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Infinitely many solutions for a class of quasilinear elliptic equations with p-Laplacian in R N

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Abstract

In this paper, we study the multiplicity of solutions for a class of quasilinear elliptic equations with p-Laplacian in R N . In this case, the functional J is not differentiable. Hence, it is difficult to work under the classical framework of the critical point theory. To overcome this difficulty, we use a nonsmooth critical point theory, which provides the existence of critical points for nondifferentiable functionals.

MSC:35J20, 35J92, 58E05.

1 Introduction and main results

Recently, the multiplicity of solutions for the quasilinear elliptic equations has been studied extensively, and many fruitful results have been obtained. For example, in [1], Shibo Liu considered the existence of multiple nonzero solutions of the Dirichlet boundary value problem

{ Δ p u = f ( x , u ) , in  Ω , u = 0 , on  Ω ,
(1.1)

where p>1, Δ p u=div( | u | p 2 u) denotes the p-Laplacian operator, Ω is a bounded domain in R N with smooth boundary Ω.

Moreover, Aouaoui studied the following quasilinear elliptic equation in [2]:

div ( A ( x , u ) u ) + 1 2 A s (x,u) | u | 2 + ( b ( x ) λ ) u=f(x,u),in  R N ,
(1.2)

and proved the multiplicity of solutions of the problem (1.2) by using the nonsmooth critical point theory. One can refer to [3, 4] and [5] for more results.

In this paper, we shall investigate the existence of infinitely many solutions of the following problem

div ( A ( x , u ) | u | p 2 u ) + 1 p A s ( x , u ) | u | p + ( b ( x ) λ ) | u | p 2 u = f ( x , u ) , in  R N ,
(1.3)

where 2<p<N, λR and A s (x,u) A s (x,s) | s = u , b(x) is a given continuous function satisfying

b(x)0for all x R N and lim | x | + b(x)=+.

In order to determine weak solutions of (1.3) in a suitable functional space E, we look for critical points of the functional J:ER defined by

J ( v ) = 1 p R N A ( x , v ) | v | p d x + 1 p R N ( b ( x ) λ ) | v | p d x R N F ( x , v ) d x , v E ,
(1.4)

where F(x,ξ)= 0 ξ f(x,t)dt. Under reasonable assumptions, the functional J is continuous, but not even locally Lipschitz. However, one can see from [4, 6] and [7] that the Gâteaux-derivative of J exists in the smooth directions, i.e., it is possible to evaluate

J ( u ) , v = lim t 0 J ( u + t v ) J ( u ) t = R N A ( x , u ) | u | p 2 u v d x + 1 p R N A s ( x , u ) | u | p v d x + R N ( b ( x ) λ ) | u | p 2 u v d x R N f ( x , u ) v d x

for all uE and vE L ( R N ).

Definition 1.1 A critical point u of the functional J is defined as a function uE such that J (u),v=0, vE L ( R N ), i.e.,

R N A ( x , u ) | u | p 2 u v d x + 1 p R N A s ( x , u ) | u | p v d x + R N ( b ( x ) λ ) | u | p 2 u v d x R N f ( x , u ) v d x = 0 , v E L ( R N ) .
(1.5)

Our approach to study (1.3) is based on the nonsmooth critical point theory developed in [8] and [9]. Dealing with this class of problems, the main difficulty is that the associated functional is not differentiable in all directions.

The main goal here is to establish multiplicity of results for (1.3), when f(x,s) is odd and A(x,s) is even in s. Such solutions for (1.3) will follow from a version of the symmetric mountain pass theorem due to Ambrosetti and Rabinowitz [10, 11]. Compared with problem (1.2) in [2], problem (1.3) is much more difficult, since the discreteness of the spectrum is not guaranteed. Therefore, we only consider the first eigenvalue λ 1 .

To state and prove our main result, we consider the following assumptions.

Suppose that N3 and p = N p N p .

