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Existence of Multiple Solutions for a Singular Elliptic Problem with Critical Sobolev Exponent
We consider the existence of multiple solutions of the singular elliptic problem , as , where , , , , , , . By the variational method and the theory of genus, we prove that the above-mentioned problem has infinitely many solutions when some conditions are satisfied.
1. Introduction and Main Results
In this paper, we consider the existence of multiple solutions for the singular elliptic problem where . and are nonnegative functions in .
In recent years, the existence of multiple solutions on elliptic equations has been considered by many authors. In , Assunção et al. considered the following quasilinear degenerate elliptic equation: where . When , where and ; the authors proved that problem (1.2) has at least two positive solutions. Rodrigues in  studied the following critical problem on bounded domain : By the variational method on Nehari manifolds [3, 4], the author proved the existence of at least two positive solutions and the nonexistence of solutions when some certain conditions are satisfied. When and , Miotto and Miyagaki in  considered the semilinear Dirichlet problem in infinite strip domains The authors also proved that problem (1.4) has at least two positive solutions by the methods of Nehari manifold. For other references, we refer to [6–11] and the reference therein. In fact, motivated by [1, 2, 5], we consider the problem (1.1). Since our problem is singular and is studied in the whole space , the loss of compactness of the Sobolev embedding renders a variational technique that is more delicate. By the variational method and the theory of genus, we prove that problem (1.1) has infinitely many solutions when some suitable conditions are satisfied.
In order to state our result, we introduce some weighted Sobolev spaces. For and in , we define the spaces and as being the set of Lebesgue measurable functions , which satisfy Particularly, when , we have We denote the completion of by with the norm of where and . It is easy to find that is a reflexive and separable Banach space with the norm .
The following Hardy-Sobolev inequality is due to Caffarelli et al. , which is called Caffarelli-Kohn-Nirenberg inequality. There exist constants such that where is called the Sobolev critical exponent.
In the present paper, we make the following assumptions: for , where ; for , where . for , where .
Then, we give some basic definitions.
Definition 1.1. is said to be a weak solution of (1.1) if for any there holds
Let be the energy functional corresponding to problem (1.1), which is defined as for all . Then the functional and for all , there holds It is well known that the weak solutions of problem (1.1) are the critical points of the functional , see . Thus, to prove the existence of weak solutions of (1.1), it is sufficient to show that admits a sequence of critical points in .
Our main result in this paper is the following.
Theorem 1.2. Let , , , . Assume – are fulfilled. Then problem (1.1) has infinitely many solutions in .
2. Preliminary Results
Our proof is based on variational method. One important aspect of applying this method is to show that the functional satisfies condition which is introduced in the following definition.
Definition 2.1. Let and be a Banach space. The functional satisfies the condition if for any such that contains a convergent subsequence in .
The following embedding theorem is an extension of the classical Rellich-Kondrachov compactness theorem, see .
Lemma 2.2. Suppose is an open bounded domain with boundary and . . Then the embedding is continuous if and , and is compact if and .
Now we prove an embedding theorem, which is important in our paper.
Lemma 2.3. Assume - and . Then the embedding is compact.
Proof. We split our proof into two cases.
(i) Consider .
By the Hölder inequality and (1.9) we have that where . Then the embedding is continuous. Next, we will prove that the embedding is compact.
Let be a ball center at origin with the radius . For the convenience, we denote by , that is, . Assume is a bounded sequence in . Then is bounded in . We choose in Lemma 2.2, then there exist and a subsequence, still denoted by , such that as . We want to prove that where . In fact, we obtain from (2.2) that The fact shows that Then (2.4) and (2.5) imply that which gives (2.3).
In the following, we will prove that strongly in .
Since is a reflexive Banach space and is bounded in . Then we may assume, up to a subsequence, that In view of (2.3), we get that for any there exists large enough such that On the other hand, due to the compact embedding in Lemma 2.2, we have that Therefore, there is such that for . Thus, the inequalities (2.8) and (2.10) show that This shows that is convergent in .
