existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient

R E S E A R C H Open AccessExistence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient Mieko Tanaka* * Correspondence: tanaka@ma.kagu.tus

R E S E A R C H Open Access

Existence of a positive solution for quasilinear elliptic equations with nonlinearity including

the gradient

Mieko Tanaka*

* Correspondence:

tanaka@ma.kagu.tus.ac.jp

Department of Mathematics, Tokyo

University of Science,

Kagurazaka 1-3, Shinjyuku-ku, Tokyo

162-8601, Japan

Abstract

We provide the existence of a positive solution for the quasilinear elliptic equation – div(a(x, |∇u|)∇u)= f (x, u, ∇u)

inunder the Dirichlet boundary condition As a special case (a(x, t) = t p–2), our

equation coincides with the usual p-Laplace equation The solution is established as

the limit of a sequence of positive solutions of approximate equations The positivity

of our solution follows from the behavior of f (x, tξ) as t is small In this paper, we do not impose the sign condition to the nonlinear term f

MSC: 35J92; 35P30 Keywords: nonhomogeneous elliptic operator; positive solution; the first

eigenvalue with weight; approximation

1 Introduction

In this paper, we consider the existence of a positive solution for the following quasilinear elliptic equation:

⎧

⎨

⎩

– div A(x, ∇u) = f (x, u, ∇u) in ,

where ⊂ R N is a bounded domain with Cboundary∂ Here, A:  × R N → RN is

a map which is strictly monotone in the second variable and satisfies certain regularity

conditions (see the following assumption (A)) Equation (P) contains the corresponding

p-Laplacian problem as a special case However, in general, we do not suppose that this

operator is (p – )-homogeneous in the second variable.

Throughout this paper, we assume that the map A and the nonlinear term f satisfy the following assumptions (A) and (f ), respectively.

(A) A(x, y) = a(x, |y|)y, where a(x, t) >  for all (x, t) ∈  × (, +∞), and there exist positive constants C, C, C, C,  < t≤  and  < p < ∞ such that

(i) A ∈ C( × R N,RN)∩ C( × (R N\ {}), RN);

(ii) |D y A(x, y) | ≤ C|y| p– for every x ∈ , and y ∈ R N\ {};

(iii) D y A(x, y)ξ · ξ ≥ C|y| p– |ξ|for every x ∈ , y ∈ R N \ {} and ξ ∈ R N;

© 2013 Tanaka; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribuAttribu-tion, and reproducAttribu-tion in any medium, provided the original work is properly cited.

(iv) |D x A(x, y) | ≤ C( +|y| p– ) for every x ∈ , y ∈ R N\ {};

(v) |D x A(x, y) | ≤ C|y| p–(– log|y|) for every x ∈ , y ∈ R N with  <|y| < t

(f ) f is a continuous function on  × [, ∞) × R N satisfying f (x, , ξ) =  for every

(x, ξ) ∈  × R N and the following growth condition: there exist  < q < p, b>  and

a continuous function fon × [, ∞) such that

–b

 + t q–

≤ f(x, t) ≤ f (x, t, ξ) ≤ b



 + t q–+|ξ| q–

()

for every (x, t, ξ) ∈  × [, ∞) × R N

In this paper, we say that u ∈ W ,p

 () is a (weak) solution of (P) if



 A(x, ∇u)∇ϕ dx =



 f (x, u, ∇u)ϕ dx

for allϕ ∈ W ,p

 ().

A similar hypothesis to (A) is considered in the study of quasilinear elliptic problems (see

[, Example ..], [–] and also refer to [, ] for the generalized p-Laplace operators).

From now on, we assume that C≤ p –  ≤ C, which is without any loss of generality as

can be seen from assumptions (A)(ii), (iii)

In particular, for A(x, y) = |y| p– y, that is, div A(x, ∇u) stands for the usual p-Laplacian

 p u, we can take C= C= p –  in (A) Conversely, in the case where C= C= p – 

holds in (A), by the inequalities in Remark (ii) and (iii), we see that a(x, t) = |t| p–whence

A(x, y) = |y| p– y Hence, our equation contains the p-Laplace equation as a special case.