(H1) Let A(,): R N ×RR be a function such that

  • for each sR, A(x,s) is measurable with respect to x;

  • for a.e. x R N , A(x,s) is a function of class C 1 with respect to s;

  • there exist 0<α<β<+ such that

    αA(x,s)β,a.e. x R N  and sR,
    (1.6)
    | A s ( x , s ) | β,a.e. x R N  and sR.
    (1.7)

(H2) There exist θ>p, 1<γ< θ p and α 1 >0 such that

0 γ p A s (x,s)s ( θ p γ ) A(x,s) α 1 ,a.e. x R N  and sR.
(1.8)

(H3) Let a Carathéodory function f(,): R N ×RR satisfy f(x,0)=0, a.e. x R N and

0<θF(x,s)f(x,s)s,a.e. x R N  and s0 in R,
(1.9)

where θ is the same as that in (H2).

(H4) There exists p1q< p 1 such that

| f ( x , s ) | c 0 | s | q ,a.e. x R N  and sR,
(1.10)

where c 0 is a positive constant.

Example 1.1 Let p=3. The following function satisfies hypotheses (H1) and (H2)

A(x,s)=2 ( sin ( | x | 2 ) + 2 ) arctan ( s 2 ) +20,

and the corresponding constants are

α=20,β=20+3π,θ=34,γ=10, α 1 =1.

Example 1.2 The following function satisfies hypotheses (H3) and (H4)

f(x,s)= | s | q 1 s,a.e. x R N  and sR.

On the other hand, we define the operator Lu= Δ p u+b()u. It follows from [12] that the discreteness of the spectrum is not guaranteed. Hence, we only consider the first eigenvalue λ 1 , where

λ 1 =inf { u p ; u E , u L p ( R N ) = 1 } .

Next, we can state the main theorem of the paper.

Theorem 1.1 Assume that A(x,s) and f(x,s) satisfy (H1)-(H4). Moreover, let A(x,s)=A(x,s) and f(x,s)=f(x,s), a.e. x R N , sR. If there exists a positive number μ such that λ(,μ λ 1 ), then problem (1.3) has infinitely many distinct solutions in E L ( R N ), i.e., there exists a sequence { u n }E L ( R N ), satisfying (1.3) and J( u n )+, as n.

To explain our result, we introduce some functional spaces. We define the reflexive Banach space E of all functions u: R N R with the norm u p = R N ( | u | p +b(x) | u | p )dx<.

Such a weighted Sobolev space has been used in many previous papers, see [13] and [14]. Now, we give an important property of the space E, which will play an essential role in proving our main results.

Remark 1.1 One can easily deduce E L z ( R N ) and E L z ( R N ) for pz< p . More details can be found in [2].

Throughout this paper, let denote the norm of E and u n u ( u n u) means that u n converges strongly (weakly) in corresponding spaces. stands for a continuous map, and means a compact embedding map. C denotes any universal positive constant unless specified.

The paper is organized as follows. In Section 2, we introduce the nonsmooth critical framework and preliminaries to our work. In Section 3, we give some lemmas to prove the main result. Finally, the proof of Theorem 1.1 is presented in Section 4.

2 Nonsmooth critical framework and preliminaries

Our results are based on the techniques of nonsmooth critical point theory. In this section, we recall some basic tools from [8] and [9].

Definition 2.1 Let (X,d) be a metric space, let I:XR be a continuous functional and uX. We denote by |dI|(u) the supremum of the σ’s in [0,+) such that there exist δ>0 and a continuous map H:B(u,δ)×[0,δ]X, satisfying

d ( H ( v , t ) , v ) tandI ( H ( v , t ) ) I(v)σt,(v,t)B(u,δ)×[0,δ].

The extended real number |dI|(u) is called the weak slope of I at u.

Note that the notion above was independently introduced in [15], as well.

Definition 2.2 Let (X,d) be a metric space, let I:XR be a continuous functional and cR. We say that I satisfies ( P S ) c , i.e., the Palais-Smale condition at level c, if every sequence { u n } in X with |dI|( u n )0 and I( u n )c admits a strongly convergent subsequence.

In order to treat the Palais-Smale condition, we need to introduce an auxiliary notion.

Definition 2.3 Let c be a real number. We say that functional I satisfies the concrete Palais-Smale condition at level c ( ( C P S ) c for short) if every sequence { u n }E satisfying

lim n + I( u n )=cand | I ( u n ) , v | ϵ n v,vE L ( R N )

possesses a strongly convergent subsequence in E, where { ϵ n } is some real number converging to zero.