(ii) Consider .
It follows from (1.8) and the Hölder inequality that where . Thus, the fact of and (2.12) imply that the embedding is continuous. Similar to the proof of (i) we can also prove that the embedding is compact for .
Similarly, we have the following result of compact embedding.
Lemma 2.4. Assume and , then the embedding is compact.
Lemma 2.5. Let . Suppose that is a sequence such that where are measures supported on and is the space of bounded measures in . Then there are the following results.(1) There exists some at most countable set , a family of distinct points in , and a family of positive numbers such that where is the Dirac measure at .(2) The following equality holds for some family satisfying (3)There hold where
Lemma 2.6. Let . Then satisfies the condition with , where is as in (1.8).
Proof. We will split the proof into three steps.
Step 1. is bounded in .
Let be a sequence of in , that is, Then, we have Since , (2.20) shows that is bounded in .
Step 2. There exists in such that in .
The inequality (1.8) shows that is bounded in . Then the above argument and the compactness embedding in Lemma 2.2 mean that the following convergence hold: It follows from Lemma 2.5 that there exist nonnegative measures and such that Thus, in order to prove it is sufficient to prove that .
For the proof of , we define the functional such that where belongs to the support of . It follows from (2.1) that Since is bounded, we can get from (1.8)-(1.9), Lemmas 2.3 and 2.5 that On the other hand, where . Then ; furthermore, (2.16) implies that or . We will prove that the later does not hold. Suppose otherwise, there exists some such that . Then (2.19) and Lemma 2.4 show that which contradicts the hypothesis of . Then .
Similarly, we define the functional as Then, the similar proof as above shows that . Thus, we can deduce from (2.22) that which implies that in .
Step 3. converges strongly in .
The following inequalities  play an important role in our proof: Our aim is to prove that is a Cauchy sequence of . In fact, let in (1.12), it follows from (2.19) that where Using the inequalities (2.31), we can get by direct computation that with some constant , independent of and .
Then the Hölder inequality together with (1.8) and (2.30) yield that Similarly, we have from the Hölder inequality, Lemmas 2.3 and 2.4 that Therefore, the above estimates imply that , that is, is a Cauchy sequence of . Then converges strongly in and we complete the proof.
Similarly, we have the following lemma.
Proof. Step 1. is bounded in .
Let be a sequence of in . Then we have from Lemma 2.3 that Since , (2.37) shows that is bounded in .
Step 2. There exists in such that in .
Similar to the proof of Lemma 2.5, we can get that or by applying the functional . Now we prove that there is no such that . Suppose otherwise, then Let Then has the unique minimum point at Then it follows from (2.38) that which contradicts the hypothesis of .
Step 3. converges strongly in .
By Lemma 2.4, this result can be similarly obtained by the method in Lemma 2.6, so we omit the proof.
3. Existence of Infinitely Solutions
In this section, we will use the minimax procedure to prove the existence of infinity many solutions of problem (1.1). Let denotes the class of such that is closed in and symmetric with respect to the origin. For , we recall the genus which is defined by If there is no mapping as above for any , then , and . The following proposition gives some main properties of the genus, see [13, 22].
Proposition 3.1. Let . Then(1)if there exists an odd map , then ,(2)if , then ,(3).(4)if is a sphere centered at the origin in , then ,(5)if is compact, then and there exists such that and , where .
Lemma 3.2. Assume (A1)–(A3). Then for any , there exists such that
Proof. For given , let be a -dimensional subspace of . If , then for we have
The fact that all the norms on finite dimensional space are equivalent implies that for all
for some constant . Then there exist large and small such that
Then is a sphere centered at the origin with radius of and
Therefore, Proposition 3.1 shows that .
If , we have Since is also a norm and all norms on the finite dimensional space are equivalent, we have Then there exist large and small such that Denote Then is a sphere centered at the origin with radius of and
Therefore, Proposition 3.1 shows that .
