Abstract and Applied Analysis
Volume 2015 (2015), Article ID 823143, 11 pages
http://dx.doi.org/10.1155/2015/823143

Research Article

Generalized Solutions for Nonlocal Elliptic Equations and Systems with Nonlinear Singularities

Youtao Wang and Guangcun Lu

Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Ministry of Education, Beijing 100875, China

Received 14 April 2015; Revised 4 June 2015; Accepted 7 June 2015

Academic Editor: Julio D. Rossi

Copyright © 2015 Youtao Wang and Guangcun Lu. 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.

Abstract.

We use the topological degree method to study the existence of solutions for nonlocal elliptic equations (systems) with a strong singular nonlinearity.

1. Introduction and Main Results

Given , an integer , and a bounded open set of with Lipschitz boundary, let be a function satisfying the following properties:

(i) with .(ii)There exists such that for any .(iii) for any .

The so-called nonlocal elliptic operator is defined by

In particular, when , is equal to the fractional Laplace operator (up to normalization factors).

For a Carathéodory function , the following problem

and its special case

have been widely studied under various contexts; see a recent survey [1] for details.

1.1. Previous Work

Motivated by the work of Caffarelli and Silvestre [2], several authors have considered an equivalent problem of (3) by means of an auxiliary variable; see [2–6]. Precisely, let denote the points in and . Take and as the completion of with respect to the norm

where is a normalization constant. For , let

and consider the problem

An energy solution to this problem is a function such that

Such an energy solution yields a function in the sense of traces, which belongs to the space and is a weak solution of (3). The converse is also true. The reader may refer to [2–6] for dealing with (3) with this method. In particular, Stinga and Torrea [6] generalized the arguments and results in [2] to the fractional powers , , of a linear second order partial differential operator that is nonnegative, densely defined, and self-adjoint in with a positive measure on .

Servadei and Valdinoci developed a variational framework to study the problem (2) in a series of papers [7–11]. They introduced the following Hilbert space in [7, 8]. Let , where , and let be the space of all Lebesgue measurable functions such that and that the map

is a Banach space endowed with the so-called Gagliardo norm

and is contained in . Consider the subspace of :

It was proved in [12, Theorem 6] that this space is the closure of in . Clearly, the space depends on . In fact, when ,   ([11, Lemma 7-b]).   can be endowed with a Hilbert space structure given by the inner product

([8, Lemma 7]) and contains ([7, Lemma 11]). Denote by the induced norm of the inner product in (10). This norm is equivalent to the restriction of to . Call a weak solution of the problem (2) if satisfies

for all . Define for the above Carathéodory function . Suppose that there exist , and , , such that

Then, the functional defined by

is of class and the critical points of are exactly the weak solutions of (2). Condition (12) is always assumed in the proofs of present several existence results on (2) via variational methods [7–11, 13–15]. Except for [13] is also assumed to be a solution in all other works. For studies of (3), there is a great deal of literature; see [2, 4, 16] and references therein.

However, all previous results cannot include the following case: and , where such that for and that for .

1.2. Main Results of This Paper

We will use the topological degree theory developed by [17] to study generalized solutions of problem (2).

Our result can apply to the example just mentioned. Without special statements, we write

(the latter plays the role of a critical Sobolev exponent).

Theorem 1. For an integer let satisfy (i)–(iii) and let be a Carathéodory function verifying the following conditions: there exist positive numbers , , , and functions and such that

Then, problem (2) has at least one generalized solution in ; that is, it satisfies

In particular, it must have a nontrivial generalized solution if is not identically zero.

Corollary 2. Under the assumptions of Theorem 1, let , and let . If either or belongs to with , then

has at least one nontrivial solution in provided is not identically zero.

Corollary 3. Under the assumptions of Theorem 1, let and with and . If is such that , then

has at least one nontrivial solution in provided is not identically zero.

In particular, this corollary includes the example at the end of Section 1.2. See Example 1 for more general cases.

Our methods can also be used to study the case of nonlocal elliptic operator systems. Let be functions satisfying conditions (i)–(iii). Given two Carathéodory functions , , consider the following problem:

Call a generalized solution of system (21) if

for every . Here is the second main result.

Theorem 4. Under the above assumptions, suppose also that there exist positive numbers , , , , and functions and such that

Then, problem (21) has at least one generalized solution in . In particular, it must have a nontrivial generalized solution if one of and is not identically zero.

