Open Access

A note on the existence of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting

Boundary Value Problems20122012:136

DOI: 10.1186/1687-2770-2012-136

Received: 13 April 2012

Accepted: 8 October 2012

Published: 22 November 2012

Abstract

In this note, we study the existence and multiplicity of solutions for the quasilinear elliptic problem as follows:

{ div ( a ( | u | ) u ) = f ( x , u ) , in  Ω ; u = 0 , on  Ω ,

where Ω R N is a bounded domain with a smooth boundary. The existence and multiplicity of solutions are obtained by a version of the symmetric mountain pass theorem.

Keywords

Orlicz-Sobolev spaces symmetric mountain pass theorem quasilinear elliptic equations

1 Introduction

In this note, we discuss the existence and multiplicity of solutions of the following boundary value problem:
{ div ( a ( | u | ) u ) = f ( x , u ) , in  Ω ; u = 0 , on  Ω ,
(1.1)
where Ω R N is a bounded domain with a smooth boundary Ω. The function a is such that p : R R defined by
p ( t ) = { a ( | t | ) t , t 0 ; 0 , t = 0 ,
is an increasing homeomorphism from R onto itself and the continuous function f ( x , t ) C ( Ω ¯ × R , R ) satisfies f ( x , 0 ) = 0 , x Ω ¯ . Especially, when a ( t ) = | t | p 2 , the problem (1.1) is the well-known p-Laplacian equation. There is a large number of papers on the existence of solutions for the p-Laplacian equation. But the problem (1.1) possesses more complicated nonlinearities. For example, it is inhomogeneous and has an important physical background, e.g.,
  1. (a)

    nonlinear elasticity: P ( t ) = ( 1 + t 2 ) γ 1 , γ > 1 2 ;

     
  2. (b)

    plasticity: P ( t ) = t α ( log ( 1 + t ) ) β , α 1 , β > 0 ;

     
  3. (c)

    generalized Newtonian fluids: P ( t ) = 0 t s 1 α ( sinh 1 s ) β d s , 0 α 1 , β > 0 .

     

So, in the discussions, some special techniques are needed, and the problem (1.1) has been studied in an Orlicz-Sobolev space and received considerable attention in recent years; see, for instance, the papers [19]. In paper [9], Fang and Tan discussed the problem (1.1) under the conditions that f ( x , t ) was odd in t. They got the first result that when h + < p , and f ( x , t ) C t q 1 for 0 < t < δ , q < p , the problem (1.1) had a sequence of solutions by genus theory. The second result is that when f ( x , t ) satisfies 0 < α F ( x , t ) t f ( x , t ) , x Ω ¯ , t 0 , α > p + and f ( x , t ) = o ( p ( | t | ) ) as | t | 0 , the problem (1.1) has infinitely many pairs of solutions which correspond to the positive critical values by the symmetric mountain pass theorem.

Motivated by their results, in this note, we discuss the problem (1.1) when f ( x , t ) is still odd in t but it satisfies weaker conditions than [9]; and furthermore, we need not know the behaviors of f ( x , t ) near the zero. If h + > p , we can get multiplicity of solutions by a version of the symmetric mountain pass theorem.

The paper is organized as follows. In Section 2, we present some preliminary knowledge on the Orlicz-Sobolev spaces and give the main result. In Section 3, we make the proof.

2 Preliminaries

Obviously, the problem (1.1) allows a nonhomogeneous function p in the differential operator defining the problem (1.1). To deal with this situation, we introduce an Orlicz-Sobolev space setting for the problem (1.1) as follows.

Let
P ( t ) = 0 t p ( s ) d s , P ˜ ( t ) = 0 t p 1 ( s ) d s , t R ,

then P and P ˜ are complementary N-functions (see [10]), which define the Orlicz spaces L P : = L P ( Ω ) and L P ˜ : = L P ˜ ( Ω ) respectively.

