Existence and multiplicity of positive solutions to a perturbed singular elliptic system deriving from a strongly coupled critical potential
© Hsu; licensee Springer 2012
Received: 12 March 2012
Accepted: 3 October 2012
Published: 17 October 2012
In this paper, we consider singular elliptic systems involving a strongly coupled critical potential and concave nonlinearities. By using variational methods and analytical techniques, the existence and multiplicity of positive solutions to the system are established.
MSC: 35J60, 35B33.
KeywordsPalais-Smale condition Nehari manifold strongly coupled elliptic system critical potential
1 Introduction and main results
where is a smooth bounded domain such that , , is the critical Sobolev exponent, is the best Hardy constant and denotes the completion of with respect to the norm and is defined as the completion of the with respect to the norm defined by for .
Definitions of strongly and weakly coupled terms are as follows.
The terms and () are weakly coupled, () is strongly coupled when L or K is a derivative operator. Thus, is strongly coupled when and are positive.
The parameters in (1.1) satisfy the following assumption.
(ℋ) , , , , , , , , .
Therefore, a weak solution of (1.1) is equivalent to a nonzero critical point of .
Regular semilinear elliptic systems have been studied extensively and many conclusions have been established. For example, Alves et al. studied in  an elliptic system and some important conclusions had been obtained. However, the elliptic systems involving the Hardy inequality have seldom been studied and we only find some results in [8–16]. Thus it is necessary for us to investigate the related singular systems deeply. Among the references above, the elliptic systems involving the Hardy inequality and concave-convex nonlinearities had been studied only in . In this paper, only the case of (1.1) involving multiple strongly-coupled critical terms is considered.
Then the main results of this paper can be concluded in the following theorems and the conclusions are new to the best of our knowledge. It can be verified that the intervals in Theorems 1.1 and 1.2 for the parameters , , μ and q are allowable.
Theorem 1.1 Suppose that (ℋ) holds and . Then problem (1.1) has at least one positive solution.
Theorem 1.2 Suppose that (ℋ) holds, , and . Then there exists such that problem (1.1) has at least two positive solutions for all and satisfying .
This paper is organized as follows. Some preliminary results and properties of the Nehari manifold are established in Sections 2 and 3, and Theorems 1.1 and 1.2 are proved in Section 4.
2 The local Palais-Smale condition
denotes the first eigenvalue of the operator L, means the norm of the space , is the dual space of E. for all and . is said to be nonnegative in Ω if and in Ω. is said to be positive in Ω if and in Ω. is a ball in . denotes a quantity satisfying , means as and is a generic infinitesimal value. In particular, the quantity means that there exist the constants such that as ε is small. We always denote positive constants as C and omit dx in integrals for convenience.
Thus, the proof is complete. □
Lemma 2.2 If is a (PS) c -sequence of the functional J, then is bounded in E.
Proof See Hsu [, Lemma 2.2]. □
which is a contradiction. Therefore, the proof of Lemma 2.3 is complete. □
3 Nehari manifold
Lemma 3.1 The functional J is coercive and bounded below on .
Thus, J is coercive and bounded below on . □
Lemma 3.2 Suppose that is a local minimizer of J on and . Then in .
Proof The proof is similar to that of  and the details are omitted. □
Lemma 3.3 for all .
which is a contradiction. □
for all .
There exists a positive constant depending on , , q, N, , and such that for all .
- (ii)Suppose that and . By (1.7), (3.1) and (3.5) we have that
where is a positive constant. □
Proof The proof is similar to that of  and is omitted. □
Then we have the following lemma.
Proof The proof is almost the same as that in [, Lemma 2.7] and is omitted here. □
4 Proof of Theorems 1.1 and 1.2
If , then the functional J has a -sequence .
Proof The proof is similar to that of  and is omitted. □
Lemma 4.2 Suppose that . Then J has a minimizer such that is a positive solution of (1.1) and .
Therefore, is a nontrivial solution of (1.1).
which is a contradiction. Since and , by Lemma 3.2 we may assume that is a nontrivial nonnegative solution of (1.1).
which is a contradiction.
Finally, from the maximum principle  we deduce that in Ω and is thus a positive solution of (1.1). □
The following results are already known.
Lemma 4.3 
Lemma 4.4 
Suppose that (ℋ) holds, is defined as in (1.6) and are the minimizers of defined as in (1.4). Then and has the minimizers , where .
In particular, for all .
Note that and as t is closed to 0. Thus, is attained at some finite with . Furthermore, , where and are the positive constants independent of ε.
where we have used the assumption .
The proof is thus complete by taking . □
Lemma 4.6 Set . Then for all , problem (1.1) has a positive solution such that and .
