TSWJ The Scientific World Journal 1537-744X Hindawi Publishing Corporation 194310 10.1155/2014/194310 194310 Research Article Infinitely Many Weak Solutions of the p -Laplacian Equation with Nonlinear Boundary Conditions http://orcid.org/0000-0003-3237-1001 Lu Feng-Yun 1,2 Deng Gui-Qian 1 Abdel-Salam E. A. Candito P. 1 Xingyi Normal University for Nationalities Xingyi Guizhou 562400 China xynun.edu.cn 2 Human Resources and Social Security Bureau Buyi and Miao Autonomous Prefecture in Southwest Guizhou Guizhou 562400 China 2014 1412014 2014 24 08 2013 27 10 2013 14 1 2014 2014 Copyright © 2014 Feng-Yun Lu and Gui-Qian Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the following p -Laplacian equation with nonlinear boundary conditions: - Δ p u + μ ( x ) | u | p - 2 u = f ( x , u ) + g ( x , u ) ,   x Ω , | u | p - 2 u / n = η | u | p - 2 u and x Ω ,   where Ω is a bounded domain in N with smooth boundary Ω . We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) and f , g do not need to satisfy the ( P . S ) or ( P . S * ) condition.

1. Introduction

In this paper, we study the following    p -Laplacian equation: (1) - Δ p u + μ ( x ) | u | p - 2 u = f ( x , u ) + g ( x , u ) , x Ω , | u | p - 2 u n = η | u | p - 2 u , x Ω , where    Ω    is a bounded domain in    N    with smooth boundary    Ω    and    / n    is the outer normal derivative,    - Δ p u = div ( | u | p - 2 u )    is the    p -Laplacian with    1 < p < N ,    η    is real parameter, and (2) μ ( x ) L ( Ω ) satisfying ess inf x Ω ¯ μ ( x ) > 0 . The perturbation functions    f , g    satisfy the following conditions:

f , g C ( Ω ¯ × , ) are odd in    u ;

there exist σ , δ ( 1 , p ) , c 1 > 0 , c 2 > 0 , c 3 > 0 such that (3) c 1 | u | σ f ( x , u ) u c 2 | u | σ + c 3 | u | δ , hhhhhhh for a . e . x Ω and u .

There exists p < q < p * (where p * = p N / N - p ) such that | g ( x , u ) | c ( 1 + | u | q ) for a.e.    x Ω    and    u . Moreover, lim u 0 g ( x , u ) / | u | p - 1 = 0 uniformly for    x Ω .

Assume that one of the following conditions hold:

lim | u | g ( x , u ) / | u | p - 2 u = 0 uniformly for    x Ω ;

lim | u | g ( x , u ) / | u | p - 2 u = - uniformly for    x Ω ; furthermore, f ( x , u ) / | u | p - 2 u and g ( x , u ) / | u | p - 2 u are decreasing in    u    for    u    is large enough;

lim | u | g ( x , u ) / | u | p - 2 u = uniformly for    x Ω ; g ( x , u ) / | u | p - 2 u is increasing in    u    for    u    is large enough; moreover, there exists α > max { σ , δ } such that (4) liminf | u | g ( x , u ) u - p G ( x , u ) | u | α c > 0 uniformly for x Ω ,

where    G ( x , u ) = 0 u g ( x , t ) d t .

Remark 1.

The above conditions were given in Zou  for the semilinear case    p = 2 .

Remark 2.

A simple example which satisfies (F1)–(F4) is (5)    f ( x , u ) + g ( x , u ) = μ | u | r - 2 u + γ | u | s - 2 u , where    1 < r < p < s < p * .

Equation (1) is posed in the framework of the Sobolev space (6) W 1 , p ( Ω ) = { u L p ( Ω ) : Ω | u | p d x < } , with the norm (7) u = ( Ω ( | u | p + μ ( x ) | u | p ) d x ) 1 / p .

The corresponding energy functional of (1) is defined by (8) Φ ( u ) = 1 p Ω ( | u | p + μ ( x ) | u | p ) d x - Ω F ( x , u ) d x - Ω G ( x , u ) d x - η p Ω | u | p d s , for    u W 1 , p ( Ω ) , where    F ( x , u ) = 0 u f ( x , t ) d t    and    d s    is the measure on the boundary. It is easy to see that Φ C 1 ( W 1 , p ( Ω ) , ) and (9) Φ ( u ) , v = Ω ( | u | p - 2 u v + μ ( x ) | u | p - 2 u v ) d x - Ω f ( x , u ) v d x - Ω g ( x , u ) v d x - η Ω | u | p - 2 u v d s , for all    u , v W 1 , p ( Ω ) . It is well-known that the weak solution of (1) corresponds to the critical point of the energy functional Φ on    W 1 , p ( Ω ) .

