(un)k+kGk(u)k.
We get unconverges strongly to u. This completes the proof.
Now let us recall the modified compactness condition introduced by Cerami
which allows rather general minimax results.
216 Z. Li — Y. Shen — Y. Zhang
Definition 3.4. Let Xbe a Banach space, a functional JC(X, R) is said
to satisfy the Cerami condition if for all cR
(a) every bounded sequence {uj} ⊂ Xsuch that {J(uj)}is bounded and
|dJ|(uj)0 possesses a convergent subsequence, and
(b) there exit δ, R, β > 0 such that for all uJ1[cδ, c+δ] with kuk ≥ R,
|dJ|(u)· kuk ≥ β.
Proposition 3.5. Assume conditions (j1)–(j5)and (g1)–(g2)hold, then I
satisfies the Cerami condition.
Proof. Firstly, according to Theorem 2.7 and Propositions 3.2 and 3.3, (a)
is obvious.
Secondly, we prove that Isatisfies (b). Suppose by contradiction. Let cR
and assume that, up to a subsequence, {un} ⊂ W1,p
0(Ω) such that I(un)c
and |dI|(un)· kunk → 0 with kunk → ∞. By Theorem 2.7, we have
(3.37) kI0(un)k1,p0≤ |df|(un).
On the other hand, by Proposition 3.1, we can take ϕ=unas test functions in
(2.6). By (3.37), we get hI0(un), uni → 0. Thus, we can argue as for (3.19) and
get a contradiction. This completes the proof.
4. Proof of main theorems
In this section, we will use the compactness results in the previous section
to prove our main results. Since we have proved that the functional Isatisfies
the (PS)ccondition, it is trivial to prove Theorems 1.1 and 1.2. For the sake of
completeness, we give the proof here.
Proof of Theorem 1.1. Firstly, Proposition 3.5 show that Isatisfies (PS)c
condition. Secondly, we prove that Isatisfies the geometrical conditions of The-
orem 2.4.
In fact, (j1) implies that for every uW1,p
0(Ω),
1
pαkukpZ
G(x, u)I(u)βkukpZ
G(x, u).
By (g3) and the definition of λ1, when uW1,p
0(Ω) small enough, we have
Z
G(x, u)<1
pαkukp.
Note that I(0) = 0, thus (2.1) of Theorem 2.4 is satisfied. Now for ϕ1W1,p
0(Ω),
the first eigenfunction of poperator, ϕ1>0, kϕ1k= 1 and tR+, by (g3)
and the definition of λ1, we have
I(1)βtpZ
|∇ϕ1|pZ
G(x, tϕ1)→ −∞.
An Application of Nonsmooth Critical Point Theory 217
Thus, we have I(1)<0 when t > 0 large enough and the condition (2.2) of
Theorem 2.4 is satisfied. Therefore, Theorem 2.4 yields the conclusion.
Proof of Theorem 1.2. Note that the Sobolev space is a separable Banach
space with infinite dimension, by [14, Theorem 7.7], there exist two sequences
{vn} ⊂ W1,p
0(Ω) and {ϕn} ⊂ W1,p0(Ω) such that
(i) < ϕn, vm>=δm
n, where δm
n= 1 when m=nand δm
n= 0, else
(ii) W1,p
0(Ω) = span{vm:mN}and W1,p0(Ω) = span{ϕn:nN}.
Without lost of generality, we assume that vmis a normalized sequence, that
is kvmk= 1, m= 1,2, . . . and vkvl, k 6=l. Denote Vm= span{vl:lm}
and V
mthe topological complementary subspace of Vmin W1,p
0(Ω) and hence
W1,p
0(Ω) = VmV
m. It is obviously that V1=W1,p
0(Ω), V
1=φ. Denote
λq,m = inf
uVm
kuk
kukq
,
where 1 < q < p. We have λq,m +as m→ ∞.
Firstly, note that C
c(Ω) is dense in Lp∗0 (Ω), then for every ε > 0, there
exist ac(x)C
c(Ω) and aε(x)Lp∗0 (Ω) with kaεkp∗ 0 εsuch that a(x) =
ac(x) + aε(x) and condition (g1) implies that
|g(x, u)| ≤ ac(x) + aε(x) + b|u|q1.
