Let , and let , , be a bounded domain with smooth boundary. We consider the following problem with singular non-linearity:
where , , , , and is the fractional p-Laplacian operator defined as
Recently, a lot of attention is given to the study of fractional and non-local operators of elliptic type due to concrete real-world applications in finance, thin obstacle problem, optimization, quasi-geostrophic flow etc.
The semilinear Dirichlet problem for the fractional Laplacian is recently studied, using variational methods, in [8, 42, 43]. The existence and multiplicity results for non-local operators, like the fractional Laplacian with a combination of convex and concave type non-linearities like , p, , is studied in [3, 5, 37, 36, 44, 45]. The eigenvalue problem for the fractional p-Laplacian and properties like the simplicity of the smallest eigenvalue is studied in [16, 32]. The Brezis–Nirenberg type existence result has been studied in . Existence and multiplicity results with convex-concave type regular non-linearities have been studied in .
In the local setting (), the paper by Crandal, Rabinowitz and Tartar  is the starting point on semilinear problems with singular non-linearity. A lot of work has been done related to the existence and multiplicity results for the Laplacian and the p-Laplacian with singular non-linearity, see [1, 11, 14, 20, 18, 23, 24]. In [11, 14], the singular problems of the type
where λ, . Here, for some and in Ω, and is a Hölder continuous function which is positive on , sublinear at and superlinear at 0. The function , for some , is non negative and non-increasing function such that . They proved several results related to existence and non existence of positive solutions of the above problems, by taking into account both the sign of the potential K and the decay rate around the origin of the singular non-linearity g. Several authors considered problems of Lane–Emden–Fowler type with singular non-linearity, see [10, 15, 19]. In addition, some bifurcation results have been proved in  for the following problem:
where , , f is non-decreasing with respect to the second variable and behaves like around the origin. The asymptotic behaviour of the solutions is shown by constructing suitable sub- and super-solutions combined with the maximum principle. We also refer to [22, 29] as a part of previous contributions to this field. For detailed study and recent results on singular problems, we refer to .
In , Giacomoni, Schindler and Taḱaĉ studied the critical growth singular problem
where and , and . Using variational methods, they proved the existence of multiple solutions with restrictions on p and q in the spirit of [13, 17]. Among the works dealing with elliptic equations with singular and critical growth terms, we cite [1, 34, 2, 9, 25, 40], see also the references therein, with no attempt to provide a complete list.
There are many works on the existence of a solution for fractional elliptic equations with regular non-linearities like , . Sub-critical growth problems are studied in [8, 42, 43], and critical exponent problems are studied in [5, 37, 36, 38]. Also, the multiplicity of solutions by the method of Nehari manifold and fibering maps has been investigated in [26, 45, 46]. For detailed study and recent results on this subject, we refer to .
In , Barrios et al. studied the singular problem
where , , , , , , is a non-negative function for , and
In particular, they studied the existence of distributional solutions for small λ using the uniform estimates of which are solutions of the regularised problem with the singular term replaced by . In , the critical growth singular problem, for and , is studied where multiplicity results are obtained using the Nehari manifold approach.
There are many works on the study of p-fractional equations with polynomial type non-linearities. In , Goyal and the second author studied the subcritical problem using the Nehari manifold and fibering maps. In , a Brezis–Nirenberg type critical exponent problem is studied. We also refer readers to [6, 27, 31] and the references therein. To the best of our knowledge, there are no works on the existence of multiplicity results with singular non-linearities involving the fractional p-Laplacian operator.
In this paper, we study existence and multiplicity results with a convex-concave type singular non-linearity. Here we follow the approach of . We obtain our results by studying the existence of minimizers that arise out of the structure of the Nehari manifold. We would like to remark that the results proved here are new even for the case . Also the existence result is sharp in the sense that we show the existence of Λ such that is the maximal range for λ for which the solution exists. We show the existence of a second solution in the sub-critical case for the maximal range of λ, where the fibering maps have two critical points along with some other condition. We also show some regularity results on the weak solutions of (Pλ).
The paper is organised as follows. In Section 2, we present some preliminaries on function spaces required for the variational setting and state the main results. In Section 3, we study the corresponding Nehari manifold and the properties of minimizers. In Sections 4 and 5, we show the existence of minimizers and solutions. In Section 6, we show some regularity results, and Section 7 is devoted to the maximal range of λ for the existence of solutions.
