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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(N≥3) 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) = ZΩ

j(x, u, ∇u)−ZΩ

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

x∈Ω, 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 W−1,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)ξiξj≥0.

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 /(N−2), ν∈(0, σ −2) such that

s

N

X

ij=1

∂saij (x, s)ξiξj≤ν

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)ξiξj≤s

N

X

i,j

∂saij (x, s)ξiξj≤0.

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 a≥0, 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|p≤0.

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

2≤p<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 x∈Ω.

We also assume that there exist α, β with β≥α > 0 and γ > 0 such that for

almost every x∈Ω and every (s, ξ)∈R×RN

α|ξ|p≤jξ(x, s, ξ)ξ≤β|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/(N−p) and a(x)∈L∞(Ω), b > 0 such

that

(g1)|g(x, s)| ≤ a(x) + b|s|q−1.

We also assume that there exist p < σ < p∗, and a0(x)∈L1(Ω), b0(x)∈Lm(Ω)

with m=p∗/(p∗−r),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 x∈Ω and every (s, ξ)∈R×RNwith |s|> R,

−µjξ(x, s, ξ)ξ≤js(x, s, ξ )s,(j3)

jξ(x, s, ξ)ξ≤pj(x, s, ξ),(j4)

and

(j5)σj(x, s, ξ)−jξ(x, s, ξ )ξ−js(x, s, ξ)s≥ν|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 x∈Ω,

(g3) lim sup

|s|→0

g(x, s)

|s|p−1< αλ1≤βλ1<lim inf

|s|→∞

g(x, s)

|s|p−2s,

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 u∈W1,p

0(Ω) of problem (1.1).

Next we study the symmetric case. In this case, we assume that for almost

every x∈Ω, j(x, −s, −ξ) = j(x, s, ξ), g(x, −s) = −g(x, s) and

(g4) lim

|s|→∞

g(x, s)

|s|p−2s=∞.

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 < p∗and b > 0 such

that

(g1)0|g(s)| ≤ b|s|q−1,

and

(g2)00< σG(x, s)≤sg(x, s).

For example, g(x, s) = |s|q−2swith p < q < p∗and σ=q.

Then we have the following result.

Theorem 1.3. Let Ω⊂RN(N≥3) 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·k−1,p0the standard

norms of W1,p

0(Ω), Lq(Ω) and W−1,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:X→Rbe a continuous function, and u∈X. 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 u∈Xand 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:X→Rbe a con-

tinuous function. A point u∈Xis a critical point of fif |df |(u) = 0. We say

that c∈Ris a critical value of fif there exists a critical point u∈Xof fwith

f(u) = c.

Definition 2.3. Let Xbe a complete metric space, f:X→Rbe a contin-

uous function and c∈R. 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:X→Ra continuous function. First, suppose that there exist w∈X,η > f(0)

and r > 0such that

f(u)> η, for all u∈X, 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

γ∈Γsup

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:X→Ra 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 V⊂Xof ﬁnite codimension

such that

f(u)≥η, for all u∈V, kuk=ρ;

(b) there exists a sequence {Rk}in (ρ, ∞)such that

f(u)≤f(0),for all u∈Wk,kuk ≥ Rk;

(c) fsatisﬁes (PS)ccondition for any c≥η.

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 yn∈W−1,p0(Ω) with yn→0

such that

(2.5) I(un)→c,

An Application of Nonsmooth Critical Point Theory 209

(2.6) hI0(un), ϕi=ZΩ

[jξ(x, un,∇un)∇ϕ+js(x, un,∇un)ϕ]−ZΩ

g(x, un)ϕ

=hyn, ϕi,

for all ϕ∈C∞

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

ϕ∈C∞

0(Ω),kϕk=1 ZΩ

[jξ(x, u, ∇u)∇ϕ+js(x, u, ∇u)ϕ−g(x, u)ϕ],

for every u∈W1,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)k−1,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 u∈W1,p

0(Ω) and assume that

there exists a w∈W−1,p0(Ω) such that for every v∈W1,p

0∩L∞(Ω)

(3.1) ZΩ

jξ(x, u, ∇u)∇v+ZΩ

js(x, u, ∇u)v=hw, vi.

Then jξ(x, u, ∇u)∇u, js(x, u, ∇u)u∈L1(Ω) and

(3.2) ZΩ

jξ(x, u, ∇u)∇u+ZΩ

js(x, u, ∇u)u=hw, ui.

Proof. Let k∈R+be ﬁxed, we deﬁne the following cutoﬀ functions:

(3.3) Tk(u) = (uif |u| ≤ k,

sgnu·kif |u|> k, Gk(u) = u−Tk(u).

