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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Existence of three solutions for a nonlocal elliptic system of ( p , q ) -Kirchhoff type

Guang-Sheng Chen1, Hui-Yu Tang1*, De-Quan Zhang2, Yun-Xiu Jiao3 and Hao-Xiang Wang4

Author Affiliations

1 Department of Construction and Information Engineering, Guangxi Modern Vocational Technology College, Hechi, Guangxi, 547000, China

2 Faculty of Science, Guilin University of Aerospace Industry, Guilin, Guangxi, 541004, China

3 Department of Common Courses, Xinxiang Polytechnic College, Xinxiang, Henan, 453006, China

4 School of Electronic Engineering, Xidian University, Xi’an, Shanxi, 710126, China

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Boundary Value Problems 2013, 2013:175  doi:10.1186/1687-2770-2013-175

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/175


Received:1 May 2013
Accepted:10 July 2013
Published:25 July 2013

© 2013 Chen et al.; licensee Springer

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.

Abstract

In this paper, we study the solutions of a nonlocal elliptic system of ( p , q ) -Kirchhoff type on a bounded domain based on the three critical points theorem introduced by Ricceri. Firstly, we establish the existence of three weak solutions under appropriate hypotheses; then, we prove the existence of at least three weak solutions for the nonlocal elliptic system of ( p , q ) -Kirchhoff type.

Keywords:
( p , q ) -Kirchhoff type system; multiple solutions; three critical points theory

1 Introduction and main results

We consider the boundary problem involving ( p , q ) -Kirchhoff

{ [ M 1 ( Ω | u | p ) ] p 1 Δ p u = λ F u ( x , u , υ ) + μ G u ( x , u , υ ) , in  Ω , [ M 2 ( Ω | υ | q ) ] q 1 Δ q υ = λ F υ ( x , u , υ ) + μ G v ( x , u , υ ) , in  Ω , u = υ = 0 , on  Ω , (1.1)

where Ω R N ( N 1 ) is a bounded smooth domain, λ , μ [ 0 , + ) , p > N , q > N , Δ p is the p-Laplacian operator Δ p u = div ( | u | p 2 u ) . F , G : Ω × R × R R are functions such that F ( , s , t ) , G ( , s , t ) are measurable in Ω for all ( s , t ) R × R and F ( x , , ) , G ( x , , ) are continuously differentiable in R × R for a.e. x Ω . F i is the partial derivative of F with respect to i, i = u , v , so is G i . M i : R + R , i = 1 , 2 , are continuous functions which satisfy the following bounded conditions.

(M) There exist two positive constants m 0 , m 1 such that

m 0 M i ( t ) m 1 , t 0 , i = 1 , 2 . (1.2)

Here and in the sequel, X denotes the Cartesian product of two Sobolev spaces W 0 1 , p ( Ω ) and W 0 1 , q ( Ω ) , i.e., X = W 0 1 , p ( Ω ) × W 0 1 , q ( Ω ) . The reflexive real Banach space X is endowed with the norm

( u , υ ) = u p + υ q , u p = ( Ω | u | p ) 1 / p , υ q = ( Ω | υ | q ) 1 / q .

Since p > N and q > N , W 0 1 , p ( Ω ) and W 0 1 , q ( Ω ) are compactly embedded in C 0 ( Ω ¯ ) . Let

C = max { sup u W 0 1 , p ( Ω ) { 0 } max x Ω ¯ { | u ( x ) | p } u p p , sup v W 0 1 , q ( Ω ) { 0 } max x Ω ¯ { | υ ( x ) | q } υ q q } , (1.3)

then one has C < + . Furthermore, it is known from [1] that

sup u W 0 1 , p ( Ω ) { 0 } max x Ω ¯ { | u ( x ) | p } u p N 1 / p π ( Γ ( 1 + N 2 ) ) 1 / N ( p 1 p N ) 1 1 / p | Ω | ( 1 / N ) ( 1 / p ) ,

where Γ is the gamma function and | Ω | is the Lebesgue measure of Ω. As usual, by a weak solution of system (1.1), we mean any ( u , υ ) X such that

[ M 1 ( Ω | u | p ) ] p 1 Ω | u | p 2 u ϕ + [ M 2 ( Ω | υ | q ) ] q 1 Ω | υ | q 2 υ ψ λ Ω ( F u ϕ + F v ψ ) d x μ Ω ( G u ϕ + G v ψ ) d x = 0 (1.4)

for all ( ϕ , ψ ) X .