Throughout this paper, we assume the following condition on P:
( p ) 1 < p : = inf t > 0 t p ( t ) P ( t ) p + : = sup t > 0 t p ( t ) P ( t ) < + .
Under the condition (p), the Orlicz space L P coincides with the set (equivalence classes) of measurable functions u : Ω R such that
Ω P ( | u | ) d x < + ,
and is equipped with the (Luxemburg) norm, i.e.,
| u | P : = inf { k > 0 : Ω P ( | u | k ) d x < 1 } .
We will denote by W 1 , P ( Ω ) the corresponding Orlicz-Sobolev space with the norm
u W 1 , P ( Ω ) : = | u | P + u P
and define W 0 1 , P ( Ω ) as the closure of C 0 in W 1 , P ( Ω ) . In this note, we will use the following equivalent norm on W 0 1 , P ( Ω ) :
u : = inf { k > 0 : Ω P ( | u | k ) d x < 1 } .
Now, we introduce the Orlicz-Sobolev conjugate P of P, which is given by
P 1 ( t ) : = 0 t p 1 ( τ ) τ N + 1 N d τ ,
where we suppose that
lim t 0 t 1 p 1 ( τ ) τ N + 1 N d τ < + , lim t 1 t p 1 ( τ ) τ N + 1 N d τ = + .

Let p : = inf t > 0 t P ( t ) P ( t ) , p + : = sup t > 0 t P ( t ) P ( t ) . Throughout this paper, we assume that p + < p . Now, we will make the following assumptions on f ( x , t ) .

( f ) There exists an odd increasing homeomorphism h from R to R, and nonnegative constants c 1 , c 2 such that
| f ( x , t ) | c 1 + c 2 h ( | t | ) , t R , x Ω ¯ ,
and lim t + H ( t ) P ( k t ) = 0 , k > 0 , where
H ( t ) : = 0 t h ( s ) d s .
Let
H ˜ ( t ) : = 0 t h 1 ( s ) d s ,

then we can obtain complementary N-functions which define corresponding Orlicz spaces L H and L H .

Similar to the condition (p), we also assume the following condition on H:
( h ) 1 < h : = inf t > 0 t h ( t ) H ( t ) h + : = sup t > 0 t h ( t ) H ( t ) < + .

In order to prove our results, we now state some useful lemmas.

Lemma 2.1 [10]

Under the condition (p), the spaces L P ( Ω ) , W 0 1 , P ( Ω ) and W 1 , P ( Ω ) are separable and reflexive Banach spaces.

Lemma 2.2 [10]

Under the condition ( f ), the embedding W 0 1 , P ( Ω ) L H ( Ω ) is compact.

Lemma 2.3 [2]

Let ρ ( u ) = Ω P ( u ) d x , we have
  1. (1)

    if | u | P < 1 , then | u | P p + ρ ( u ) | u | P p ;

     
  2. (2)

    if | u | P > 1 , then | u | P p ρ ( u ) | u | P p + ;

     
  3. (3)

    if 0 < t < 1 , then t p + P ( u ) P ( t u ) t p P ( u ) ;

     
  4. (4)

    if t > 1 , then t p P ( u ) P ( t u ) t p + P ( u ) .

     

Lemma 2.4 [1113]

Let E = V + X , where E is a real Banach space and V is finite dimensional. Suppose I C 1 ( E , R ) is an even functional satisfying I ( 0 ) = 0 and

( I 1 ) there is a constant ρ > 0 such that I | B ρ X 0 ;

( I 2 ) there is a subspace W of E with dim V < dim W < and there is M > 0 such that max u W I ( u ) < M ;

( I 3 ) considering M > 0 given by ( I 2 ), I satisfies (PS) c for 0 c M .

Then I possesses at least dim W dim V pairs of nontrivial critical points.

Using the version of the symmetric mountain pass theorem mentioned above, we can state our result as follows.

Theorem 2.1 Assume that f ( x , t ) is odd in t, satisfies ( f ) with p < h + p + and the following assumptions:

( f 1 ) there exist η > p + and 1 < σ < p , and a 1 , a 2 > 0 , such that 1 η f ( x , t ) t F ( x , t ) a 1 a 2 | t | σ for every t R , a.e. in Ω.

( f 2 ) there is Ω 0 Ω with | Ω 0 | > 0 such that lim inf | t | F ( x , t ) / | t | p + = uniformly a.e. in Ω 0 .

Then for any given k N , the problem (1.1) possesses at least k pairs of nontrivial solutions.

3 Main results and proofs

In this section, we assume that N 1 and E = W 0 1 , P ( Ω ) , u E is called a weak solution of the problem (1.1) if
Ω a ( | u | ) u ϕ d x = Ω f ( x , u ) ϕ d x , ϕ E .
Set
I ( u ) = Ω P ( | u | ) d x Ω F ( x , u ) d x , u E

and we know that the critical points of I are just the weak solutions of the problem (1.1).