Now choose uVmwith kuk= 1, from condition (j1), we have
I(u)1
pαkuk2(kack2kuk2+kaεkp∗0 kukp+bkukq
q)1
pαc1
λ2,m
+c2ε+b
λq
q,m .
We can choose εsmall enough and mlarge enough to such that
c1
λ2,m
+c2ε+b
λq
q,m
<α
p,
this implies the geometrical condition (a) of Theorem 2.5 is satisfied.
Secondly, because V
mis a finite dimensional subspace, since all norms in
a finite dimensional space are all equivalent, we know that there exists a C2>0
such that for every uV
m,kuk ≤ C2kukp. From conditions (j1) and (g4), we
have
I(u)βZ
|∇u|pZ
G(x, u)→ −∞,
when kuk → ∞, this implies the geometrical condition (b) of Theorem 2.5 is
satisfied. Therefore, there exist a sequence {un}of critical points of Isuch that
I(un)→ ∞. This completes the proof.
Before we prove Theorem 1.3, we give the Pohozaev identity in [12]. Let
F(x, u, r): ×R×RNbe a functional of class C1, we consider the following
218 Z. Li — Y. Shen — Y. Zhang
equation
(F) div{Fr(x, u, Du)}=Fu(x, u, Du).
Here we write Du = (∂u/∂x1, . . . , ∂u/∂xN), Fn=F/∂u,Fxi=∂F /∂xiand
Fri=∂F /∂ri,r= (r1, . . . , rN). Assume that F(x, 0,0) = 0. Let a(x) and h(x)
be two functions of class C1(Ω) C(Ω) and uC1(Ω) C(Ω) be a solution of
problem (F), then we have the following Pohozaev identity.
Z
(F(x, 0, Du)DiuFri(x, 0, Du))(h·ν)ds =Z
F(x, u, Du)div h
+Z
hiFxi(x, u, Du)Z
(DjuDihj+uDia(x))Fri(x, u, Du)
Z
a(x)(DiuFri(x, u, Du) + uFu(x, u, Du)).
We refer also to [9], where the above variational relation is proved for C1solu-
tions.
Proof of Theorem 1.3. Assume on the contrary, uC1(Ω) C(Ω) is
a weak solution of equation (1.2), let
F(x, u, Du) = j(u, u)G(u)
and let abe independent on x,h=x. We get
Z
j(0,u)(x·ν)ds =np
p+aZ
jξ(u, u)u
+Z
js(u, u)uZ
(nG(u)aug(u)).
We take a=N/σ, note that Ω is a star shape region, by (g2)0, we get
(4.1) 1σ
pZ
jξ(u, u)u+Z
js(u, u)u0.
Therefore if usatisfies (j6), then (4.1) implies that u0. This completes the
proof.
Example 4.1. Let RN(N3) be a open bounded domain, and a0,
λ > 0, 2 p < N ,p < q < p=N p/(Np). Let
J(u) = 1
pZ1 + 1
1 + |u|a|∇u|pλ
qZ
|u|q.
Then by Theorem 1.2, there exist a sequence {un} ⊂ W1,p
0(Ω) of nontrivial
critical points of Jsuch that J(un)→ ∞ as n→ ∞.
Acknowledgments. The authors warmly thank the referee for careful read-
ing and helpful comments.
An Application of Nonsmooth Critical Point Theory 219
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Manuscript received June 1, 2009
Zhouxin Li
School of Mathematical Sciences
Peking University
Beijing, 100871, P.R. China
E-mail address: lzx@math.pku.edu.cn
Yaotian Shen and Yimin Zhang
Department of Mathematics
South China Univveristy of Tech.
Guangzhou, 510640, P.R. China
TMNA : Volume 35 – 2010 – No2
... As for as we know, (a 3 ) was first introduced in [26] to study (1.1) on bounded domain Ω when q = 2, and then was extended to the case 2 ≤ q < 2 * in [20]. There are many equations like (1.3) with κ < 0, see also Example 4.1 in [21], that do not satisfy the sign condition. Parameter κ plays an important role in (1.3) especially when it is negative. ...
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