In the case where f does not depend on the gradient of u, there are many existence results because our equation has the variational structure (cf [, , ]) Although there are

a few results for our equation (P) with f including ∇u, we can refer to [, ] and [] for the

existence of a positive solution in the case of the (p, q)-Laplacian or m-Laplacian ( < m <

N ) In particular, in [] and [], the nonlinear term f is imposed to be nonnegative The

results in [] and [] are applied to the m-Laplace equation with an (m – )-superlinear

term f w.r.t u Here, we mention the result in [] for the p-Laplacian Faria, Miyagaki

and Motreanu considered the case where f is (p – )-sublinear w.r.t u and ∇u, and they

supposed that f (x, u, ∇u) ≥ cu r for some c >  and  < r < p –  The purpose of this paper

is to remove the sign condition and to admit the condition like f (x, u, ∇u) ≥ λu p– + o(u p–)

for largeλ >  as u → + Concerning the condition for f as |u| → , Zou in [] imposed

that there exists an L >  satisfying f (x, u, ∇u) = Lu m– +o(|u| m–+|∇u|m–) as|u|, |∇u| → 

for the m-Laplace problem Hence, we cannot apply the result of [] and [] to the case of

f (x, u, ∇u) = λm(x)u p– + ( – u p–)|∇u| r– + o(u p– ) as u → + for  < r < p and m ∈ L∞()

(admitting sign changes), but we can do our result ifλ >  is large.

In [], the positivity of a solution is proved by the comparison principle However, since

we are not able to do it for our operator in general, after we provide a non-negative and

non-trivial solution as a limit of positive approximate solutions (in Section ), we obtain

the positivity of it due to the strong maximum principle for our operator

1.1 Statements

To state our first result, we define a positive constant A pby

A p:= C

p – 



C

C

p–

which is equal to  in the case of A(x, y) = |y| p– y (i.e., the case of the p-Laplacian) because

we can choose C= C= p –  Then, we introduce the hypothesis (f) to the function

f(x, t) in (f ) as t is small.

(f) There exist m ∈ L∞() and b>μ(m)A psuch that the Lebesgue measure of

{x ∈ ; m(x) > } is positive and

lim inf

t→+

f(x, t)

where fis the continuous function in (f ) andμ(m) is the first positive eigenvalue

of the p-Laplacian with the weight function m obtained by

μ(m) := inf

 |∇u| p dx; u ∈ W ,p

 () and



 m |u| p dx = 

Theorem  Assume (f) Then equation (P) has a positive solution u ∈ int P, where

P :=

u ∈ C

(); u(x) ≥  in  , intP :=

u ∈ C

(); u(x) >  in  and ∂u/∂ν <  on ∂ ,

Next, we consider the case where A is asymptotically (p – )-homogeneous near zero in

the following sense:

(AH) There exist a positive function a∈ C(, (, +∞)) and

a(x, t) ∈ C( × [, +∞), R) such that

A(x, y) = a(x)|y| p– y + a



x, |y|y for every x ∈ , y ∈ R N and ()

lim

t→+

a(x, t)

Under (AH), we can replace the hypothesis (f) with the following (f):

(f) There exist m ∈ L∞() and b>λ(m) such that () and the Lebesgue measure of {x ∈ ; m(x) > } is positive, where λ(m) is the first positive eigenvalue of – div(a(x)|∇u| p– ∇u) with a weight function m obtained by

λ(m) := inf

 a(x) |∇u| p dx; u ∈ W ,p

 () and



 m |u| p dx = 

Theorem  Assume (AH) and (f) Then equation (P) has a positive solution u ∈ int P.

Throughout this paper, we may assume that f (x, t, ξ) =  for every t ≤ , x ∈  and ξ ∈

RN because we consider the existence of a positive solution only In what follows, the

norm on W,p( p, where q denotes the usual norm of L q()

for u ∈ L q() ( ≤ q ≤ ∞) Moreover, we denote u±:= max{±u, }.