For E is a separable and reflexive Banach space, then there exist (see [9]) { e n } n = 1 E and { e n } n = 1 E such that
e n ( e m ) = δ n , m = { 1 , if  n = m ; 0 , if  n m . and e n ( v ) = α n for  v = i = 1 α i e i E .
Now, we set V j = { u W 0 1 , P ( Ω ) : e i ( u ) = 0 , i > j } , X j = { u W 0 1 , P ( Ω ) : e i ( u ) = 0 , i j } , so
W 0 1 , P ( Ω ) = V j X j .
(3.1)

Lemma 3.1 Given δ > 0 , there is j N such that for all u X j , | u | H δ u .

Proof We prove the lemma by contradiction. Suppose that there exist δ > 0 and u j X j for every j N such that | u j | H δ u j . Taking v j = u j | u j | H , we have | v j | H = 1 for every j N and v j 1 δ . Hence, { v j } W 0 1 , P ( Ω ) is a bounded sequence, and we may suppose, without loss of generality, that v j v in W 0 1 , P ( Ω ) . Furthermore, e n ( v ) = 0 for every n N since e n ( v j ) = 0 for all j n . This shows that v = 0 . On the other hand, by the compactness of embedding W 0 1 , P ( Ω ) L H ( Ω ) , we conclude that | v | H = 1 . This proves the lemma. □

Lemma 3.2 Suppose f satisfies ( f ), then there exist j N and ρ , α > 0 such that
I | B ρ X j α .
Proof Now suppose that u > 1 . From ( f ), we know that
I ( u ) = Ω P ( | u | ) d x Ω F ( x , u ) d x u p C 1 | u | H h + C 2 .
Consequently, considering δ > 0 to be chosen posteriorly by Lemma 3.1, we have for all u X j and j sufficiently large,
I ( u ) u p ( 1 C 1 δ h + u h + p ) C 2 .

Now, taking u = ρ ( δ ) = ( 1 2 C δ h + ) 1 h + p and noting that ρ ( δ ) + , if δ 0 , we can choose δ > 0 such that 1 2 ρ p > C 2 , ρ > 1 , and I ( u ) > 0 for every u X j , u = ρ , the proof is complete. □

Lemma 3.3 Suppose f satisfies ( f 2 ). Then given m N , there exist a subspace W of W 0 1 , P ( Ω ) and a constant M m > 0 such that dim W = m and max u W I ( u ) < M m .

Proof Let x 0 Ω 0 and r 0 > 0 be such that B ( x 0 , r 0 ) ¯ Ω , and 0 < | B ( x 0 , r 0 ) ¯ Ω 0 | < | Ω 0 | 2 . First, we take v 1 C 0 ( Ω ) with supp ( v 1 ) = B ( x 0 , r 0 ) ¯ . Considering Ω 1 = Ω 0 [ B ( x 0 , r 0 ) ¯ Ω 0 ] Ω ˆ 0 = Ω B ( x 0 , r 0 ) ¯ , we have | Ω 1 | | Ω 0 | 2 > 0 . Let x 1 Ω 1 and r 1 > 0 be such that B ( x 1 , r 1 ) ¯ Ω ˆ 0 , and 0 < | B ( x 1 , r 1 ) ¯ Ω 1 | < | Ω 1 | 2 . Next, we take v 2 C 0 ( Ω ) with supp ( v 2 ) = B ( x 1 , r 1 ) ¯ . After a finite number of steps, we get v 1 , v 2 , , v m such that supp ( v i ) supp ( v j ) = , i j , and | supp ( v j ) Ω 0 | > 0 for all i , j { 1 , 2 , , m } . Let W = span { v 1 , v 2 , , v m } , by construction, dim W = m , and Ω | v | p + d x > 0 for every v W { 0 } .

Since max u W { 0 } I ( u ) = max t > 0 , v W B 1 ( 0 ) ( Ω P ( t | v | ) d x Ω F ( x , t v ) d x ) , if t > 1 , then I ( t v ) t p + Ω F ( x , t v ) d x = t p + ( 1 1 t p + Ω F ( x , t v ) d x ) . Now, it suffices to verify that
lim t 1 t p + Ω F ( x , t v ) d x > 1 .
From the condition ( f 2 ), given L > 0 , there is C > 0 such that for every s R , a.e. x in Ω 0 ,
F ( x , s ) L | s | p + C .
Consequently, for v B 1 ( 0 ) W and t > 1 ,
Ω F ( x , t v ) d x L t p + Ω 0 | v | p + d x C t h + Ω Ω 0 H ( v ) d x C 2 ,
and
lim t Ω f ( x , t v ) d x t p + L Ω 0 | v | p + d x C Ω Ω 0 H ( v ) d x L r C R ,

where r = min { Ω 0 | v | p + d x , v B 1 ( 0 ) W } and R = max { Ω Ω 0 H ( v ) d x , v B 1 ( 0 ) W } . Observing that W is finite dimensional and we have R < + , r > 0 , the inequality is obtained by taking L > 1 r ( 1 + C R ) ; the proof is complete. □