Let . It is easy to check that . We define It is not difficult to find that and for any since is coercive and bounded below. Furthermore, we define the set
Then, is compact and we have the following important lemma, see .
Lemma 3.3. All the are critical values of . Moreover, if , then .
Proof of Theorem 1.2. In view of Lemmas 2.6 and 2.7, satisfies the condition in . Furthermore, as the standard argument of [13, 22, 23], Lemma 3.3 gives that has infinity many critical points with negative values. Thus, problem (1.1) has infinitely many solutions in , and we complete the proof.
The author would like to express his sincere gratitude to the anonymous reviewers for the valuable comments and suggestions.
- R. B. Assunção, P. C. Carrião, and O. H. Miyagaki, “Critical singular problems via concentration-compactness lemma,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 137–154, 2007.
- R. D. S. Rodrigues, “On elliptic problems involving critical Hardy-Sobolev exponents and sign-changing function,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 4, pp. 857–880, 2010.
- Z. Nehari, “On a class of nonlinear second-order differential equations,” Transactions of the American Mathematical Society, vol. 95, pp. 101–123, 1960.
- K. J. Brown, “The Nehari manifold for a semilinear elliptic equation involving a sublinear term,” Calculus of Variations and Partial Differential Equations, vol. 22, no. 4, pp. 483–494, 2005.
- M. L. Miotto and O. H. Miyagaki, “Multiple positive solutions for semilinear Dirichlet problems with sign-changing weight function in infinite strip domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3434–3447, 2009.
- R. B. Assunção, P. C. Carrião, and O. H. Miyagaki, “Subcritical perturbations of a singular quasilinear elliptic equation involving the critical Hardy-Sobolev exponent,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 6, pp. 1351–1364, 2007.
- J. V. Gonçalves and C. O. Alves, “Existence of positive solutions for -Laplacian equations in involving critical Sobolev exponents,” Nonlinear Analysis: Theory, Methods & Applications, vol. 32, no. 1, pp. 53–70, 1998.
- T.-F. Wu, “On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 253–270, 2006.
- T.-F. Wu, “On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function,” Communications on Pure and Applied Analysis, vol. 7, no. 2, pp. 383–405, 2008.
- L. Iturriaga, “Existence and multiplicity results for some quasilinear elliptic equation with weights,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1084–1102, 2008.
- H. Q. Toan and Q.-A. Ngô, “Multiplicity of weak solutions for a class of nonuniformly elliptic equations of -Laplacian type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 4, pp. 1536–1546, 2009.
- L. Caffarelli, R. Kohn, and L. Nirenberg, “First order interpolation inequalities with weights,” Compositio Mathematica, vol. 53, no. 3, pp. 259–275, 1984.
- P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.
- B. Xuan, “The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 4, pp. 703–725, 2005.
- P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case. I,” Annales de l'Institut Henri Poincaré Analyse Non Linéaire, vol. 1, no. 2, pp. 109–145, 1984.
- P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case. II,” Annales de l'Institut Henri Poincaré Analyse Non Linéaire, vol. 1, no. 4, pp. 223–283, 1984.
- P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The limit case. I,” Revista Matemática Iberoamericana, vol. 1, no. 1, pp. 145–201, 1985.
- P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The limit case. II,” Revista Matemática Iberoamericana, vol. 1, no. 2, pp. 45–121, 1985.
- G. Bianchi, J. Chabrowski, and A. Szulkin, “On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent,” Nonlinear Analysis: Theory, Methods & Applications, vol. 25, no. 1, pp. 41–59, 1995.
- A. K. Ben-Naoum, C. Troestler, and M. Willem, “Extrema problems with critical Sobolev exponents on unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 4, pp. 823–833, 1996.
- J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, vol. 106, Pitman, Boston, Mass, USA, 1985, Elliptic Equations.
- M. Struwe, Variational Methods, vol. 34, Springer, New York, NY, USA, 3rd edition, 2000.
- I. Kuzin and S. Pohozaev, Entire Solutions of Semilinear Elliptic Equations, Birkhäuser, 1997.
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