1.2 Properties of the mapA

Remark  The following assertions hold under condition (A):

(i) for all x ∈ , A(x, y) is maximal monotone and strictly monotone in y;

(ii) |A(x, y)| ≤ C|y| p– for every (x, y) ∈  × R N;

(iii) A(x, y)y≥ C

p– |y| p for every (x, y) ∈  × R N,

where Cand Care the positive constants in (A)

Proposition  ([, Proposition ]) Let A: W ,p

 () → W ,p

 ()∗be a map defined by



A(u), v

=



 A(x, ∇u)∇v dx

for u, v ∈ W ,p

prop-erty, that is, any sequence {u n } weakly convergent to u with lim sup n→∞A(u n ), u n – u

strongly converges to u.

2 Constructing approximate solutions

Choose a function ψ ∈ P \ {} In this section, for such ψ and ε > , we consider the

following elliptic equation:

⎧

⎨

⎩

– div A(x, ∇u) = f (x, u, ∇u) + εψ(x) in ,

In [], the caseψ ≡  in the above equation is considered.

Lemma  Suppose (f) or (f) Then there exists λ>  such that f (x, t, ξ)t + λt p ≥  for

every x ∈ , t ≥  and ξ ∈ R N

Proof From the growth condition of fand (), it follows that

f(x, t)t ≥ –b ∞t p – bt p for every (x, t) ∈  × [, ∞)

holds, where bis a positive constant independent of (x, t) Therefore, for λ≥ b ∞+

b, we easily see that f (x, t, ξ)t +λt p ≥ f(x, t)t + λt p ≥  for every x ∈ , t ≥  and ξ ∈ R N

Proposition  If u ε ∈ W ,p

 () is a non-negative solution of (P; ε) for ε ≥ , then u ε∈

L∞() Moreover, for any ε> , there exists a positive constant D >  such that ε ∞≤

Proof Set p∗= Np/(N – p) if N > p, and in the case of N ≤ p, p∗> p is an arbitrarily fixed

constant Let u ε be a non-negative solution of (P; ε) with  ≤ ε ≤ ε(someε> ) For

r > , choose a smooth increasing function η(t) such that η(t) = t r+if ≤ t ≤ , η(t) =

dt if t ≥ d andη(t) ≥ d>  if ≤ t ≤ d for some  < d<  < d, d Defineξ M (u) :=

M r+ η(u/M) for M > .

If u ε ∈ L r+p(), then by taking ξ M (u ε) as a test function (note thatηis bounded), we have

C

p – 



 |∇u ε|p ξ

M (u ε ) dx

≤

 A(x, ∇u ε)∇uε ξ

M (u ε ) dx







f (x, u ε,∇u ε) +εψξ M (u ε ) dx

≤ b







 + u q– ε +ε ∞

M r+ η(u ε /M) dx + b



 |∇u ε|q– ξ M (u ε ) dx

≤ dd



b+ε ∞

ε r+q r+q+ ε r+ r+

+ b



 |∇u ε|q– ξ M (u ε ) dx ()

due to Remark (iii) and M r+ η(t/M) ≤ ddt r+ Putting β := p/(p – q + ) < p, we see

that (ξ M (u ε))/(ξ

M (u ε))(q–)/p = u r+

ε /((r + )u r

ε)(q–)/p ≤ u +r/ β

ε provided  < u ε < M (note

r > ) Similarly, if M ≤ u ε ≤ dM, then ( ξ M (u ε))/(ξ

M (u ε))(q–)/p ≤ ddM r+ /(dM r)(q–)/p=

ddd (–q)/p M +r/ β ≤ ddd (–q)/p u +r/ β

ε , and if u ε > dM, then (ξ M (u ε))/(ξ

M (u ε))(q–)/p=

d/β M r/β u ε ≤ d/β

 u +r/ β

ε (note d> ) Thus, according to Young’s inequality, for everyδ > ,

there exists C δ>  such that



 |∇u ε|q– ξ M (u ε ) dx ≤ δ



 |∇u ε|p ξ

M (u ε ) dx + C δ



u ε>

(ξ M (u ε))β (ξ

M (u ε))(q–) β/p dx

≤ δ



 |∇u ε|p ξ

M (u ε ) dx + C δ d



 u r+β ε dx, () whereβ := p/(p – q + ) < p and d= max{ddd (–q)/p , d/β} (> ) As a result, because of