Then for every v∈W1,p

0(Ω), we have Tk(v)∈W1,p

0∩L∞(Ω). Thus, we can take

Tk(u) as a test function in (3.1) and get

(3.4) ZΩ

jξ(x, u, ∇u)∇Tk(u) + ZΩ

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|tξ|pt−1dt =α

p|ξ|p,

j(x, s, ξ) = Z1

0

jξ(x, s, tξ)ξ dt ≤Z1

0

β|tξ|pt−1dt =β

p|ξ|p.

This means that

(3.5) α

p|ξ|p≤j(x, s, ξ)≤β

p|ξ|p

for almost every x∈Ω and every (s, ξ)∈R×RN. Let Rbe given by (j3), we

denote

(3.6) AR:= {x∈Ω : |u|> R}, BR:= Ω \AR.

Then by (j2), we have

(3.7) |js(x, u, ∇u)u| ≤ Rγ|∇u|ponBR.

Denote

Ω+:= {x∈Ω : 0 ≤js(x, u, ∇u)u},(3.8)

Ω−:= {x∈Ω : −µjξ(x, u, ∇u)∇u≤js(x, u, ∇u)u≤0}.(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∩Ω+,

by (j3) and (j1), we have

(3.11) |js(x, u, ∇u)u| ≤ µjξ(x, u, ∇u)∇u≤µβ|∇u|pon AR∩Ω−.

Combining (3.7), (3.10) and (3.11), we have

|js(x, u, ∇u)u| ≤ Rγ +σ

p+µβ|∇u|p.

This means that js(x, u, ∇u)u∈L1(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:R→Rdeﬁned by

ϑε(s) =

0 for 0 ≤s≤R,

(1 + ε)(s−R) for R≤s≤Rε,

sfor Rε≤s,

−ϑε(−s) for s≤0,

An Application of Nonsmooth Critical Point Theory 211

where Ris given in (j3) and Rε= (1 + ε)R/ε. Then, for every u∈W1,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∩Ω+

n

[σj(x, un,∇un)−jξ(x, un,∇un)∇ϑε(un)−js(x, un,∇un)ϑε(un)]

≥ZAR,n∩Ω+

n

[σj(x, un,∇un)−(1 + ε)jξ(x, un,∇un)∇un−js(x, un,∇un)un]

≥νZAR,n∩Ω+

n

|∇un|p−εβ ZΩ

|∇un|p,

and by (j4), (3.5),

ZAR,n∩Ω−

n

[σj(x, un,∇un)−jξ(x, un,∇un)∇ϑε(un)−js(x, un,∇un)ϑε(un)]

≥ZAR,n∩Ω−

n

[σj(x, un,∇un)−(1 + ε)jξ(x, un,∇un)∇un]

≥σ−p

pαZAR,n∩Ω−

n

|∇un|p−εβ ZΩ

|∇un|p,

where Ω+

nand Ω−

nare deﬁned as in (3.8) and (3.9). Thus we get

(3.14) min ν, σ−p

pαZAR,n

|∇un|p−2εβ ZΩ

|∇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|p≤ZBR,n

j(x, un,∇un),

where BR,n is deﬁned as in (3.6). Now let

ν0:= min ν, σ−p

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

ν0ZΩ

|∇un|p−2εβ ZΩ

|∇un|p≤σZΩ

j(x, un,∇un)(3.16)

−ZAR,n

[jξ(x, un,∇un)ϑε(un) + js(x, un,∇un)ϑε(un)]

=σI(un) + ZΩ

G(x, un)− hyn, ϑε(un)i − ZΩ

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{x∈Ω: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

2kunkp≤C+1 + ν0

4βkynk−1,p0kunk+kb0kmkunkr,

where C=C(R, ε, c), cis given by (2.5). Note that 1 < r < p and kynk−1,p0→0,

(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 u∈W1,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 < p∗and unconverges to ualmost everywhere in Ω.

For k∈R+be ﬁxed, without lost of generality, we assume that k > R, where

Ris given by (j3), we denote

Ak,n ={x∈Ω : |un|> k}, Bk,n = Ω \Ak,n,

A+

k,n ={x∈Ω : un> k}, A−

k,n ={x∈Ω : 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)(un−k)

=ZA+

k,n∩Ω+

n1−k

unjs(x, un,∇un)un

+ZA+

k,n∩Ω−

n1−k

unjs(x, un,∇un)un

≥ZA+

k,n∩Ω−

n

js(x, un,∇un)un

≥ − µZA+

k,n∩Ω−

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)k2≤ZAk,n

|g(x, un)||Gk(un)|dx +kynk−1kGk(un)k.

Since g(x, un)→g(x, u) and yn→0 in W−1,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) ZΩ

g(x, un)ϕ(vn)dx +hyn, ϕ(vn)i

=ZΩ

ϕ0(vn)jξ(x, un,∇un)∇vn+ZΩ

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)|p−1|∇vn|

≤βϕ0(2k)kGk(un)kp−1k∇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)|p−2∇Tk(un)∇vn.