2 Preliminaries and main results
The motivation for defining the function space comes from . In , Goyal and the second author discussed the Dirichlet boundary value problem involving the p-fractional Laplace operator using variational techniques. Due to non-localness of the fractional p-Laplace operator, they introduced the function space . The space X is defined as
where and . The space X is endowed with the norm
Then we define . There exists a constant such that for all . Hence, is a norm on , and is a Hilbert space. Note that the norm involves the interaction between Ω and . We denote as and for the norm in . Now, for each , we set
Then = Lebesgue measure of Ω and for all . From the embedding results in , we know that is continuously and compactly embedded in , where and the embedding is continuous but not compact if . We define the best constant S of the embedding as
We say that is a positive weak solution of (Pλ) if in Ω and
for all .
The functional associated to (Pλ), , is defined by
where is the function defined by
For each , we set and . Notice that if and if is, for example, of . We will need the following important lemma.
For each , there exists a sequence in such that strongly in , where and has compact support in Ω for each k.
The proof here is adopted from . Let and let be a sequence in such that is non negative and converges strongly to w in . Define . Then strongly in . Now, we set where is such that . Then strongly in , thus we can find such that . We set , and get strongly as . Consequently, by induction, we set to obtain the desired sequence, since we can see that has compact support for each k and , which says that converges strongly to w in . ∎
Let be the eigenfunction of corresponding to the smallest eigenvalue . This is obtained as minimizer of the minimization problem
We assume . With these preliminaries, we state our main results.
For each , we define the fiber map by .
Assume . In the case , assume also . Let be a constant defined by
For all , (Pλ) has at least two distinct solutions in when and at least one solution in the critical case when .
We say that is a weak sub-solution of (Pλ) if in Ω and
for all . Similarly, is said to be a weak super-solution to (Pλ) if in the above, the reverse inequalities hold.
Next we study that the existence of a solution with the parameter in the maximal interval. For this, we minimize the functional over the convex set , where and are sub- and super-solutions, respectively. Using truncation techniques as in , we show that the minimizer is a weak solution.
Let and . Then there exists such that (Pλ) has a solution for all and no solution for .
3 Nehari manifold and fibering maps
We denote for simplicity now. One can easily verify that the energy functional I is not bounded below on the space . We will show that it is bounded on the manifold associated to the functional I. In this section, we study the structure of this manifold. We define
The functional I is coercive and bounded below on .
In the case , since , using the embedding of in , we get
for some constants and . This says that I is coercive and bounded below on .
In the case , using the inequality and embedding results for , we can similarly get I as bounded below. ∎
From the definition of fiber map , we have
It is easy to see that the points in are corresponding to the critical points of at . So, it is natural to divide into three sets corresponding to local minima, local maxima and points of inflexion. Therefore, we define
There exists such that for each , there are unique and with the property that , and for all .
Define and . Let . Then we have
and we define . Since , we can easily see that attains its maximum at
Now, if and only if , and we see that
if and only if
where and are defined in (2.1).
In the case , we see that if and only if . So for , there exist exactly two points with and , that is, and . Thus, has a local minimum at and a local maximum at . In other words, is decreasing in and increasing in .
In the case , since and , with similar reasoning as above, we get .
In both cases, has exactly two critical points and such that , and . Thus, , . ∎
We have for all .
Let and . Then , that is, is a critical point of . By Lemma 3.2, has critical points corresponding to either local minima or local maxima. So, is the critical point corresponding to either local minima or local maxima of . Thus, either or , which is a contradiction. ∎
We can now show that I is bounded below on and as follows.
The following hold:
and for each .
Moreover, and .
Let and . Then we have
Thus, we obtain
This implies that
If , using , we get
Suppose that and are minimizers of I over and , respectively. Then, for each , the following hold:
there exists such that for each ,
as , where is the unique positive real number satisfying .
(1) Let , i.e., and . We set
for each . Then, using the continuity of ρ, and since , , there exists such that for . Since, for each , there exists such that , it follows that as , and for each , we have
(2) We define by
for . Then h is a function. Therefore, we have
and for each ,
Thus, by the implicit function theorem, there exists an open neighbourhood and containing 1 and , respectively, such that for all , has a unique solution , where is a continuous function. So,
Thus, by the continuity of g, we get as . ∎
Suppose that and are minimizers of I on and , respectively. Then, for each , we have ,
Let . For sufficiently small , by Lemma 3.5,
We can easily verify that as ,
which imply that . Also, for each ,
which increases monotonically as and
Next we will show these properties for v. For each , there exists with . By Lemma 3.5 (2), for sufficiently small , we have
which implies . Thus, we have
As , . Thus, using similar arguments as above, we obtain and (3.4) follows. ∎
Let be such that satisfies
for all (i.e., ϕ is a sub-solution of (Pλ)) and for each . Then we have the following lemma.