System (1.1) is related to the stationary version of a model established by Kirchhoff [2]. More precisely, Kirchhoff proposed the following model:

ρ 2 u t 2 ( P 0 h + E 2 L 0 L | u x | 2 d x ) 2 u x 2 = 0 , (1.5)

which extends D’Alembert’s wave equation with free vibrations of elastic strings, where ρ denotes the mass density, P 0 denotes the initial tension, h denotes the area of the cross-section, E denotes the Young modulus of the material, and L denotes the length of the string. Kirchhoff’s model considers the changes in length of the string produced during the vibrations.

Later, (1.1) was developed into the following form:

u t t M ( Ω | u | 2 ) Δ u = f ( x , u ) in  Ω , (1.6)

where M : R + R is a given function. After that, many authors studied the following problem:

M ( Ω | u | 2 ) Δ u = f ( x , u ) in  Ω , u = 0 on  Ω , (1.7)

which is the stationary counterpart of (1.6). By applying variational methods and other techniques, many results of (1.7) were obtained, the reader is referred to [3-13] and the references therein. In particular, Alves et al. [[3], Theorem 4] supposed that M satisfies bounded condition (M) and f ( x , t ) satisfies the condition

0 < υ F ( x , t ) f ( x , t ) t , | t | R , x Ω  for some  v > 2  and  R > 0 , (AR)

where F ( x , t ) = 0 t f ( x , s ) d s ; one positive solution for (1.7) was given.

In [14], using Ekeland’s variational principle, Corrêa and Nascimento proved the existence of a weak solution for the boundary problem associated with the nonlocal elliptic system of p-Kirchhoff type

{ [ M 1 ( Ω | u | p ) ] p 1 Δ p u = f ( u , υ ) + ρ 1 ( x ) , in  Ω , [ M 2 ( Ω | υ | p ) ] p 1 Δ p υ = g ( , u , υ ) + ρ 2 ( x ) , in  Ω , u η = υ η = 0 , on  Ω , (1.8)

where η is the unit exterior vector on Ω, and M i , ρ i ( i = 1 , 2 ), f, g satisfy suitable assumptions.

In [15], when μ = 0 in (1.1), Bitao Cheng et al. studied the existence of two solutions and three solutions of the following nonlocal elliptic system:

{ [ M 1 ( Ω | u | p ) ] p 1 Δ p u = λ F u ( x , u , υ ) , in  Ω , [ M 2 ( Ω | υ | q ) ] q 1 Δ q υ = λ F υ ( x , u , υ ) , in  Ω , u = υ = 0 , on  Ω . (1.9)

In this paper, our objective is to prove the existence of three solutions of problem (1.1) by applying the three critical points theorem established by Ricceri [16]. Our result, under appropriate assumptions, ensures the existence of an open interval Λ [ 0 , + ) and a positive real number ρ such that, for each λ Λ , problem (1.1) admits at least three weak solutions whose norms in X are less than ρ. The purpose of the present paper is to generalize the main result of [15].