Recall that for p≥2 and all x, y ∈RN[10, Lemma 4.1, p. 5709],

(3.28) (|x|p−2x− |y|p−2y, x −y)≥Cp|x−y|p,

where Cp>0 is a constant. Note that

ZBk,n

ϕ0(vn)|∇Tk(u)|p−2(∇Tk(u))∇vn→0.

Let x=∇Tk(un), y =∇Tk(u) in (3.28), by direct computation, we get

(3.29) IV ≥C1ZBk,n

ϕ0(vn)|∇vn|p−εn,

where C1>0 is a constant. Combining (3.26) and (3.29), we get

(3.30) I ≥C1ZBk,n

ϕ0(vn)|∇vn|p−εn.

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+ 2p−1γZBk,n

|ϕ(vn)||∇Tk(u)|p

+ 2p−1γZBk,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 = 2p−1γZBk,n

|ϕ(vn)||∇Tk(u)|p→0.

Moreover, by Step 1 and note that |ϕ(vn)| ≤ ϕ(2k), we have

(3.33) V = γZAk,n

|ϕ(vn)||∇Gk(un)|p≤εn.

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

aϕ0(t)−b|ϕ(t)| ≥ a/ for every t∈Rwith 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|p≤ZΩ

g(x, un)ϕ(vn)dx +εn→0.

On the other hand, since vn=Tk(un)−Tk(u) = signun·k−Tk(u) in Ak,n, we

have

(3.36) ZAk,n

|∇vn|p=ZAk,n

|∇Tk(u)|p→0.

Thus (3.35) and (3.36) imply that kTk(un)−Tk(u)k → 0.

Finally, since for any ﬁxed k∈R+,

kun−uk ≤ 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 J∈C(X, R) is said

to satisfy the Cerami condition if for all c∈R

(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 u∈J−1[c−δ, 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 c∈R

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)k−1,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 u∈W1,p

0(Ω),

1

pαkukp−ZΩ

G(x, u)≤I(u)≤βkukp−ZΩ

G(x, u).

By (g3) and the deﬁnition of λ1, when u∈W1,p

0(Ω) small enough, we have

ZΩ

G(x, u)<1

pαkukp.

Note that I(0) = 0, thus (2.1) of Theorem 2.4 is satisﬁed. Now for ϕ1∈W1,p

0(Ω),

the ﬁrst eigenfunction of −∆poperator, ϕ1>0, kϕ1k= 1 and t∈R+, by (g3)

and the deﬁnition of λ1, we have

I(tϕ1)≤βtpZΩ

|∇ϕ1|p−ZΩ

G(x, tϕ1)→ −∞.

An Application of Nonsmooth Critical Point Theory 217

Thus, we have I(tϕ1)<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} ⊂ W−1,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:m∈N}and W−1,p0(Ω) = span{ϕn:n∈N}.

Without lost of generality, we assume that vmis a normalized sequence, that

is kvmk= 1, m= 1,2, . . . and vk⊥vl, k 6=l. Denote Vm= span{vl:l≥m}

and V⊥

mthe topological complementary subspace of Vmin W1,p

0(Ω) and hence

W1,p

0(Ω) = Vm⊕V⊥

m. It is obviously that V1=W1,p

0(Ω), V⊥

1=φ. Denote

λq,m = inf

u∈Vm

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|q−1.

Now choose u∈Vmwith kuk= 1, from condition (j1), we have

I(u)≥1

pαkuk2−(kack2kuk2+kaεkp∗0 kukp∗+bkukq

q)≥1

pα−c1

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 u∈V⊥

m,kuk ≤ C2kukp. From conditions (j1) and (g4), we

have

I(u)≤βZΩ

|∇u|p−ZΩ

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): Ω×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 u∈C1(Ω) ∩C(Ω) be a solution of

problem (F), then we have the following Pohozaev identity.

Z∂Ω

(F(x, 0, Du)−DiuFri(x, 0, Du))(h·ν)ds =ZΩ

F(x, u, Du)div h

+ZΩ

hiFxi(x, u, Du)−ZΩ

(DjuDihj+uDia(x))Fri(x, u, Du)

−ZΩ

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, u∈C1(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∂Ω

j(0,∇u)(x·ν)ds =n−p

p+aZΩ

jξ(u, ∇u)∇u

+ZΩ

js(u, ∇u)u−ZΩ

(nG(u)−aug(u)).

We take a=−N/σ, note that Ω is a star shape region, by (g2)0, we get

(4.1) 1−σ

p∗ZΩ

jξ(u, ∇u)∇u+ZΩ

js(u, ∇u)u≥0.

Therefore if usatisﬁes (j6), then (4.1) implies that u≡0. This completes the

proof.

Example 4.1. Let Ω ⊂RN(N≥3) be a open bounded domain, and a≥0,

λ > 0, 2 ≤p < N ,p < q < p∗=N p/(N−p). Let

J(u) = 1

pZΩ1 + 1

1 + |u|a|∇u|p−λ

qZΩ

|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