Remark 2.1 Under assumptions (H1)-(H4), if the functional J satisfies (1.4), then J is continuous, and for every uE we have

|dJ|(u)sup { J ( u ) , v ; v E L ( R N ) , v 1 } ,

where |dJ|(u) denotes the weak slope of J at u.

Remark 2.2 Let c be a real number. If J satisfies ( C P S ) c , then J satisfies ( P S ) c .

Proof Let { u n }E be a sequence such that

lim n + |dJ|( u n )=0and lim n + J( u n )=c.

Note that for vE L ( R N ),

| J ( u n ) , v | J ( u n ) v=sup { J ( u n ) , v , v 1 } v.

By Remark 2.1, we have |dJ|( u n )sup{ J ( u n ),v,v1}. Taking ϵ n =sup{ J ( u n ),v,v1}, the conclusion follows. □

3 Basic lemmas

To derive our main theorem, we need the following lemmas. The first lemma is the version of the Ambrosetti-Rabinowitz mountain pass lemma [10, 11] and [16].

Lemma 3.1 Let X be an infinite-dimensional Banach space, and let I:XR be a continuous even functional satisfying ( P S ) c for every cR. Assume that

  1. (i)

    there exist ϱ>0, α>I(0) and a subspace VX of finite codimension such that

    u { V : u = ϱ } I(u)α,
  2. (ii)

    for every finite-dimensional subspace WX, there exists R>0 such that

    u { W : u = R } I(u)I(0).

Then there exists a sequence { c h } of critical values of I with c h .

Lemma 3.2 If uE is a critical point of J, then u L ( R N ).

Proof For r>1, M>0, consider the real functions T r , U M and W M defined in by

T r (s)={ s r , if  s > r , 0 , if  r s r , s + r , if  s < r ,
(3.1)

U M =min( T r (s),M) and W M =max( T r (s),M). Denoting s + =max(s,0) and s =min(s,0), we can take v= U M ( u + )E L ( R N ) as a test function in (1.5). Therefore,

R N A ( x , u ) | u | p 2 u ( U M ( u + ) ) d x + 1 p R N A s ( x , u ) | u | p U M ( u + ) d x + R N ( b ( x ) λ ) | u | p 2 u U M ( u + ) d x = R N f ( x , u ) U M ( u + ) d x .

Noting that u + U M ( u + )0 and A s (x, u + ) U M ( u + )0, we get

R N A ( x , u + ) | u + | p 2 u + ( U M ( u + ) ) d x | λ | R N | u + | p 2 u + | U M ( u + ) | d x + R N | f ( x , u + ) | | U M ( u + ) | d x .

From (1.10) and the fact | U M ( u + )| u + we deduce

{ u + > r } A ( x , u + ) | u + | p 2 u + ( U M ( u + ) ) dxC { u + > r } ( u + ) q + 1 dx.

Since U M ( u + ) T r ( u + ) a.e. in R N and U M ( u + ) T r ( u + ) in E as M+. It follows from M+ that

{ u + > r } A ( x , u + ) | u + | p dxC { u + > r } ( u + ) q + 1 dx.

Denote Ω r + ={x R N , u + (x)>r}. If mes( Ω r + )=0, then the result is true. In the following discussion, mes( Ω r + )0 is assumed. By (1.6), we obtain

Ω r + | u + | p dxC Ω r + ( u + r ) q + 1 dx+C r q + 1 mes ( Ω r + ) .
(3.2)

Note that ( Ω r + ( u + r ) q + 1 d x ) 1 p q + 1 C, then we can get

Ω r + ( u + r ) q + 1 dxC ( Ω r + ( u + r ) q + 1 d x ) p q + 1 .
(3.3)

On the other hand, we have

Ω r + r p dx Ω r + | u | p dx=C,

which implies that

r p C mes ( Ω r + ) .
(3.4)

Eventually, one can deduce from (3.2)-(3.4) that

Ω r + | u + | p dxC ( Ω r + ( u + r ) q + 1 d x ) p q + 1 +C r p mes ( Ω r + ) 1 q + 1 p + p p ,r>1.
(3.5)

By Theorem 5.2 of [17], we get that u + L ( R N ). Replacing U M ( u + ) by W( u ), we can similarly prove that u L ( R N ). We conclude that u L ( R N ), and the proof of Lemma 3.2 is completed. □

Lemma 3.3 Let { u n } be a bounded sequence in E with

J ( u n ) , v ϵ n v,vE L ( R N ) ,
(3.6)

where { ϵ n } is a sequence of real numbers converging to zero. Then there exists uE such that u n u a.e. in R N and, up to a subsequence, { u n } is weakly convergent to u in E. Moreover, we have

J ( u ) , v =0,vE L ( R N ) ,
(3.7)

i.e., u is a critical point of J.