Suppose and are minimizers of I on and , respectively. Then and in Ω. In particular, .
By Lemma 2.2, let be a sequence in such that supp is compact, for each k and strongly converges to in . Then
where . Since converges to strongly, we get a subsequence of , still calling it , such that pointwise almost everywhere in Ω, and we write for each , as . Then
Further, we can see that
where and . Now, we separately estimate each integrals and to begin with, first we see that
Next we see that
using , and . Now, consider
and, similarly, we get
Since , for each , using (3.7), we get
and letting , we get . Thus, we showed . Similarly, we can show . ∎
4 Existence of minimizer on
In this section, we will show that the minimum of I on is achieved in . Also, we show that this minimizer is also a solution of (Pλ).
For all , there exist satisfying .
Assume and . We show that there exists such that . Let be a sequence such that as . Now, by (3.1), we can assume that there exists such that weakly in (up to a subsequence). First we will show that . Let . We have , which gives
Therefore, using , we obtain
This proves that . Case (I): and . First, we claim that . When , if , then , which is a contradiction. In the case , the sequence is bounded, since the sequence and is bounded. So, using Fatou’s lemma and the fact that for each k, we get
which implies and thus, in both cases we have shown . We claim that strongly in . Suppose not. Then, we may assume . Using the Brezis–Lieb lemma and the embedding results for in the subcritical case, we have
which implies , using the fact that for each k. Since , there exist (by fibering map analysis) such that and . By (4.1), we have , which gives two cases: or . When , we have
which is a contradiction. Thus, we have . We set for . From (4.1), we get , and since , . So, f is increasing in , and we obtain
which gives a contradiction. Hence, , and thus strongly in . Since , we have , so we obtain and . Case (II): and . We set and claim that strongly in . Suppose and as . Since , using the Brezis–Lieb lemma, we get
We claim that . Suppose . If and , then , which is a contradiction, and if , then
which is again a contradiction. In the case , the sequence is bounded, since the sequences and are bounded. So using Fatou’s lemma and , for each k, we get
which implies . Thus, in both cases we have shown that . So, there exist such that and . Then, the following three cases arise:
(i) Let for . By (4.2), we get and
which implies that h increases in . Then we get
which is a contradiction.
(ii) In this case, we have and . Since , we have
which gives a contradiction.
Consequently, only (iii) holds true, where we have
Clearly, this holds when and , which yields and . Thus, strongly in as and . ∎
is a positive weak solution of (Pλ).
Since is arbitrary, we conclude that is a positive weak solution of (Pλ). ∎
We recall the following comparison principle from .
Let be such that in and
for all . Then in Ω.
The proof follows by taking and using the equality
As a consequence, we have the following.
We have .
Then is a super solution of the eigenvalue problem
Also we can choose r small enough such that is a subsolution of (Qϵ).
Then, using the boundedness of (see Theorem 6.4) and , we can further choose r small enough such that .
Now, we consider the following monotone iterations:
Then, by the weak comparison Lemma 4.3, we get
Therefore, the sequence is bounded in , and hence has a weakly convergent subsequence, still calling it , that converges to in . Thus, is a weak solution of (Qϵ). Since is arbitrary, we get a contradiction to the simplicity and isolatedness of . ∎
5 Existence of minimizer on
In this section we show the existence of second solution for (Pλ) in the subcritical case. We assume throughout this section.
For all , there exist satisfying .
Assume and . We will show that there exists with . Let be a sequence such that . Using Lemma 3.4, we can assume that weakly as in . We claim that . When , if , then converges strongly to 0, which contradicts Lemma 3.4. If , we similarly have as in the earlier section. So, in both cases we get . Next we claim that converges strongly to in . Suppose not. Then, we may assume and we have the following:
and for each k and .
By (2), we have and . So, there exists such that and . Thus, . Define as for . From (2), we get and, since , . So, g is increasing on . Now, we obtain
which gives a contradiction. Hence, , and thus converges strongly to in . Since , we have . Therefore, we obtain and . This completes the proof of this proposition for the subcritical case. ∎
For , is a positive weak solution of (Pλ).