Similarly, consider the following problem:

or

Assume

(A) are Carathéodory functions, ;(B)there exist constants

 and measurable functions , , such that

(C)there exist measurable functions and constants such that, for any ,

 Combing the proof of [17] and that of Theorem 4, we can prove the following.

Theorem 5. Under the conditions (A), (B), and (C), if or is not zero, then the equation systems (26) and (27) have at least a nontrivial generalized solution .

Finally, let us point out that the corresponding results of Theorems 1 and 4 can be also proved if the operator is replaced by in [6, (1.10)]. They will be given in other places.

The arrangements of this paper are as follows. In Section 2, we give some necessary preliminaries. The proof of Theorem 1 will be completed in Section 3. In Section 4, we will prove Corollaries 2 and 3 and give an example. Theorem 4 will be proved in Section 5.

2. Preliminaries

Firstly, we review the topological degree theory for mappings of class developed in [17]. Let be a Hilbert space with inner product and let be a strictly increasing sequence of finite dimensional subspace of such that is dense in . Denote by the orthogonal projection from onto for every integer . Let be an open bounded set in and let be a mapping from , the closure of in , into . Put

Since and induce equivalent topologies on all finite dimensional spaces the subset has the same closure in , , and , denoted by .

Definition 6. Under the above assumptions, is said to be of class on if and only if the following conditions are satisfied:

(a) is a continuous mapping for each .(b)There is not any sequence in such that the sequence is weakly convergent in and for all and in .

Lemma 7 (see [17, Lemma 2.3]). Let and be as in Definition 6. Assume that is of class on . Then, there exists an integer such that the Leray-Schauder degree is defined and

It follows that

is defined. It was the topological degree of on at   in [17]. The corresponding versions with usual properties of the Leray-Schauder degree were given in [17, Theorem 2.1]. In particular, the identity map is of class (), and if . Moreover, the following proposition is key for the proof of our main results.

Proposition 8 (see [17, Corollary 2.1]). Let and be as in Definition 6. Let be a mapping from into such that is continuous on for any . Suppose that contains 0 and

Then there is a weakly Cauchy sequence in such that

Next, we need the following results on the space .

Lemma 9.    and are continuously embedded in and , respectively ([8, Lemma 5]).
If is a bounded open subset with continuous boundary, the embedding is compact for any   ([8, Lemma 8] and [11, Lemma 9-a]).
The embedding is continuous for ([11, Lemma 9-b]).
The embedding is continuous for any ([18, Theorem 6.5]).
If is an open set in of class with bounded boundary, then there exist continuous embeddings and for any and ([18, Proposition 2.2]).

Lemma 10. Let be a bounded open set in with boundary of class . Then, the space is separable. Furthermore, there exists a sequence in such that is a maximal orthogonal set of .

Proof. By Proposition 9(f) of [9], there exists a Hilbert basis of , which implies separability of . So is a dense countable subset in . Let denote this countable set. Since is dense in by [12, Theorem 6], for each we can take such that . Then, is also dense in . Let be a maximal subset of such that any finite elements in are linearly independent. Then, is dense in . Making the Hilbert-Schmidt orthogonalization procedure for , we obtain an orthogonal set , which is also a maximal in .

3. Proof of Theorem 1

Take an increasing sequence of open subsets of , , such that each of them has -boundary and that

By Lemma 10, we may choose a sequence in such that is a maximal orthogonal set of . Then, we can find a sequence in such that is a maximal orthogonal set of . By the mathematical induction, it is easy to find the set in such that is a maximal orthogonal set of for every . Let us rewrite the countable set as a sequence . Let be the vector subspace of spanned by , and . For conveniences we set and denote by the orthogonal projection from onto .

Lemma 11.    is dense in .
For each , there are and a sequence in such that the supports of all are contained in and that in as .
For every and for every given , there exists a unique in such that

Moreover, if , then

Suppose that a sequence weakly converges to in . Then, weakly converges to in for .
For every given (the support of must be contained in some by the construction of ), there exists a unique such that

   is continuous for every .
There exists a constant such that

Proof. (a) Since is a maximal orthogonal set of , is dense in .
(b) For a given , by the choices of , there exists such that the support of is contained in . Let be the maximal orthogonal set of as constructed above. Then, is dense in . Hence, we can find a sequence in such that as .
(c) By (16), we can write . Then, , and (15) implies

Note that and . Moreover, since we get

For given , by [19, Theorem 3.2.4] (see also [20, page 30]) we have a continuous mapping from into , where