Now, for every x 0 Ω and choosing R 1 , R 2 with R 2 > R 1 > 0 , such that B ( x 0 , R 2 ) Ω , where B ( x , R ) = { y R N : | y x | < R } , put

α 1 = α 1 ( N , p , R 1 , R 2 ) = C 1 / p ( R 2 N R 1 N ) 1 / p R 2 R 1 ( π N / 2 Γ ( 1 + N / 2 ) ) 1 / p , (1.10)

α 2 = α 2 ( N , q , R 1 , R 2 ) = C 1 / q ( R 2 N R 1 N ) 1 / q R 2 R 1 ( π N / 2 Γ ( 1 + N / 2 ) ) 1 / q . (1.11)

Moreover, let a, c be positive constants and define

y ( x ) = a R 2 R 1 ( R 2 { i = 1 N ( x i x 0 i ) 2 } 1 / 2 ) , x B ( x 0 , R 2 ) B ( x 0 , R 1 ) , A ( c ) = { ( s , t ) R × R : | s | p + | t | q c } , M + = max { m 1 p 1 p , m 1 q 1 q } , M = min { m 0 p 1 p , m 0 q 1 q } .

Our main result is stated as follows.

Theorem 1.1Assume that R 2 > R 1 > 0 such that B ( x 0 , R 2 ) Ω , and suppose that there exist four positive constantsa, b, γandβwith γ < p , β < q , ( a α 1 ) p + ( a α 2 ) q > b M + / M , and a function α ( x ) L ( Ω ) such that

(j1) F ( x , s , t ) 0 for a.e. x Ω B ( x 0 , R 1 ) and all ( s , t ) [ 0 , a ] × [ 0 , a ] ;

(j2) [ ( a α 1 ) p + ( a α 2 ) q ] | Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) < b B ( x 0 , R 1 ) F ( x , a , a ) d x ;

(j3) F ( x , s , t ) α ( x ) ( 1 + | s | γ + | t | β ) for a.e. x Ω and all ( s , t ) R × R ;

(j4) F ( x , 0 , 0 ) = 0 for a.e. x Ω .

Then there exist an open interval Λ [ 0 , ) and a positive real numberρwith the following property: for each λ Λ and for two Carathéodory functions G u , G v : Ω × R × R R satisfying

(j5) sup { | s | ξ , | t | ξ } ( | G u ( , s , t ) | + | G v ( , s , t ) | ) L 1 ( Ω ) for all ξ > 0 ,

there exists δ > 0 such that, for each μ [ 0 , δ ] , problem (1.1) has at least three weak solutions w i = ( u i , υ i ) X ( i = 1 , 2 , 3 ) whose norms w i are less thanρ.

2 Proof of the main result

First we recall the modified form of Ricceri’s three critical points theorem (Theorem 1 in [16]) and Proposition 3.1 of [17], which is our primary tool in proving our main result.

Theorem 2.1 ([16], Theorem 1)

Suppose that X is a reflexive real Banach space and that Φ : X R is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X , and that Φ is bounded on each bounded subset ofX; Ψ : X R is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact; I R is an interval. Suppose that

lim x + ( Φ ( x ) + λ Ψ ( x ) ) = +

for all λ I , and that there exists h R such that

sup λ I inf x X ( Φ ( x ) + λ ( Ψ ( x ) + h ) ) < inf x X sup λ I ( Φ ( x ) + λ ( Ψ ( x ) + h ) ) . (2.1)

Then there exist an open interval Λ I and a positive real numberρwith the following property: for every λ Λ and every C 1 functional J : X R with compact derivative, there exists δ > 0 such that, for each μ [ 0 , δ ] , the equation

Φ ( x ) + λ Ψ ( x ) + μ J ( x ) = 0

has at least three solutions inXwhose norms are less thanρ.

Proposition 2.1 ([17], Proposition 3.1)

Assume thatXis a nonempty set and Φ, Ψ are two real functions onX. Suppose that there are r > 0 and x 0 , x 1 X such that

Φ ( x 0 ) = Ψ ( x 0 ) = 0 , Φ ( x 1 ) > 1 , sup x Φ 1 ( [ , r ] ) Ψ ( x ) < r Ψ ( x 1 ) Φ ( x 1 ) .

Then, for eachhsatisfying

sup x Φ 1 ( [ , r ] ) Ψ ( x ) < h < r Ψ ( x 1 ) Φ ( x 1 ) ,

one has

sup λ 0 inf x X ( Φ ( x ) + λ ( Ψ ( x ) + h ) ) < inf x X sup λ 0 ( Φ ( x ) + λ ( Ψ ( x ) + h ) ) .