r + p > r + q, r + β, according to Hölder’s inequality and the monotonicity of t rwith respect

to r on [, ∞), taking a  < δ < C/b(p – ) and setting u M

ε (x) := min {u ε (x), M}, we obtain

b

rp

max

, ε r+p r+p ≥rp

 |∇u ε|p ξ

M (u ε ) dx≥rp

∇u M ε p

u M εr

dx

=u M

εrp

≥ C∗u M

ε rp

¯p∗= C∗u M

εr+p

¯p∗r ()

provided u ε ∈ L r+p() by () and (), where r=  + r/p, C∗ comes from the

continu-ous embedding of W,p() into L p∗

() and dis a positive constant independent of u ε,

ε and r Consequently, Moser’s iteration process implies our conclusion In fact, we

de-fine a sequence{r m}m by r:= p∗– p and r m+ := p∗(p + r m )/p – p Then, we see that u ε∈

L p∗(p+rm)/p

() = L p+r m+() holds if u ε ∈ L p+r m() by applying Fatou’s lemma to () and

letting M → ∞ Here, we also see r m+ = p∗r m /p + p∗– p ≥ (p∗/p) m+ r→ ∞ as m → ∞.

Therefore, by the same argument as in Theorem C in [], we can obtain u ε ∈ L∞() and

Lemma  Suppose (f) or (f) If u ε ∈ W ,p

 () is a solution of (P; ε) for ε > , then u ε ∈ int P.

Proof Taking –(u ε)–as a test function in (P; ε), we have

C

p – ∇(uε)–p

p≤



 A(x, ∇u ε)

–∇(u ε)–

dx = – ε



 ψ(u ε)–dx≤ 

because of f (x, t, ξ) =  if t ≤  and by Remark (iii) Hence, u ε≥  follows Because

Propo-sition  guarantees that u ε ∈ L∞(), we have u ε ∈ C,α

 () (for some  < α < ) by the

regu-larity result in [] Note that u ε ≡  because of ε >  and ψ ≡  In addition, Lemma 

im-plies the existence ofλ>  such that – div A(x, ∇u ε) +λu p– ε ≥  in the distribution sense

Therefore, according to Theorem A and Theorem B in [], u ε>  in and ∂u ε/∂ν <  on

The following result can be shown by the same argument as in [, Theorem .]

Proposition  Suppose (f) or (f) Then, for every ε > , (P; ε) has a positive solution u ε∈

intP.

Proof Fix any ε >  and let {e, , e m, } be a Schauder basis of W ,p

 () (refer to

[] for the existence) For each m ∈ N, we define the m-dimensional subspace V m of

W,p() by V m:= lin.sp.{e, , e m } Moreover, set a linear isomorphism T m: Rm → V m

by T m(ξ, ,ξ m) :=m

i= ξ i e i ∈ V m , and let T m∗ : V m∗→ (Rm)∗be a dual map of T m By iden-tifyingRm and (Rm)∗, we may consider that T m∗ maps from V m∗ toRm Define maps A m

and B m from V m to V m∗ as follows:



A m (u), v

:=



 A(x, ∇u)∇v dx and B m (u), v

:=



 f (x, u, ∇u)v dx + ε



for u, v ∈ V m We claim that for every m ∈ N, there exists u m ∈ V m such that A m (u m) –