Since is arbitrary, we conclude that is a positive weak solution of (Pλ). ∎
To prove the existence of second positive solution in the critical case, one requires to know the classification of exact solutions of the problem
These are the minimizers of S, the best constant of the embedding into . In [6, 38], several estimates on these minimizers were obtained, and it was conjectured that the solutions are dilations and translations of the radial function
where . In the case , these classifications were proved in , where, in addition, it was proved that all solutions are classified by dilations and translations of . Using these classifications, in , it was shown that
where , , and is the minimizer on . Then, by carefully analysing the related fiber maps, it was shown that for large t. From this it follows
Then the existence of a minimizer of was shown using the analysis of fibering maps in Lemma 3.2.
6 Regularity of weak solutions
In this section, we shall prove some regularity properties of positive weak solutions of (Pλ). We begin with the following lemma.
Suppose u is a weak solution of (Pλ). Then, for each , and
Let u be a weak solution of (Pλ) and . By Lemma 2.2, we get a sequence such that strongly in . Each has compact support in Ω and . Therefore, u is a positive weak solution of (Pλ) and, for each k, we get
Using the monotone convergence theorem, we get and
If , then and . Since we proved the lemma for each , we obtain the conclusion. ∎
Before proving our next result, let us recall some estimates or inequalities from .
Let and be a convex function. If , and , then
Let and be an increasing function. Then we have
where for .
Let u be a positive solution of (Pλ). Then .
The proof here is adopted from Brasco and Parini . Let be very small and define
which is smooth, convex and Lipschitz. Let and take as the test function in (6.1). By taking the choices
in Lemma 6.2, we get
As , and we have . So using Fatou’s lemma, we let in the above inequality to get
Then, using Lemma 6.3 with the function , we get
Now, from the support of , we have
where and . Using the Sobolev inequality, given in [33, Theorem 1], we get
where is a nonnegative constant and the last inequality follows from the triangle inequality and . Using all these estimates, we now have
where is a constant. By the convexity of the map , we can show that
Using this, we can also check that
Hence, we have
for constant. We now choose
and let be such that
In addition, if we let and , then the above inequality reduces to
At this stage, if we take , then we can say that for all m. This will imply that for all m. Now, we iterate (6.3) using , and let which gives
Taking limit as in (6.4), we finally get
Since , using the triangle inequality in the above inequality, we get
for some constant . If we now let , we get
Hence, in particular, we have that . ∎
Let u be a positive solution of (Pλ). Then there exists such that .
Let . Then, using Lemma 3.7 and the above regularity result, for any , we get
for some constant , since we can find such that on . Thus, we have weakly in . So, using [31, Theorem 4.4] and applying a covering argument on the above inequality as in [31, Corollary 5.5], we can prove that there exists such that for all . Therefore, we get . ∎
7 Global existence of solution
In this section, we show the existence of solution for maximal range of λ. Let us define
We have .
The proof follows similarly to the proof of Lemma 4.4. ∎
In the following lemmas, we will show the existence of a solution of (Pλ) when .
If is a weak sub-solution and is a weak super-solution of (Pλ) such that a.e. in Ω, then there exists a weak solution satisfying .
We follow . We know that the functional I is non- differentiable in . Let . Then M is closed and convex, and I is weakly lower semicontinuous on M. We can see that if and weakly in as , then we may assume pointwise a.e. in Ω (along a subsequence). Since , and , so, by Lebesgue’s dominated convergence theorem,
So, . Thus, there exists such that . We claim that u is a weak solution of (Pλ). For and , define , where and . For , and we have
Now, we consider
Let and . Then, using the technique of Lemma 3.7, we get
where we used the inequality for and . Thus,
since as . Similarly, as , we can show that
Therefore, taking in (7.1), we get
Since is arbitrary, for all , we get
For , (Pλ) has a weak solution .
We fix . By the definition of Λ, there exists such that (P) has a solution , say. Then becomes a super-solution of (Pλ). Now, consider the function as the eigenfunction of corresponding to the smallest eigenvalue . Then and
Let us choose such that and . If we define , then
We remark that using the method in Lemma 7.2, we can show the existence of solution for the following pure singular problem:
where . We define u to be a positive weak solution of (7.2) if in Ω, and
Also, we say that is a positive weak sub-solution of (7.2) if and
We define the functional by
where is as defined in Section 2. One can easily see that is coercive, bounded below and weakly lower semicontinuous in . Thus, there exist such that . We claim that is a positive weak solution of (7.2). We choose such that in Ω and is a sub-solution of (7.2) ( is defined in Proposition 7.3). Let us define , where is a weak sub-solution of (7.2). Then , and following the proof of Lemma 7.2 with , where , and , we can show that is a positive weak solution of (7.2).