Before proving Theorem 1.1, we define a functional and give a lemma.

The functional H : X R is defined by

H ( u , v ) = Φ ( u , v ) + λ J ( u , v ) + μ ψ ( u , v ) = 1 p M ˆ 1 ( Ω | u | p ) + 1 q M ˆ 2 ( Ω | υ | q ) λ Ω F ( x , u , v ) d x μ Ω G ( x , u , v ) d x (2.2)

for all ( u , υ ) X , where

M ˆ 1 = 0 t [ M 1 ( s ) ] p 1 d s , M ˆ 2 = 0 t [ M 2 ( s ) ] q 1 d s . (2.3)

By conditions (M) and (j3), it is clear that H C 1 ( X , R ) and a critical point of H corresponds to a weak solution of system (1.1).

Lemma 2.2Assume that there exist two positive constantsa, bwith ( a α 1 ) p + ( a α 2 ) q > b M + / M such that

(j1) F ( x , s , t ) 0 , for a.e. x Ω B ( x 0 , R 1 ) and all ( s , t ) [ 0 , a ] × [ 0 , a ] ;

(j2) [ ( a α 1 ) p + ( a α 2 ) q ] | Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) < b B ( x 0 , R 1 ) F ( x , a , a ) d x .

Then there exist r > 0 and u 0 W 0 1 , p ( Ω ) , υ 0 W 0 1 , q ( Ω ) such that

Φ ( u 0 , v 0 ) > r

and

| Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) b M + C Ω F ( x , u 0 , υ 0 ) d x Φ ( u 0 , υ 0 ) .

Proof We put

w 0 ( x ) = { 0 , x Ω ¯ B ( x 0 , R 2 ) , a R 2 R 1 ( R 2 { i = 1 N ( x i x 0 i ) } 1 / 2 ) , x B ( x 0 , R 2 ) B ( x 0 , R 1 ) , a , x B ( x 0 , R 1 ) ,

and u 0 ( x ) = υ 0 ( x ) = w 0 ( x ) . Then we can verify easily ( u 0 , υ 0 ) X and, in particular, we have

u 0 p p = ( R 2 N R 1 N ) π N / 2 Γ ( 1 + N / 2 ) ( a R 2 R 1 ) p , (2.4)

and

υ 0 q q = ( R 2 N R 1 N ) π N / 2 Γ ( 1 + N / 2 ) ( a R 2 R 1 ) q . (2.5)

Hence, we obtain from (1.10), (1.11), (2.4) and (2.5) that

u 0 p p = w 0 p p = ( a α 1 ) p C , υ 0 q q = w 0 q q = ( a α 2 ) q C . (2.6)

Under condition (M), by a simple computation, we have

M ( u p p + υ q q ) Φ ( u , υ ) M + ( u p p + υ q q ) . (2.7)

Setting r = b M + C and applying the assumption of Lemma 2.2

( a α 1 ) p + ( a α 2 ) q > b M + / M ,

from (2.6) and (2.7), we obtain

Φ ( u 0 , v 0 ) M ( u 0 p p + v 0 q q ) = M C [ ( a α 1 ) p + ( a α 2 ) q ] > M C b M + M = r .

Since, 0 u 0 a , 0 v 0 a for each x Ω , from condition (j1) of Lemma 2.2, we have

Ω B ( x 0 , R 2 ) F ( x , u 0 , υ 0 ) d x + B ( x 0 , R 2 ) B ( x 0 , R 1 ) F ( x , u 0 , υ 0 ) d x 0 .