Remark 3.

Under condition (2), it is easy to check that norm (7) is equivalent to the usual one, that is, the norm defined in (7) with μ ( x ) 1 .

In , the author researched (1) ( η = 0 ) and obtained the existence of infinitely many weak solutions. Moreover, the existence of three solutions for (1) ( η = 0 , p > N ) was researched in  by using a three-critical-point theorem due to Ricceri . Also, some authors researched and obtained the existence of infinitely many weak solution without requiring any symmetric conditions and also with discontinuous nonlinearities; see [5, 6]. Recently, this equation was studied by J.-H. Zhao and P.-H. Zhao  via Bartsch’s dual fountain theorem in  and obtained the existence of infinitely many weak solutions for (1) under the case of Remark 2. They obtained the following theorem.

Theorem A.

Let    f ( x , t ) + g ( x , t ) = μ | u | r - 2 u + γ | u | s - 2 u , where    1 < r < p < s < p * . Then there exists a constant    Λ > 0    such that, for any    η < Λ ,

for any    γ > , μ , (1) has a sequence of solutions    u k    such that    Φ ( u k )    as    k ;

for any    μ > 0 , γ , (1) has a sequence of solutions    v k    such that    Φ ( v k ) 0 -    as    k .

The main ingredient for the proof of the above theorem is a dual fountain theorem in . It should be noted that the    ( P . S )    or    ( P . S * )    condition and its variants play an important role in this theorem and its application. While the variant fountain theorem in Zou  does not need not the    ( P . S )    or    ( P . S * )    condition, we obtain the following generalized result by using Zou’s theorem.

Theorem 4.

Assume that (F1)–(F4) hold; then there exists a constant    Λ > 0    such that, for any    η < Λ , (1) has infinitely many weak solutions    { u k }    satisfying (10) Φ ( u k ) 0 - a s k .

This paper is organized as follows. In Section 2, we recall some preliminary theorems and lemmas. In Section 3, we give the proof of Theorem 4.

2. Preliminaries

In what follows, we make use of the following notations:    E    (or    W 1 , p ( Ω ) ) denotes Banach space with the norm    · ;    E *    denotes the conjugate space for    E ;    L p ( Ω )    denotes Lebesgue space with the norm | · | p ; · , · is the dual pairing of the spaces E * and    E ; we denote by       (resp., ) the strong (resp., weak) convergence; c , c 1 , c 2 , denote (possibly different) positive constants.

For completeness, we first recall the variant fountain theorem in Zou . Let    E    be a Banach space with norm · and E = j N X j ¯ with dim X j < for any    j N . Set    Y k = j = 0 k X j , Z k = j = k X j ¯ .

Theorem 5 (see [<xref ref-type="bibr" rid="B6">1</xref>, Theorem 2.2]).

The C 1 -functional Φ λ : E defined by Φ λ ( u ) = A ( u ) - λ B ( u ) ,    λ [ 1,2 ] , satisfies

Φ λ maps bounded sets to bounded sets uniformly for    λ [ 1,2 ] ; furthermore, Φ λ ( - u ) = Φ λ ( u ) for all ( λ , u ) [ 1,2 ] × E .

B ( u ) 0 for all    u E ; B ( u ) as u    on any finite dimensional subspace of    E .

There exists ρ k > r k > 0 such that (11) a k ( λ ) : = inf u Z k , u = ρ k Φ λ ( u ) 0 > b k ( λ ) : = max u Y k , u = r k Φ λ ( u ) ;

for all    λ [ 1,2 ] , (12) d k ( λ ) : = inf u Z k , u ρ k Φ λ ( u ) 0 a s k u n i f o r m l y f o r λ [ 1,2 ] .

Then there exist λ n 1 , u ( λ n ) Y n , such that (13) Φ λ n Y n ( u ( λ n ) ) = 0 , Φ λ n ( u ( λ n ) ) c k [ d k ( 2 ) , b k ( 1 ) ] hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhi as n .

Particularly, if { u ( λ n ) } has a convergent subsequence for every    k , then    Φ 1    has infinitely many nontrivial critical points { u k } E { 0 } satisfying Φ 1 ( u k ) 0 - as k .

Remark 6.

Obviously, the sequence    { u ( λ n ) }    is a    ( P . S * )    sequence.

For our working space E = W 1 , p ( Ω ) , E is a reflexive and separable Banach space; then there are    e j E    and e j * E * such that (14)    E = span { e j : j = 1,2 , } ¯ , E * = span { e j * : j = 1,2 , } ¯ , e j * , e j = { 1 , i = j , 0 , i j .