Adimurthi and J. Giacomoni, Multiplicity of positive solutions for a singular and critical elliptic problem in , Commun. Contemp. Math. 8 (2006), no. 5, 621–656. Google Scholar
G. Arioli and F. Gazzola, Some results on p-Laplace equations with a critical growth term, Differential Integral Equations 11 (1998), no. 2, 311–326. Google Scholar
B. Barrios, E. Coloradoc, R. Servadei and F. Soriaa, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), 875–900. CrossrefGoogle Scholar
B. Brändle, E. Colorado, A. de Pablo and U. Sànchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 39–71. Web of ScienceCrossrefGoogle Scholar
L. Brasco, S. Mosconi and M. Squassina, Optimal decay of extremals for the fractional Sobolev inequality, Calc. Var. Partial Differential Equations 55 (2016), no. 2, Article ID 23. Web of ScienceGoogle Scholar
L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9 (2016), no. 4, 323–355. Google Scholar
F. Cîrstea, M. Ghergu and V. Rădulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane–Emden–Fowler type, J. Math. Pures Appl. (9) 84 (2005), 493–508. CrossrefGoogle Scholar
G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.) 5 (2014), no. 2, 373–386. Google Scholar
J. P. Garcia Azorero and I. Peral Alonso, Some results about the existence of a second positive solution in a quasilinear critical problem, Indiana Univ. Math. J. 43 (1994), 941–957. CrossrefGoogle Scholar
M. Ghergu and V. Rădulescu, Multiparameter bifurcation and asymptotics for the singular Lane–Emden–Fowler equation with a convection term, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), 61–84. CrossrefGoogle Scholar
M. Ghergu and V. Rădulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Ser. Math. Appl. 37, Oxford University Press, Oxford, 2008. Google Scholar
J. Giacomoni, I. Schindler and P. Taḱaĉ, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 117–158. Google Scholar
S. Goyal and K. Sreenadh, On multiplicity of positive solutions for N-Laplacian with singular and critical nonlinearity, Complex Var. Elliptic Equ. 59 (2014), no. 12, 1636–1649. Web of ScienceCrossrefGoogle Scholar
S. Goyal and K. Sreenadh, Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function, Adv. Nonlinear Anal. 4 (2015), no. 1, 37–58. Web of ScienceGoogle Scholar
S. Goyal and K. Sreenadh, The Nehari manifold for non-local elliptic operator with concave-convex nonlinearities and sign-changing weight functions, Proc. Indian Acad. Sci. Math. Sci. 125 (2015), no. 4, 545–558. CrossrefGoogle Scholar
J. Hernández, F. J. Mancebo and J. M. Vega, On the linearization of some singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 777–813. CrossrefGoogle Scholar
N. Hirano, C. Saccon and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations 9 (2004), no. 1–2, 197–220. Google Scholar
A. Iannizzotto, S. Mosconi and Marco Squassina, Global Hölder regularity for the fractional p-Laplacian, preprint (2014), https://arxiv.org/abs/1411.2956.
G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia Math. Appl. 162, Cambridge University Press, Cambridge, 2016. Google Scholar
G. Molica Bisci and R. Servadei, Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent, Adv. Differential Equations 20 (2015), 635–660. Google Scholar
S. Mosconi, K. Perera, M. Squassina and Y. Yang, A Brezis–Nirenberg result for the fractional p-Laplacian, preprint (2015), https://arxiv.org/abs/1508.00700v1.
T. Mukherjee and K. Sreenadh, Critical growth fractional elliptic problem with singular nonlinearities, Electron. J. Differential Equations 54 (2016), 1–23. Google Scholar
S. Prashanth and K. Sreenadh, Multiplicity results in a ball for p-Laplace equation with positive nonlinearity, Adv. Differential Equations 7 (2002), no. 7, 876–897. Google Scholar
G. Rosen, Minimal value for c in the Sobolev inequality, SIAM J. Appl. Math. 21 (1971), 30–33. Google Scholar
J. Zhang, X. Liu and H. Jiao, Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity, preprint (2015), https://arxiv.org/abs/1502.02222.
About the article
Published Online: 2016-12-02
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 52–72, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0100.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0