Hence, based on condition (j2), we get

| Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) < b ( a α 1 ) p + ( a α 2 ) q B ( x 0 , R 1 ) F ( x , a , a ) d x = b M + C B ( x 0 , R 1 ) F ( x , a , a ) d x M + ( ( a α 1 ) p + ( a α 2 ) q ) / C b M + C Ω B ( x 0 , R 1 ) F ( x , u 0 , υ 0 ) d x + B ( x 0 , R 1 ) F ( x , u 0 , υ 0 ) d x M + ( u 0 p p + υ 0 q q ) b M + C Ω F ( x , u 0 , υ 0 ) d x Ψ ( u 0 , υ 0 ) .

 □

Now, we can prove our main result.

Proof of Theorem 1.1 For each ( u , v ) X , let

Φ ( u , v ) = M ˆ 1 ( u p p ) p + M ˆ 2 ( v q q ) q , Ψ ( u , υ ) = Ω F ( x , u , υ ) d x , J ( u , v ) = Ω G ( x , u , v ) d x .

From the assumption of Theorem 1.1, we know that Φ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional. Additionally, the Gâteaux derivative of Φ has a continuous inverse on X . Since p > N , q > N , Ψ and J are continuously Gâteaux differential functionals whose Gâteaux derivatives are compact. Obviously, Φ is bounded on each bounded subset of X. In particular, for each ( u , v ) , ( ξ , η ) X ,

Φ ( u , v ) , ( ξ , η ) = [ M 1 ( Ω | u | p ) ] p 1 Ω | u | p 2 u ξ + [ M 2 ( Ω | υ | q ) ] q 1 Ω | υ | q 2 υ η , Ψ ( u , v ) , ( ξ , η ) = Ω F u ( x , u , v ) ξ d x Ω F v ( x , u , v ) η d x , J ( u , v ) , ( ξ , η ) = Ω G u ( x , u , v ) ξ d x Ω G v ( x , u , v ) η d x .

Hence, the weak solutions of problem (1.1) are exactly the solutions of the following equation:

Φ ( u , v ) + λ Ψ ( u , v ) + μ J ( u , v ) = 0 .

From (j3), for each λ > 0 , one has

lim ( u , v ) + ( λ Φ ( u , v ) + μ Ψ ( u , v ) ) = + , (2.8)

and so the first condition of Theorem 2.1 is satisfied. By Lemma 2.2, there exists ( u 0 , υ 0 ) X such that

Φ ( u 0 , v 0 ) = M ˆ 1 ( u 0 p p ) p + M ˆ 2 ( v 0 q q ) q M ( u 0 p p + v 0 q q ) = M C [ ( a α 1 ) p + ( a α 2 ) q ] > M C b M + M = b M + C > 0 = Φ ( 0 , 0 ) , (2.9)

and

| Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) b M + C Ω F ( x , u 0 , υ 0 ) d x Φ ( u 0 , υ 0 ) . (2.10)

From (1.3), we have

max x Ω ¯ { | u ( x ) | p } C u p p , max x Ω ¯ { | υ ( x ) | q } C υ q q

for each ( u , υ ) X . We obtain

max x Ω ¯ { | u ( x ) | p p + | v ( x ) | q q } C { u p p p + v q q q } (2.11)

for each ( u , υ ) X . Let r = b M + C for each ( u , υ ) X such that

Φ ( u , υ ) = M ˆ 1 ( u p p ) p + M ˆ 2 ( v q q ) q r .

From (2.11), we get

| u ( x ) | p + | υ ( x ) | q C ( u p p + υ q q ) C r M = C M b M + C = b M + M . (2.12)

Then, from (2.10) and (2.12), we find

sup ( u , υ ) Φ 1 ( , r ) ( Ψ ( u , υ ) ) = sup { ( u , υ ) | Φ ( u , υ ) r } Ω F ( x , u , υ ) d x sup { ( u , υ ) | | u ( x ) | p + | υ ( x ) | q b M + / M } Ω F ( x , u , υ ) d x Ω sup ( s , t ) A ( b M + / M ) F ( x , s , t ) d x | Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) b M + C Ω F ( x , u 0 , υ 0 ) d x Φ ( u 0 , υ 0 ) = r Ψ ( u 0 , υ 0 ) Φ ( u 0 , υ 0 ) .