We write    X j : = span { e j } ; then    Y k , Z k    can be defined as that in the beginning of Theorem 5. Consider    Φ λ : E defined by (15) Φ λ ( u ) : = 1 p u p - Ω G ( x , u ) d x - η p Ω | u | p d s - λ Ω F ( x , u ) d x : = A ( u ) - λ B ( u ) , λ [ 1,2 ] .

Then    B ( u ) 0    for all    u E ;    B ( u )    as    u    on any finite dimensional subspace of    E ;    Φ λ ( - u ) = Φ λ ( u )    for all    λ [ 1,2 ] , u E .    We need the following lemmas.

Lemma 7 (see [<xref ref-type="bibr" rid="B5">7</xref>, Lemma 3.5]).

If    1 q < p * , then one has (16) β k : = sup u Z k , u = 1 | u | q 0 a s k .

3. Proof of Theorem <xref ref-type="statement" rid="thm1.1">4</xref>

First, we check the condition of Theorem 5.

Lemma 8.

Assume (F1)–(F3); then (A1)–(A3) hold.

Proof.

(A1) and (A2) are obvious. Let n > k > 2 ; we assume that σ δ    and define (17) β k ( σ ) : = sup u Z k , u = 1 | u | σ , β k ( δ ) : = sup u Z k , u = 1 | u | δ .

Observe that (18) | u | σ β k ( σ ) u , | u | δ β k ( δ ) u , for any u Z k . Note that q < p * ; there exists a constant c > 0 such that (19) | u | q c u . By the Sobolev trace imbedding inequality, we have (20) | u | L p ( Ω ) p K u p . Then we take    Λ * = 1 / 4 K    such that, for all    η < Λ * , (21) η p | u | L p ( Ω ) p 1 4 p u p . By (F3), for any    ɛ > 0 , there exists    c ɛ    such that (22)    | G ( x , u ) | ɛ | u | p + c ɛ | u | q . Then, by (F1)–(F3) and (18)–(21), we obtain (23) Φ λ ( u ) = 1 p u p - Ω G ( x , u ) d x - η p Ω | u | p d s - λ Ω F ( x , u ) d x 3 4 p u p - ɛ | u | p p - c ɛ | u | q q - c | u | σ σ - c | u | δ δ 3 4 p u p - ɛ c u p - c u q - c β k ( σ ) σ u σ - c β k ( δ ) δ u δ . Note that p < q ; we may choose ɛ > 0 and R > 0 sufficiently small that (24) 1 4 p u p - ɛ c u p - c u q 0 holds true for any    u W 1 , p ( Ω )    with    u R . So we have (25) Φ λ ( u ) 1 2 p u p - c β k ( σ ) σ u σ - c β k ( δ ) δ u δ , for any u Z k    with u R . Choosing (26) ρ k : = ( 4 p c β k ( σ ) σ + 4 p c β k ( δ ) δ ) 1 / ( p - δ ) , by Lemma 7, for    β k ( σ ) 0 ,    β k ( δ ) 0    as    k , it follows that    ρ k 0    as    k , so there exists    k 0    such that ρ k R when k k 0 . Thus, for k k 0 , u Z k , and u = ρ k , we have Φ λ ( u ) ρ k p / 4 p > 0 ; then a k ( λ ) 0 for all λ [ 1,2 ] .

On the other hand, if u Y k    with u being small enough, since all the norms are equivalent on the finite dimensional space and    σ < p , then    b k ( λ ) < 0    for all    λ [ 1,2 ] .

Furthermore, if u Z k with u ρ k , k k 0 , we see that (27)    Φ λ ( u ) - c β k ( σ ) σ ρ k σ - c β k ( δ ) δ ρ k δ 0 as k . Therefore, d k ( λ ) 0 as k . Thus, (A3) holds.

By Theorem 5, we have the following lemma.

Lemma 9.

There exist λ n 1 and u ( λ n ) Y n such that (28) Φ λ n | Y n ( u ( λ n ) ) = 0 , Φ λ n ( u ( λ n ) ) c k [ d k ( 2 ) , b k ( 1 ) ] a s n .

In order to complete our proof of Theorem 4, by a standard argument (see the proof of Lemma 3.4 in Zhao ), we only need to show that { u ( λ n ) } is bounded.

Lemma 10.

{ u ( λ n ) } is bounded in    W 1 , p ( Ω ) .

Proof.

Since Φ λ n | Y n ( u ( λ n ) ) = 0 , then (29) 1 - η Ω | u ( λ n ) | p u ( λ n ) p d s = Ω λ n f ( x , u ( λ n ) u ( λ n ) ) + g ( x , u ( λ n ) ) u ( λ n ) u ( λ n ) p d x .

