Research Article | Open Access

# Quasilinear Elliptic Equations with Hardy-Sobolev Critical Exponents: Existence and Multiplicity of Nontrivial Solutions

**Academic Editor:**Tai-Ping Chang

#### Abstract

We study the existence of positive solutions and multiplicity of nontrivial solutions for a class of quasilinear elliptic equations by using variational methods. Our obtained results extend some existing ones.

#### 1. Introduction and Main Results

Let us consider the following problem: where denotes the -Laplacian differential operator, is an open bounded domain in with smooth boundary and , , is the Hardy-Sobolev critical exponent, and is the Sobolev critical exponent. Here, we let which is equivalent to the usual norm of Sobolev space due to the Poincaré inequality. Let which is the best Hardy-Sobolev constant.

In the case where and hold, then (1) reduces to the quasilinear elliptic problem: Gonçalves and Alves [1] have studied (4) in involving , and to obtain existence of positive solutions where , , or and a suitable . We should mention that problem (4) with has been widely studied since Brézis and Nirenberg; see [2–4] and the references therein.

Ghoussoub and Yuan [5] have studied (1) with , where . They obtained a positive solution in the case where , , and (in particular if ) hold. They also obtained a sign-changing solution in the case where , , and (in particular if ) hold. For other relevant papers, see [6–12] and the references therein.

We should mention that the energy functional associated with (1) is defined on , which is not a Hilbert space for . Due to the lack of compactness of the embedding in and , we cannot use the standard variational argument directly. The corresponding energy functional fails to satisfy the classical Palais-Smale ( for short) condition in . However, a local () condition can be established in a suitable range. Then the existence result is obtained via constructing a minimax level within this range and the Mountain Pass Lemma due to Ambrosetti et al. [2] and Rabinowitz [13].

In this paper, we study (1) with a general nonlinearity by using a variational method; besides, we also considerably generalize the results obtained in [5]. In what follows, we always assume that the nonlinearity satisfies . Let , . To state our main results, we still need the following assumptions., , and uniformly for .There exists a constant with such that , , and uniformly for .There exists a constant with such that

Now, our main results read as follows.

Theorem 1. *Suppose that , , , and hold. If , , and
**
hold, then (1) has at least one positive solution.*

Theorem 2. *Suppose that , , , and hold. If , , and (7) hold, then (1) has at least two distinct nontrivial solutions.*

Noting that and imply that (7) holds, therefore, we have the following corollaries.

Corollary 3. *Suppose that , , , and . Moreover, and hold; then (1) has at least one positive solution.*

Corollary 4. *Suppose that , , , and . Moreover, and hold; then (1) has at least two distinct nontrivial solutions.*

*Remark 5. *Theorem 1 generalizes Theorem 1.3 in [5], where the author only studied the special situation that , . There are functions satisfying the assumptions of our Theorem 1 and not satisfying those in [5]. Let
where , , , and . Obviously, satisfies all the conditions of Theorem 1 in this paper, while it does not satisfy the conditions of Theorem 1.3 in [5].

The rest of this paper is organized as follows. In Section 2, we give some preliminary lemmas, which are useful in the proofs of our main results. In Section 3, we give the detailed proofs of our main results.

#### 2. Preliminaries

In what follows, we let denote the norm in . It is obvious that the values of for are irrelevant in Theorem 1, so we may define We firstly consider the existence of nontrivial solutions to the problem: The energy functional corresponding to (10) is given by By Hardy-Sobolev inequalities (see [5, 14]) and , we know . Now it is well known that there exists a one-to-one correspondence between the weak solutions of (10) and the critical points of on . More precisely we say that is a weak solution of (10), if, for any , there holds

Lemma 6 (see [15]). *If a.e. in and for all and some , then
*

Lemma 7. *For any , , and , we have .*

*Proof. *Let
Clearly, , , so .

Lemma 8 (see [5]). *If hold, then we have*(i)* is independent of , and will henceforth be denoted by ;*(ii)* is attained when by the functions
**for some . Moreover, the functions are the only positive radial solutions of
**
in , and satisfy
*

Lemma 9. *If , , and hold, then satisfies condition.*

*Proof. *Suppose that is a sequence in . By , we have
where . Hence we conclude that is a bounded sequence in . So there exists ; going if necessary to a subsequence, we have
By the continuity of embedding, we have . From [5], going if necessary to a subsequence, one can get that
as . By , we know that for any there exists such that
Set . When , , we get
It follows from Vitali’s theorem that
Similarly, we can also get
Since , we have
Let , which together with Lemma 6 implies
From (20), we can obtain
Note that as , which together with Lemma 6 implies
Therefore, one gets that
From (26) and (27), we have
then as . Otherwise, there exists a subsequence (still denoted by ) such that
By (3), we have
then . That is, . It follows from (29) and that . However, we have by (27) and (). We get a contradiction. Therefore, we can obtain
From the discussion above, satisfies condition.