B m (u m ) =  in V m∗ Indeed, by the growth condition of f , Remark (iii) and Hölder’s

in-equality, we easily have



A m (u) – B m (u), u

≥ C

p – 

p – b

+ q q+ q– p β

for every u ∈ V m, whereβ = p/(p – q + ) < p This implies that A m – B m is coercive on V m

by q < p Set a homotopy H m (t, y) := ty + ( – t)T m∗(A m (T m (y)) – B m (T m (y))) for t∈ [, ] and

y∈ Rm By recalling that A m – B m is coercive on V m , we see that there exists an R >  such

equivalent on V m Therefore, we have

 = deg

I m , B R(), 

= deg

H m(,·), B R(), 

= deg

H m(,·), B R(), 

= deg

T m∗◦ (A m – B m)◦ T m , B R(), 

,

where I mis the identity map onRm , B R() :={y ∈ R m;|y| < R} and deg(g, B, ) denotes the

degree onRm for a continuous map g : B→ Rm (cf []) Hence, this yields the existence

of y m∈ Rm such that (T m∗◦ (A m – B m)◦ T m )(y m ) = , and so the desired u mis obtained by

setting u m = T m (y m)∈ V m since T m∗ is injective

Because () with u = u m ∈ W ,p

assume, by choosing a subsequence, that u m converges to some uweakly in W,p() and

strongly in L p() Let P m be a natural projection onto V m , that is, P m u =m

i= ξ i e i for u =

∞

i= ξ i e i Since u m , P m u∈ V m and A m (u m ) – B m (u m ) =  in V m∗, by noting that A m = A on

V m for a map A defined in Proposition , we obtain



A(u m ), u m – u

 +

A(u m ), u– P m u



=

A m (u m ), u m – P m u



=

B m (u m ), u m – P m u









f (x, u m,∇u m) +εψ(u m – u) dx

+







f (x, u m,∇u m) +εψ(u– P m u) dx→ 

 ()∗ is bounded, by the boundedness of m

∞ holds As a result, it follows from the (S)+property of A that u m → uin W,p() as

Finally, we shall prove that uis a solution of (P; ε) Fix any l ∈ N and ϕ ∈ V l For each

m ≥ l, by letting m → ∞ in A m (u m), m (u m),



 A(x, ∇u)∇ϕ dx =



 f (x, u,∇u)ϕ dx + ε



Since l is arbitrary, () holds for every ϕ ∈l≥V l Moreover, the density of

l≥V lin

W,p() guarantees that () holds for every ϕ ∈ W ,p

 () This means that uis a solution

of (P; ε) Consequently, our conclusion u∈ int P follows from Lemma . 

3 Proof of theorems

Lemma  Let ϕ, u ∈ int P Then



 A(x, ∇u)∇



ϕ p

u p–

holds, where A p is the positive constant defined by ().

Proof Because of ϕ, u ∈ int P, there exist δ>δ>  such thatδu ≥ ϕ ≥ δu in  Thus,

δ≥ ϕ/u ≥ δ and /δ≥ u/ϕ ≥ /δin Hence, u/ϕ, ϕ/u ∈ L∞() hold Therefore, we

have



ϕ p

u p–

= p



ϕ u

p–

A(x, ∇u)∇ϕ – (p – )



ϕ u

p

A(x, ∇u)∇u

≤ pC

p – 

ϕ

u

p–

|∇u| p– |∇ϕ| – C

ϕ

u

p

|∇u| p

p – 

/p ϕ

u |∇u|

p–

p

p – 

/p

CC(–p)/p |∇ϕ|

– C



ϕ u

p

Lemma  Assume that a∈ C(, [, ∞)) and let ϕ, u ∈ int P Then



 a(x) |∇ϕ| p– ∇ϕ∇

ϕ p

– u p

ϕ p–

dx –



 a(x) |∇u| p– ∇u∇

ϕ p

– u p

u p–

holds.