We can choose 0 < Λ < Λ * and if η < Λ such that 1 - η K > 0 . If, up to a subsequence, u ( λ n ) as n , then, by (F2), (30) 1 + | η | K Ω g ( x , u ( λ n ) ) u ( λ n ) u ( λ n ) p d x 1 2 ( 1 - η K ) , for n is large enough. Obviously, it is a condition if (F4)(1) holds.

Otherwise, we set w n = u ( λ n ) / u ( λ n ) ; then, up to a subsequence, (31) w n w in E , w n w in L t ( Ω ) for 1 t < p * , w n ( x ) w ( x ) a . e . x Ω . If w 0 in E and lim | u | g ( x , u ) / | u | p - 2 u = - in (F4)(2), then, for n is large enough, by Fatou’s Lemma, we have that (32) - 1 2 ( 1 - η K ) Ω - g ( x , u ( λ n ) ) u ( λ n ) | u ( λ n ) | p | w n | p d x c + { x Ω : w 0 , | u ( λ n ) | c } - g ( x , u ( λ n ) ) u ( λ n ) | u ( λ n ) | p | w n | p d x ; this is a contradiction. It is similar if lim | u | g ( x , u ) / | u | p - 2 u = in (F4)(3). Thus, w = 0 .

Let t n [ 0,1 ] such that (33) Φ λ n ( t n u ( λ n ) ) : = max t [ 0,1 ] Φ λ n ( t u ( λ n ) ) . For any    c > 0    large enough, and w ¯ n : = ( 2 p c ) 1 / p w n , for n is large enough, we have that (34) Φ λ n ( t n u ( λ n ) ) Φ λ n ( w ¯ n ) = 2 c - Ω G ( x , w ¯ n ) d x - η p Ω | w ¯ n | p d s - λ n Ω F ( x , w ¯ n ) d x c , which implies that lim n Φ λ n ( t n u ( λ n ) ) . Obviously, (35) Φ λ n ( t n u ( λ n ) ) , t n u ( λ n ) = 0 .

It follows that (36) = lim n ( Φ λ n ( t n u ( λ n ) ) - 1 p Φ λ n ( t n u ( λ n ) ) , t n u ( λ n ) ) lim n λ n Ω ( 1 p f ( x , t n u ( λ n ) ) t n u ( λ n ) - F ( x , t n u ( λ n ) ) 1 p ) d x + Ω ( 1 p g ( x , t n u ( λ n ) ) t n u ( λ n ) - G ( x , t n u ( λ n ) ) ) d x .

If (F4)(2) holds, we have that ( 1 / p ) f ( x , u ) u - F ( x , u ) and ( 1 / p ) g ( x , u ) u - G ( x , u ) are decreasing in    u    for u is large enough. Therefore, (37)    1 p f ( x , s u ) s u - F ( x , s u ) + 1 p g ( x , s u ) s u - G ( x , s u ) c for all s > 0 and u ; it is a contradiction.

If (F4)(3) holds, then we have that (38) c Ω | u ( λ n ) | σ d x + Ω ( 1 p g ( x , u ( λ n ) ) u ( λ n ) - G ( x , u ( λ n ) ) ) d x , which implies (39) Ω ( 1 p g ( x , u ( λ n ) ) u ( λ n ) - G ( x , u ( λ n ) ) ) d x .

On the other hand, by the property of u ( λ n ) , for n is large enough, since α > max { δ , σ } , we have that (40) b k ( 1 ) λ n Ω ( 1 p f ( x , u ( λ n ) ) u ( λ n ) - F ( x , u ( λ n ) )    ) d x + Ω ( 1 p g ( x , u ( λ n ) ) u ( λ n ) - G ( x , u ( λ n ) ) ) d x 1 2 Ω ( 1 p g ( x , u ( λ n ) ) u ( λ n ) - G ( x , u ( λ n ) ) ) d x + 1 2 c Ω | u ( λ n ) | α d x - 1 2 c Ω | u ( λ n ) | δ d x - 1 2 c Ω | u ( λ n ) | σ d x c Ω ( 1 p g ( x , u ( λ n ) ) u ( λ n ) - G ( x , u ( λ n ) ) ) d x - c ; this implies that Ω ( ( 1 / p ) g ( x , u ( λ n ) ) u ( λ n ) - G ( x , u ( λ n ) ) ) d x is bounded, which contradicts (39).

By the above arguments, we have that { u ( λ n ) } is bounded.

Remark 11.

In fact, our result still holds if we consider a weaker condition than (F4)(2); that is, there is c > 0 such that (41) H ( x , t ) H ( x , s ) + c , H ¯ ( x , t ) H ¯ ( x , s ) + c for all 0 < s < t or t < s < 0 ,    x Ω , where H ( x , t ) = ( 1 / p ) f ( x , t ) t - F ( x , t ) and H ¯ ( x , t ) = ( 1 / p ) g ( x , t ) t - G ( x , t ) .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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