Proof Since { u n } is bounded in E, and there is a uE (see [18]) such that, up to a subsequence,

u n uin E, u n uin  L p ( R N ) , u n ua.e. in  R N .

Moreover, since { u n } satisfies (3.6), by Theorem 2.1 of [19], we have, up to a further subsequence, u n u a.e. in R N .

We will use the device of [20]. We consider the test functions

v n =φexp { M u n + } ,
(3.8)

where φE L ( R N ), φ0 and M>0. According to (1.6) and (1.7), we have

MA(x, u n ) 1 p | A s ( x , u n ) | .

Since (3.6) holds by density for every vE L ( R N ), we can put v= v n in (3.6) and obtain that

R N A ( x , u n ) | u n | p 2 u n φ exp { M u n + } d x + R N ( 1 p A s ( x , u n ) | u n | p M A ( x , u n ) | u n | p 2 u n u n + ) φ exp { M u n + } d x + R N ( b ( x ) λ ) | u n | p 2 u n φ exp { M u n + } d x R N f ( x , u n ) φ exp { M u n + } d x ϵ n v .
(3.9)

On the other hand, note that

R N ( 1 p A s ( x , u n ) | u n | p M A ( x , u n ) | u n | p 2 u n u n + ) φexp { M u n + } dx0.
(3.10)

One can deduce from (3.10) and Fatou’s lemma that

R N A ( x , u ) | u | p 2 u φ exp { M u + } d x + R N ( 1 p A s ( x , u ) | u | p M A ( x , u ) | u | p 2 u u + ) φ exp { M u + } d x + R N ( b ( x ) λ ) | u | p 2 u φ exp { M u + } d x R N f ( x , u ) φ exp { M u + } d x 0 .
(3.11)

We consider the test functions φ n =φg( u n )exp{M u + } with φE L ( R N ), φ0 and g:RR, g C 1 (R), 0g1,

g=1on  [ 1 p , 1 p ] ,g=0on [,1][1,].

This together with (3.11) can prove that

R N A ( x , u ) | u | p 2 u φ d x + 1 p R N A s ( x , u ) | u | p φ d x + R N ( b ( x ) λ ) | u | p 2 u φ d x R N f ( x , u ) φ d x 0 , φ E L ( R N ) , as  n .
(3.12)

In a similar way, by considering the test functions v n =φexp{M u n }, it is possible to prove that

R N A ( x , u ) | u | p 2 u φ d x + 1 p R N A s ( x , u ) | u | p φ d x + R N ( b ( x ) λ ) | u | p 2 u φ d x R N f ( x , u ) φ d x 0 , φ E L ( R N ) , as  n .
(3.13)

From (3.12) and (3.13), it follows that

R N A ( x , u ) | u | p 2 u φ d x + 1 p R N A s ( x , u ) | u | p φ d x + R N ( b ( x ) λ ) | u | p 2 u φ d x R N f ( x , u ) φ d x = 0 , φ E L ( R N ) .
(3.14)

Finally, we can deduce (3.7) from (3.14). □

Remark 3.1 (see [21])

Let { u n } be a sequence in E satisfying (3.6). Then λ | u n | p f(x, u n ) u n L 1 ( R N ) and

| R N ( A ( x , u n ) | u n | p + 1 p A s ( x , u n ) u n | u n | p + ( b ( x ) λ ) | u n | p f ( x , u n ) u n ) d x | ϵ n u n , n N .
(3.15)

In the following lemma, we will prove the boundedness of a ( C P S ) c sequence { u n }E under (1.6), (1.8) and (1.9).

Lemma 3.4 Let cR and { u n } be a sequence in E satisfying (3.6) and

lim n + J( u n )=c.
(3.16)

Then { u n } is bounded in E.