For , we have and by Lemma 9(b) and (c). Thus, and

Using Lemma 9(c) again, there exists a constant such that for all . It follows that

Hence, Riesz representative theorem yields a unique such that

If , for each integer we deduce

(d) Let be a sequence weakly converging to in . Since

from Lemma 9(b), we deduce that converges to in . Then, the continuity of the map implies that converges to in . Moreover, for , we have . Recall that and . We deduce that

and hence by (46).
(e) Since , and , by (46) we deduce that

This shows that is a continuous linear functional. Using the Riesz representative theorem again we obtain a unique such that

Clearly, for all .
(f) By the construction of , we have an integer such that each has a support contained in . Let converge to . By (d), weakly converges to in for every . Then, (e) implies that as . In particular, since satisfies , we have

This shows that weakly converges to in . However, the strong converge and the weak ones on finitely dimensional space are equivalent. Hence, as .
(g) Since , by Lemma 9(b) and (c) there is a constant such that . It follows from this and (17) that

This leads to (g).

Proof of Theorem 1. Let , , and be as above. Since , we have such that

Let . Define by

Let us prove that is of class on . Note that has the same closure in , , and , denoted by . Let be defined by for each . Suppose that a sequence converges to . Lemma 11(f) implies in . Hence, is continuous. By the proof of Lemma 11(g) and (54) we deduce that

This implies that Definition 6(b) is satisfied. Hence, is of class on . Moreover, it also shows that satisfies the conditions of Proposition 8. Thus, we have a weakly Cauchy sequence such that

Let be the weak limit of in . For a given , the support of it is contained in some , and thus

by Lemma 11(c). For each fixed , there exists such that . Hence, Lemma 11(e) yields

Taking in both sides of and using (57) and Lemma 11(d), we deduce

For any given , by Lemma 11(b), we have an integer and a sequence such that for any and that in . So (60) leads to

that is, . Letting , we get

which shows that is a generalized solution. Note that might be zero! But is not a solution if takes nonzero values on a nonzero measure set. The proof is completed.

4. Proofs of Corollaries and Examples

Proof of Corollary 2. Let . It suffices to check that satisfies (15)–(17). Clearly, we can assume . Let . Since , we have and thus

Then, and

By Young’s inequality, we obtain

Note that and imply . Let , which sits in . From these and (15), it follows that

for some constants and , where Young’s inequality is used again. So satisfies (15). Moreover, ; that is, satisfies (16).
Finally, let us check that (17) holds for . If , then

For another case, observe that

Since , we may choose a real number in . Let . By Young’s inequality, we have

Note that since . Moreover, and

Let , which belongs to . Using Young’s inequality, we can derive

for some constants and . Hence, for a.e. and all , we have

where belongs to as above. This shows that (17) is true for . The desired conclusion follows from Theorem 1 immediately.

Proof of Corollary 3. Let . Since , we see that satisfies (17) from the above proof. It remains to prove that satisfies (15)-(16). Let . Note that

By Young’s inequality, we have

Now, implies and because . We obtain . As in the proof of Corollary 2, using Young’s inequality, we may derive from this and (73)-(74) that satisfies (15) and (16).

Example 1. Let and be as above. Consider

where belongs to with , is absolutely continuous, , , and , a.e., , . Then, (75) has a nontrivial generalized solution.
In fact, taking in Corollary 3, we should require . Since , there is sufficiently small such that

This means that we can take . Moreover, , and

Hence, (15)–(17) are satisfied for .

5. Proof of Theorem 4

Consider the product Hilbert space equipped with inner product

for . The induced norm is

Let and be as in Section 3. For every integer let and . Denote by the orthogonal projection from onto . Corresponding to Lemma 11, we have the following.

Lemma 13.    is dense in .
For each , there are and a sequence in such that the supports of all are contained in and that in as .
For every , and for every given , there exists a unique in such that

for any . Moreover, if , then .
Suppose that a sequence weakly converges to in . Then, weakly converges to in for .
For every given (the support of must be contained in some by the construction of ), there exists a unique such that

   is continuous for every .
There exists a constant such that

for all

Proof. (a) and (b) follow from Lemma 11(a) and (b) immediately.
(c) By (24), we can write as in the proof of Lemma 11(c); then, , and

Moreover, for every given and , we have continuous mappings

where for , . Recall that . For , by Lemma 9(b) and (c), we have

Thus, , , and

Using Lemma 9(c) again, there exist constants such that for all , . It follows that

By the Riesz representative theorem, for each , we have a unique such that

Setting , we obtain

Another claim can be proved as that of Lemma 11(c).
(d) Let for each . Then, weakly converges to in and weakly converges to in . For each , by Lemma 11(d), weakly converges to in and weakly converges to in . Thus, weakly converges to in for .
(e) Since , , and , for any integer and , we derive from (89) that

These show that

are two continuous linear functionals. Using the Riesz representative theorem again we obtain a unique such that

for all . Set ; then,

Clearly, for all .
(f) follows the above (e) and Lemma 11(f) directly.
(g) Since , by Lemma 9(b) and (c) there is a constant such that

It follows from this and (25) that

Here, . This leads to (g).