Proof First, we note that u/ϕ, ϕ/u ∈ L∞() hold by the same reason as in Lemma 

Ap-plying Young’s inequality to the second term of the right-hand side in () (refer to ()

with C= C= p – ), we obtain

a(x) |∇ϕ| p– ∇ϕ∇



ϕ p – u p

ϕ p–

≥ a(x)



|∇ϕ| p – p



u ϕ

p–

|∇ϕ| p– |∇u| + (p – )



u ϕ

p

|∇ϕ| p

()

≥ a(x)

|∇ϕ| p–|∇u| p

()

in Similarly, we also have

a(x)|∇u| p– ∇u∇

ϕ p

– u p

u p–

≤ a(x)

|∇ϕ| p–|∇u| p

Under (f) or (f), we denote a solution u ε ∈ int P of (P; ε) for each ε >  obtained by

Proposition 

Lemma  Assume (f) or (f) Let I := (, ] Then {u ε}ε∈I is bounded in W,p().

Proof Taking u ε as a test function in (P; ε), we have

C

p

p≤



 A(x, ∇u ε)∇uε dx =



 f (x, u ε,∇u ε )u ε dx + ε



 ψu ε dx

≤ b



ε + ε q

q+ ε q–

p ε β

≤ b





ε ε q

by Remark (iii), the growth condition of f , Hölder’s inequality and the continuity of the

embedding of W,p() into L p(), where β = p/(p–q+) (< p) and b

is a positive constant

independent of u ε Because of q < p, this yields the boundedness of ε ε p) 

Lemma  Assume (f) or (f) Then |∇u ε |/u ε ∈ L p( ε |/u ε p p ≤ λ||/Chold

for every ε > , where || denotes the Lebesgue measure of , and where Cand λare

positive constants as in (A) and Lemma , respectively.

Proof Fix any ε >  and choose any ρ >  By taking (u ε+ρ) –pas a test function, we obtain

( – p)





A(x, ∇u ε)∇uε

(u ε+ρ) p dx =





f (x, u ε,∇u ε) +εψ

(u ε+ρ) p– dx ≥ –λ





u p– ε

(u ε+ρ) p– dx

by Lemma  andεψ ≥  On the other hand, by Remark (iii) and  – p < , we have

( – p)





A(x, ∇u ε)∇uε

(u ε+ρ) p dx ≤ –C





|∇u ε|p

Therefore, () and () imply the inequality

 |∇u ε|p /(u ε+ρ) p dx ≤ λ||/Cfor every

Lemma  Assume (f) and (AH) Let ϕ ∈ int P If u ε →  in C

() as ε → +, then

lim

ε→+



 a



x, |∇u ε|∇u ε∇

ϕ p

– u p ε

u p– ε

dx

 = 

holds, where ais a continuous function as in (AH).

Proof Note that u ε/ϕ, ϕ/u ε ∈ L∞() hold (as in the proof of Lemma ) Because we easily

see that| a(x, |∇u|)|∇u|dx p p for every u ∈ W ,p

 () with some C > 

inde-pendent of u (see ()), it is sufficient to show| a(x, |∇u ε |)∇u ε ∇(ϕ p /u p– ε ) dx| →  as

ε → + Here, we fix any δ >  By the property of a(see ()) and because we are

assum-ing that u ε →  in C

() as ε → +, we have |a(x, |∇u ε |)| ≤ δ|∇u ε|p– for every x ∈ 

provided sufficiently smallε >  Therefore, for such sufficiently small ε > , we obtain



 a



x, |∇u ε|∇u ε∇

 ϕ p

u p– ε

dx



≤ p





|a(x, |∇u ε |)||∇u ε ||∇ϕ|ϕ p–





|a(x, |∇u ε |)||∇u ε|ϕ p

p

C() p







|∇u ε|

u ε

p–

dx + (p – )







|∇u ε|

u ε

p

dx

p

C

 () ||p( λ/C)–/p + (p – )( λ/C) because of |∇u ε |/u ε ∈ L p() by Lemma  Since δ >  is arbitrary, our conclusion is