Lemma 3.4 Suppose f satisfies ( f 1 ), then I satisfies the (PS) condition.

Proof We suppose that u n > 1 ,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equy_HTML.gif

Noting that 1 < σ < p , η > p + , { u n } is bounded. By [9], Lemma 3.1, we know that I satisfies the (PS) condition. □

Proof of Theorem 2.1 First, we recall that W 0 1 , P ( Ω ) = V j X j , where V j and X j are defined in (3.1). Invoking Lemma 3.2, we find j N , and I satisfies I 1 with X = X j . Now, by Lemma 3.3, there is a subspace W of W 0 1 , P ( Ω ) with dim W = k + j = k + dim V j and such that I satisfies ( I 2 ). Since I ( 0 ) = 0 and I is even, we may apply Lemma 2.4 to conclude that I possesses at least k pairs of nontrivial critical points. The proof is complete. □

Declarations

Acknowledgements

Project supported by Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK201209), the China Scholarship Council, the Fundamental Research Funds for the Central Universities (No. JUSRP11118) and Foundation for young teachers of Jiangnan University (No. 2008LQN008).

Authors’ Affiliations

(1)
School of Science, Jiangnan University
(2)
School of Mathematics Science, Nanjing Normal University

References

  1. Clément PH, García-Huidobro M, Manásevich R, Schmitt K: Mountain pass type solutions for quasilinear elliptic equations. Calc. Var. Partial Differ. Equ. 2000, 11: 33–62. 10.1007/s005260050002View ArticleGoogle Scholar
  2. Fukagai N, Ito M, Narukawa MK:Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on R N . Funkc. Ekvacioj 2006, 49: 235–267. 10.1619/fesi.49.235MathSciNetView ArticleGoogle Scholar
  3. Fukagai N, Narukawa K: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Ann. Mat. Pura Appl. 2007, 186: 539–564. 10.1007/s10231-006-0018-xMathSciNetView ArticleGoogle Scholar
  4. García-Huidobro M, Le V, Manásevich R, Schmitt K: On the principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting. Nonlinear Differ. Equ. Appl. 1999, 6: 207–225. 10.1007/s000300050073View ArticleGoogle Scholar
  5. Tan, Z, Fang, F: Orlicz-Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations. Preprint
  6. Mihǎilescu M, Rădulescu V: Nonhomogeneous Neumann problems in Orlicz-Sobolev spaces. C. R. Math. 2008, 346: 401–406. 10.1016/j.crma.2008.02.020View ArticleGoogle Scholar
  7. Bonanno G, Bisci GM, Rǎdulescu VD: Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces. Nonlinear Anal. 2012, 75: 4441–4456. 10.1016/j.na.2011.12.016MathSciNetView ArticleGoogle Scholar
  8. Černý R: Generalized n -Laplacian: quasilinear nonhomogenous problem with critical growth. Nonlinear Anal. 2011, 74: 3419–3439. 10.1016/j.na.2011.03.002MathSciNetView ArticleGoogle Scholar
  9. Fang F, Tan Z: Existence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting. J. Math. Anal. Appl. 2012, 389: 420–428. 10.1016/j.jmaa.2011.11.078MathSciNetView ArticleGoogle Scholar
  10. Adams RA, Fournier JJF: Sobolev Spaces. 2nd edition. Academic Press, Amsterdam; 2003.Google Scholar
  11. Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349–381. 10.1016/0022-1236(73)90051-7MathSciNetView ArticleGoogle Scholar
  12. Bartolo P, Benci V, Fortunato D: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. TMA 1983, 7: 981–1012. 10.1016/0362-546X(83)90115-3MathSciNetView ArticleGoogle Scholar
  13. Silva, EAB: Critical point theorems and applications to differential equations. PhD thesis, University of Wisconsin-Madison (1988)Google Scholar

Copyright

© Yang and Zhang; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.