Proof Calculating θJ( u n )γ J ( u n ), u n , from (3.15) and (3.16), we obtain

( θ p γ ) R N A ( x , u n ) | u n | p d x γ p R N A s ( x , u n ) | u n | p u n d x + ( θ p γ ) R N ( b ( x ) λ ) | u n | p d x + R N ( γ f ( x , u n ) u n θ F ( x , u n ) ) d x C ( 1 + u n ) , n N .

From (1.8) and (1.9), it follows that

α 1 R N | u n | p d x + ( θ p γ ) R N ( b ( x ) λ ) | u n | p d x + θ ( γ 1 ) R N F ( x , u n ) d x C ( 1 + u n ) , n N .
(3.17)

Moreover, there exist M>0 and C 1 (λ)>0 such that

( θ p γ ) R N ( b ( x ) λ ) | u n | p dx ( θ p γ ) R N b ( x ) 2 | u n | p dx C 1 { | x | < M } | u n | p dx.

Therefore, denoting D M ={x R N ,|x|<M}, we obtain from (3.17) that

α 1 R N | u n | p d x + ( θ p γ ) R N b ( x ) 2 | u n | p d x + θ ( γ 1 ) R N F ( x , u n ) d x C ( 1 + u n ) + C 1 u n L p ( D M ) p , n N .
(3.18)

By virtue of hypothesis (H3), we know that there exist a 0 >0 and b 0 >0 such that

F(x,s) a 0 | s | θ b 0 ,a.e. x D M  and sR.
(3.19)

From (3.18) and (3.19), it follows that

min ( α 1 , 1 2 ( θ p γ ) ) u n p + θ ( γ 1 ) a 0 u n L θ ( D M ) θ C ( 1 + u n ) + b 0 θ ( γ 1 ) mes ( D M ) + C 1 u n L p ( D M ) p .
(3.20)

On the other hand, by Hölder’s inequality and Young’s inequality, for all ϵ>0, there exists C ϵ >0 such that

C 1 u n L p ( D M ) p C u n L θ ( D M ) p C ϵ +ϵ u n L θ ( D M ) θ .
(3.21)

Using (3.20) and (3.21), we get

min ( α 1 , 1 2 ( θ p γ ) ) u n p C ( 1 + u n ) + b 0 θ ( γ 1 ) mes ( D M ) + C ϵ + ( ϵ θ ( γ 1 ) a 0 ) u n L θ ( D M ) θ .
(3.22)

Choosing 0<ϵ<θ(γ1) a 0 in (3.22), we find that { u n } is bounded in E. □

Lemma 3.5 Let { u n } be the same as that in Lemma 3.3. Then { u n }, up to a subsequence, converges strongly to u in E.

Proof By Lemma 3.3, we know that u is a critical point of the functional J. Then, from Lemma 3.2, we get u L ( R N ). Therefore, taking v=u as a test function in (3.7), we get

R N A ( x , u ) | u | p d x + 1 p R N A s ( x , u ) u | u | p d x + R N ( b ( x ) λ ) | u | p d x = R N f ( x , u ) u d x .
(3.23)

By virtue of { u n } is bounded in E, we can assume that there exists uE satisfying

u n uin Eand u n uin  L p ( R N ) and u n ua.e. x R N .

By Lemma 3.3, u n u a.e. in R N . Then by Fatou’s lemma, we have

R N A s (x,u)u | u | p dx lim inf n + R N A s (x, u n ) u n | u n | p dx.
(3.24)

Moreover, by E L p ( R N ) and E L q + 1 ( R N ), we get

lim n + R N f(x, u n ) u n dx= R N f(x,u)udx,
(3.25)
lim n + R N | u n | p dx= R N | u | p dx.
(3.26)

By using (3.23)-(3.26) and passing to limit in (3.15), we obtain

lim sup n + ( R N A ( x , u n ) | u n | p d x + R N b ( x ) | u n | p d x ) R N A ( x , u ) | u | p d x + R N b ( x ) | u | p d x .
(3.27)