This implies that and . From the strong maximum principle  it follows that is a positive solution of (1.1). □
Proof of Theorems 1.1 and 1.2. By Lemma 4.2, we obtain that (1.1) has a positive solution for all . On the other hand, from Lemma 4.6, we can get the second positive solution for all . Since , this implies that and are distinct. □
The author was grateful for the referee’s helpful suggestions and comments.
- Rabinowitz P CBMS Regional Conference Series in Mathematics 65. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc, Providence; 1986.Google Scholar
- Hardy G, Littlewood J, Polya G: Inequalities. Cambridge University Press, Cambridge; 1952.Google Scholar
- Terracini S: On positive solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differ. Equ. 1996, 2: 241-264.MathSciNetGoogle Scholar
- Kang D, Peng S: Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy potential. Appl. Math. Lett. 2005, 18: 1094-1100. 10.1016/j.aml.2004.09.016MathSciNetView ArticleGoogle Scholar
- Cao D, Kang D: Solutions of quasilinear elliptic problems involving a Sobolev exponent and multiple Hardy-type terms. J. Math. Anal. Appl. 2007, 333: 889-903. 10.1016/j.jmaa.2006.12.005MathSciNetView ArticleGoogle Scholar
- Kang D: On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms. Nonlinear Anal. 2008, 68: 1973-1985. 10.1016/j.na.2007.01.024MathSciNetView ArticleGoogle Scholar
- Alves C, Filho D, Souto M: On systems of elliptic equations involving subcritical or critical Sobolev exponents. Nonlinear Anal. 2000, 42: 771-787. 10.1016/S0362-546X(99)00121-2MathSciNetView ArticleGoogle Scholar
- Abdellaoui B, Felli V, Peral I:Some remarks on systems of elliptic equations doubly critical in the whole . Calc. Var. Partial Differ. Equ. 2009, 34: 97-137. 10.1007/s00526-008-0177-2MathSciNetView ArticleGoogle Scholar
- Bouchekif M, Nasri Y: On elliptic system involving critical Sobolev-Hardy exponents. Mediterr. J. Math. 2008, 5: 237-252. 10.1007/s00009-008-0147-0MathSciNetView ArticleGoogle Scholar
- Bouchekif M, Nasri Y: On a singular elliptic system at resonance. Ann. Mat. Pura Appl. 2010, 189: 227-240. 10.1007/s10231-009-0106-9MathSciNetView ArticleGoogle Scholar
- Cai M, Kang D: Elliptic systems involving multiple strongly-coupled critical terms. Appl. Math. Lett. 2012, 25: 417-422. 10.1016/j.aml.2011.09.026MathSciNetView ArticleGoogle Scholar
- Hsu TS: Multiplicity of positive solutions for critical singular elliptic systems with concave-convex nonlinearities. Adv. Nonlinear Stud. 2009, 9: 295-311.MathSciNetGoogle Scholar
- Huang Y, Kang D: On the singular elliptic systems involving multiple critical Sobolev exponents. Nonlinear Anal. 2011, 74: 400-412. 10.1016/j.na.2010.08.051MathSciNetView ArticleGoogle Scholar
- Huang Y, Kang D: Elliptic systems involving the critical exponents and potentials. Nonlinear Anal. 2009, 71: 3638-3653. 10.1016/j.na.2009.02.024MathSciNetView ArticleGoogle Scholar
- Liu Z, Han P: Existence of solutions for singular elliptic systems with critical exponents. Nonlinear Anal. 2008, 69: 2968-2983. 10.1016/j.na.2007.08.073MathSciNetView ArticleGoogle Scholar
- Wang L, Wei Q, Kang D: Existence and multiplicity of positive solutions to elliptic systems involving critical exponents. J. Math. Anal. Appl. 2011, 383: 541-552. 10.1016/j.jmaa.2011.05.053MathSciNetView ArticleGoogle Scholar
- Brézis H, Lieb E: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 1983, 88: 486-490.View ArticleGoogle Scholar
- Han P: The effect of the domain topology on the number of positive solutions of elliptic systems involving critical Sobolev exponents. Houst. J. Math. 2006, 32: 1241-1257.Google Scholar
- Brown KJ, Zhang Y: The Nehari manifold for a semilinear elliptic equation with a sign-changing weigh function. J. Differ. Equ. 2003, 193: 481-499. 10.1016/S0022-0396(03)00121-9MathSciNetView ArticleGoogle Scholar
- Brown KJ, Wu TF: A semilinear elliptic system involving nonlinear boundary condition and sign-changing weigh function. J. Math. Anal. Appl. 2008, 337: 1326-1336. 10.1016/j.jmaa.2007.04.064MathSciNetView ArticleGoogle Scholar
- Wu TF: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 2006, 318: 253-270. 10.1016/j.jmaa.2005.05.057MathSciNetView ArticleGoogle Scholar
- Vazquez J: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 1984, 12: 191-202. 10.1007/BF01449041MathSciNetView ArticleGoogle Scholar
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.