3.1 Proof of main results

Proof of Theorems

independent ofε ∈ (, ] Hence, there exist M>  and  <α <  such that u ε ∈ C,α

 () and

ε C,α

 () ≤ Mfor everyε ∈ (, ] by the regularity result in [] Because the embedding

of C,α() into C

() is compact and by u ε ∈ int P, there exists a sequence {ε n } and u∈ P

such thatε n → + and u n := u ε n → uin C() as n → ∞ If u=  occurs, then u∈

shall prove u=  by contradiction for each theorem So, we suppose that u= , whence

u n →  in C

() as n → ∞.

eigenvalueμ(m) (cf [, ], it is well known that we can obtain ϕ as the minimizer of ()),

namely,ϕ is a positive solution of – p u = μ(m)m(x)|u| p– u in  and u =  on ∂ Since

p-Laplacian is (p – )-homogeneous, we may assume that ϕ satisfies m(x)ϕ p dx = , and

hence p=μ(m)

 m(x) ϕ p dx = μ(m) holds by taking ϕ as a test function Choose

ρ >  satisfying b– A p μ(m) > p p (note that b– A p μ(m) >  as in (f)) Due to (f),

there exists aδ >  such that f(x, t) ≥ (bm(x) – ρ)t p–for every ≤ t ≤ δ and x ∈  Since

we are assuming u n →  in C

( n ∞≤ δ occurs for sufficiently large n.

Then, for such sufficiently large n, according to Lemma , () and ψ ≥ , we obtain

A p μ(m) = A p p p≥



 A(x, ∇u n)∇

 ϕ p

u p– n

dx =





f (x, u n,∇u n) +εψ

u p– n

ϕ p dx

≥





f(x, u n)

u p– n ϕ p dx ≥ b



 m(x) ϕ p dx – p p = b– p p > A p μ(m).

Proof of Theorem  Since∞ > supx ∈ a(x)≥ infx ∈ a(x) >  holds, by the standard

ar-gument as in the p-Laplacian, we see that λ(m) >  and it is the first positive eigenvalue

of – div(a(x) |∇u| p– ∇u) = λm(x)|u| p– u in  and u =  on ∂ Therefore, by the

well-known argument, there exists a positive eigenfunctionϕ∈ int P corresponding to λ(m)

(we can obtainϕ as the minimizer of ()) Hence, by takingϕ as a test function, we

have  <

 a(x) |∇ϕ|p dx = λ(m)

 m(x) ϕ p

dx Thus,

 m(x) ϕ p

dx >  follows Because

u n ∈ int P is a solution of (P; ε n) andϕ∈ int P is an eigenfunction corresponding to λ(m),

according to Lemma  and Lemma  (note A(x, y) = a|y| p– y + a(x, |y|)y as in (AH)),

we obtain

≤



 a(x)|∇ϕ|p– ∇ϕ∇

ϕ p

 – u p n

ϕ p–



dx –



 a(x)|∇u n|p– ∇u n∇

ϕ p

 – u p n

u p– n

dx

≤ λ(m)



 m

ϕ p

 – u p n

dx –





f(x, u n)

u p– n

ϕ p

dx

+



 a



x, |∇u n|∇u n∇

ϕ p

 – u p n

u p– n

dx +



 f (x, u n,∇u n )u n dx + ε n



 ψu n dx

= –







f(x, u n)

u p– n

– bm(x)

ϕ p

dx –

b–λ(m) 

m(x)ϕ p

as n → ∞ since we are assuming u n →  in C

(), where we use the facts that ψ ≥  and

ϕ>  in Furthermore, by Fatou’s lemma and (), we have

lim inf

n→∞







f(x, u n)

u p– n

– bm(x)

ϕ p

dx≥ 

As a result, by taking a limit superior with respect to n in (), we have  ≤ –(b–

λ(m))

m(x)ϕ p

Competing interests

The author declares that she has no competing interests.

Acknowledgements

The author would like to express her sincere thanks to Professor Shizuo Miyajima for helpful comments and

encouragement The author thanks Professor Dumitru Motreanu for giving her the opportunity of this work The author

thanks referees for their helpful comments.

Received: 15 May 2013 Accepted: 10 July 2013 Published: 24 July 2013

Therefore, according to Theorem A and Theorem...