In the following, we shall give some estimates for the extremal functions. Let Define a function such that for , for , , where . Set so that . Then, by using the argument as used in [5], we can get the following results: Moreover, by using the Sobolev embedding theorem and (36), one can deduce

Lemma 10. *Suppose that . If , and (7) hold, then there exists , , such that
*

*Proof. *We consider the functions
Since , , and for small enough, is attained for some . Therefore, we have
and hence
Therefore, we obtain
By (), we can easily get
Hence, we can get
By (36)–(38), when is small enough, we conclude that
On the one hand, from Lemma 7 and (36), it follows that
On the other hand, the function attains its maximum at and is increasing in the interval . Note that () implies , which together with (36), (46), and (47) implies that
Furthermore, from (7) and (37), we get
By (7), we have , which implies
Therefore, by choosing small enough, we have
Hence, the proof of the lemma is completed by taking .

#### 3. Proofs of Main Results

*Proof of Theorem 1. *Let . From the Sobolev and Hardy-Sobolev inequalities, we can easily get
It follows from () that
uniformly for all . Therefore, we deduce that
for all and for . Then one gets
for all and for . By (52) and (55) we have
for small enough. So there exists such that
By Lemma 10, there exists with such that
It follows from the nonnegativity of that
Therefore, , so we can choose such that
By virtue of the Mountain Pass Lemma in [16], there is a sequence satisfying
where
Note that
By Lemma 9 we can assume that in . From the continuity of , we know that is a weak solution of problem (10). Then , where . Thus . Therefore, is a nonnegative solution of (1). By the Strong Maximum Principle [17], is a positive solution of problem (1). Therefore, Theorem 1 holds.

*Proof of Theorem 2. *By Theorem 1, we know that (1) has a positive solution . Set
It follows from Theorem 1 that the equation
has at least one positive solution . Let ; then is a solution of
It is obvious that , , and . So (1) has at least two nontrivial solutions. Therefore, Theorem 2 holds.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author thanks the referees and the editors for their helpful comments and suggestions. Research is supported by the Tianyuan Fund for Mathematics of NSFC (Grant no. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant no. 13A110015).

#### References

- J. V. Gonçalves and C. O. Alves, “Existence of positive solutions for $m$-Laplacian equations in ${\mathbb{R}}^{N}$ involving critical Sobolev exponents,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 32, no. 1, pp. 53–70, 1998. View at: Publisher Site | Google Scholar | MathSciNet - A. Ambrosetti, H. Brezis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,”
*Journal of Functional Analysis*, vol. 122, no. 2, pp. 519–543, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Brézis and L. Nirenberg, “Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,”
*Communications on Pure and Applied Mathematics*, vol. 36, no. 4, pp. 437–477, 1983. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Jannelli, “The role played by space dimension in elliptic critical problems,”
*Journal of Differential Equations*, vol. 156, no. 2, pp. 407–426, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. Ghoussoub and C. Yuan, “Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents,”
*Transactions of the American Mathematical Society*, vol. 352, no. 12, pp. 5703–5743, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Chen and S. Ma, “On the quasilinear elliptic problem with a Hardy-Sobolev critical exponent,”
*Dynamics of Partial Differential Equations*, vol. 8, no. 3, pp. 225–237, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Chen and S. Ma, “Existence and multiplicity of solutions for quasilinear elliptic equations,” preprinted. View at: Google Scholar
- G. Chen and S. Ma, “Multiple positive solutions for a quasilinear elliptic equation involving Hardy term and Hardy-Sobolev critical exponent,” preprinted. View at: Google Scholar
- Y. Deng and L. Jin, “Multiple positive solutions for a quasilinear nonhomogeneous Neumann problems with critical Hardy exponents,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 67, no. 12, pp. 3261–3275, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Egnell, “Positive solutions of semilinear equations in cones,”
*Transactions of the American Mathematical Society*, vol. 330, no. 1, pp. 191–201, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. Ghoussoub and X. S. Kang, “Hardy-Sobolev critical elliptic equations with boundary singularities,”
*Annales de l'Institut Henri Poincaré*, vol. 21, no. 6, pp. 767–793, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. A. B. Silva, J. C. N. Padua, and S. H. M. Soares, “Positive solutions of critical semilinear problems involving a sublinear term at the origin,”
*Cadernos de Mathematica*, vol. 5, pp. 245–262, 2004. View at: Google Scholar - 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, Washington, DC, USA, 1986. View at: MathSciNet - L. Caffarelli, R. Kohn, and L. Nirenberg, “First order interpolation inequalities with weights,”
*Compositio Mathematica*, vol. 53, no. 3, pp. 259–275, 1984. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Brézis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals,”
*Proceedings of the American Mathematical Society*, vol. 88, no. 3, pp. 486–490, 1983. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Struwe,
*Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems*, vol. 34 of*Results in Mathematics and Related Areas (3)*, Springer, Berlin, Germany, 2nd edition, 1996. View at: MathSciNet - J. L. Vázquez, “A strong maximum principle for some quasilinear elliptic equations,”
*Applied Mathematics and Optimization*, vol. 12, no. 3, pp. 191–202, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2014 Guanwei Chen. 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.