Proof of Theorem 4. We replace the space in the proof of Theorem 1 by . Since for , as in (54), we have such that

Then repeating the proof of Theorem 1, we get a such that for any ; namely, (22) holds.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referee for the valuable comments and suggestions on the paper. This work is partially supported by the NNSF 10971014 and 11271044, of China, and PCSIRT.

References

X. Ros-Oton, “Nonlocal elliptic equations in bounded domains: a survey,” http://arxiv.org/abs/1504.04099

L. A. Caffarelli and L. Silvestre, “An extension problem related to the fractional Laplacian,” Communications in Partial Differential Equations, vol. 32, no. 7–9, pp. 1245–1260, 2007.

B. Barrios, E. Colorado, A. De Pablo, and U. Sanchez, “On some critical problems for the fractional Laplacian operator,” Journal of Differential Equations, vol. 252, no. 11, pp. 6133–6162, 2012.

X. Cabré and Y. Sire, “Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates,” Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 31, no. 1, pp. 23–53, 2014.

X. Cabré and J. Tan, “Positive solutions of nonlinear problems involving the square root of the Laplacian,” Advances in Mathematics, vol. 224, no. 5, pp. 2052–2093, 2010.

P. R. Stinga and J. L. Torrea, “Extension problem and Harnack's inequality for some fractional operators,” Communications in Partial Differential Equations, vol. 35, no. 11, pp. 2092–2122, 2010.

R. Servadei and E. Valdinoci, “Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,” Revista Matemática Iberoamericana, vol. 29, no. 3, pp. 1091–1126, 2013.

R. Servadei and E. Valdinoci, “Mountain pass solutions for non-local elliptic operators,” Journal of Mathematical Analysis and Applications, vol. 389, no. 2, pp. 887–898, 2012.

R. Servadei and E. Valdinoci, “Variational methods for non-local operators of elliptic type,” Discrete and Continuous Dynamical Systems, Series A, vol. 33, no. 5, pp. 2105–2137, 2013.

R. Servadei and E. Valdinoci, “On the spectrum of two different fractional operators,” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 144, no. 4, pp. 831–855, 2014.

R. Servadei and E. Valdinoci, “The Brezis-Nirenberg result for the fractional Laplacian,” Transactions of the American Mathematical Society, vol. 367, no. 1, pp. 67–102, 2015.

A. Fiscella, R. Servadei, and E. Valdinoci, “Density properties for fractional Sobolev spaces,” Annales Academiae Scientiarum Fennicae Mathematica, vol. 40, pp. 235–253, 2015.

A. Fiscella, “Saddle point solutions for non-local elliptic operators,” Topological Methods in Nonlinear Analysis, vol. 44, no. 2, pp. 527–538, 2014.

A. Fiscella, R. Servadei, and E. Valdinoci, “A resonance problem for non-local elliptic operators,” Zeitschrift für Analysis und Ihre Anwendungen, vol. 32, no. 4, pp. 411–431, 2013.

Y. H. Wei and X. F. Su, “Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian,” Calculus of Variations and Partial Differential Equations, vol. 52, no. 1-2, pp. 95–124, 2015.

P. d'Avenia, M. Squassina, and M. Zenari, “Fractional logarithmic Schrödinger equations,” http://arxiv.org/abs/1412.0270

D. M. Duc, N. H. Loc, and P. V. Tuoc, “Topological degree for a class of operators and applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 57, no. 4, pp. 505–518, 2004.

E. Di Nezza, G. Palatucci, and E. Valdinoci, “Hitchhiker's guide to the fractional Sobolev spaces,” Bulletin des Sciences Mathématiques, vol. 136, no. 5, pp. 521–573, 2012.

P. Drabek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser, Basel, Switzerland, 2007.

M. A. Krasnosel'kii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford, UK, 1964.