Hence, we have

sup { ( u , v ) | Φ ( u , v r } ( Ψ ( u , v ) ) < r Ψ ( u 0 , υ 0 ) Φ ( u 0 , υ 0 ) . (2.13)

Fix h such that

sup { ( u , v ) | Φ ( u , v r } ( Ψ ( u , v ) ) < h < r Ψ ( u 0 , υ 0 ) Φ ( u 0 , υ 0 ) ,

by (2.9), (2.13) and Proposition 2.1, with ( u 1 , v 1 ) = ( 0 , 0 ) and ( u , v ) = ( u 0 , v 0 ) , we obtain

sup λ 0 inf x X ( Φ ( x ) + λ ( h + Ψ ( x ) ) ) < inf x X sup λ 0 ( Φ ( x ) + λ ( h + Ψ ( x ) ) ) , (2.14)

and so assumption (2.1) of Theorem 2.1 is satisfied.

Now, with I = [ 0 , ) , from (2.8) and (2.14), all the assumptions of Theorem 2.1 hold. Hence, our conclusion follows from Theorem 2.1. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. This work was supported by the Scientific Research Project of Guangxi Education Department (no. 201204LX672).

References

  1. Talenti, G: Some inequalities of Sobolev type on two-dimensional spheres. In: Walter W (ed.) General Inequalities, pp. 401–408. Birkhäuser, Basel (1987)

  2. Kirchhoff, G: Mechanik, Teubner, Leipzig (1883)

  3. Alves, CO, Corrêa, FJSA, Ma, TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl.. 49(1), 85–93 (2005). Publisher Full Text OpenURL

  4. Cheng, B, Wu, X: Existence results of positive solutions of Kirchhoff type problems. Nonlinear Anal., Theory Methods Appl.. 71(10), 4883–4892 (2009). Publisher Full Text OpenURL

  5. Cheng, B, Wu, X, Liu, J: Multiplicity of nontrivial solutions for Kirchhoff type problems. Bound. Value Probl.. 2010, (2010) Article ID 268946

  6. Chipot, M, Lovat, B: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal., Theory Methods Appl.. 30(7), 4619–4627 (1997). Publisher Full Text OpenURL

  7. D’Ancona, P, Spagnolo, S: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math.. 108(2), 247–262 (1992)

  8. He, X, Zou, W: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal., Theory Methods Appl.. 70(3), 1407–1414 (2009). Publisher Full Text OpenURL

  9. Ma, T, Muñoz Rivera, JE: Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett.. 16(2), 243–248 (2003). Publisher Full Text OpenURL

  10. Ma, T: Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal., Theory Methods Appl.. 63, 1967–1977 (2005). Publisher Full Text OpenURL

  11. Mao, A, Zhang, Z: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal., Theory Methods Appl.. 70(3), 1275–1287 (2009). Publisher Full Text OpenURL

  12. Perera, K, Zhang, Z: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ.. 221(1), 246–255 (2006). Publisher Full Text OpenURL

  13. Zhang, Z, Perera, K: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl.. 317(2), 456–463 (2006). Publisher Full Text OpenURL

  14. Corrêa, FJSA, Nascimento, RG: On a nonlocal elliptic system of p-Kirchhoff-type under Neumann boundary condition. Math. Comput. Model.. 49(3-4), 598–604 (2009). Publisher Full Text OpenURL

  15. Cheng, B, Wu, X, Liu, J: Multiplicity of solutions for nonlocal elliptic system of ( p , q ) -Kirchhoff type. Abstr. Appl. Anal.. 2011, (2011) Article ID 526026

    Article ID 526026

    Publisher Full Text OpenURL

  16. Ricceri, B: A three critical points theorem revisited. Nonlinear Anal.. 70(9), 3084–3089 (2009). Publisher Full Text OpenURL

  17. Ricceri, B: Existence of three solutions for a class of elliptic eigenvalue problems. Math. Comput. Model.. 32(11-13), 1485–1494 (2000). Publisher Full Text OpenURL