On the other hand, by Lebesgue’s dominated convergence theorem and the weak convergence of u n to u in E, we get

lim n + R N A(x, u n ) u n | u | p 2 udx= R N A(x,u) | u | p dx,
(3.28)
lim n + R N A(x, u n ) | u | p dx= R N A(x,u) | u | p dx,
(3.29)
lim n + R N b(x) | u | p 2 u u n dx= R N b(x) | u | p dx.
(3.30)

Moreover, since A(x, u n ) | u n | p 2 u n and b ( x ) p 1 p | u n | p 2 u n are bounded in L p p 1 ( R N ), then we have

A ( x , u n ) | u n | p 2 u n A ( x , u ) | u | p 2 u in  L p p 1 ( R N ) , b ( x ) p 1 p | u n | p 2 u n b ( x ) p 1 p | u | p 2 u in  L p p 1 ( R N ) .

Therefore, from the definition of weak convergence, we obtain

lim n + R N A(x, u n ) | u n | p 2 u n udx= R N A(x,u) | u | p dx,
(3.31)
lim n + R N b(x) | u n | p 2 u n udx= R N b(x) | u | p dx.
(3.32)

Combining (3.27)-(3.32), it follows that

lim sup n + ( R N A ( x , u n ) ( | u n | p 2 u n | u | p 2 u ) ( u n u ) d x + R N b ( x ) ( | u n | p 2 u n | u | p 2 u ) ( u n u ) d x ) 0 .

It is well known that the following inequality

( | ξ | t 2 ξ | η | t 2 η ) (ξη)>0
(3.33)

holds for any t>1, ξ,η R N and ξη. Therefore,

lim sup n + ( R N A ( x , u n ) | u n u | p d x + R N b ( x ) | u n u | p d x ) 0.

According to (1.6), we conclude that { u n } converges strongly to u in E. □

Lemma 3.6 For every real number c, the functional J satisfies ( C P S ) c .

Proof Let { u n } be a sequence in E satisfying (3.6) and (3.16). By Lemma 3.4, { u n } is bounded in E. Therefore, the conclusion can be deduced from Lemma 3.5. □

4 Proof of Theorem 1.1

It is easy to check that the functional J is continuous and even. Moreover, by Remark 2.2 and Lemma 3.6, J satisfies ( P S ) c for every cR.

On the other hand, from (1.4), (1.6), (1.9) and (1.10), for uE, we have

J(u) min ( 1 , α ) p u p λ p R N | u | p dxC u q + 1 .
(4.1)

We discuss (4.1) in the following two cases:

In case λ0, we get

J(u) min ( 1 , α ) p u p C u q + 1 .

In case λ>0, by the definition of λ 1 , we get

J(u) 1 p ( min ( 1 , α ) λ λ 1 ) u p C u q + 1 ,

i.e., μ=min(1,α). Therefore, if λ satisfies λ<min(1,α) λ 1 , there exist ϱ>0 small enough and δ>0 such that

J(u)δfor u=ϱ.

Hence, condition (i) of Lemma 3.1 holds with V=E.

Now we consider a finite-dimensional subspace W of E. Let uW and J(u)0. From (1.6), we have

0J(u)max(1,β) u p λ p u L p ( R N ) p R N F(x,u)dx.
(4.2)

By virtue of (1.9) and (1.10), we know that there exist z(x) L ( R N ), satisfying z(x)>0 a.e. x R N and a positive constant C 2 such that

F(x,s)z(x) | s | θ C 2 | s | p ,a.e. x R N  and sR.
(4.3)

Combining (4.2)-(4.3), we have

max(1,β) u p R N z(x) | u | θ dx C 2 R N | u | p dx.
(4.4)

Since W is finite-dimensional, then all norms of W are equivalent. From (4.4), there exists C 3 >0 such that

u θ C 3 u p .

In view of θ>p , we deduce that the set {uW,J(u)0} is bounded in E and condition (ii) of Lemma 3.1 holds. By Lemma 3.1, the conclusion follows.

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Acknowledgements

The authors express their sincere thanks to the referees for their valuable criticism of the manuscript and for helpful suggestions. This work has been supported by the Natural Science Foundation of China (No. 11171220) and Shanghai Leading Academic Discipline Project (XTKX2012).

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Correspondence to Gao Jia.

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Keywords

  • quasilinear elliptic equations
  • nondifferentiable functional
  • p-Laplacian
  • multiple solutions