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In this paper, we consider a class of nonhomogeneous fractional elliptic equations involving critical Hardy Sobolev exponents as follows where il C MN is a bounded domain, O < s < 1, A>Oisa parameter. We prove the existence of multiple solutions using the variational methods and the Nehari manifold decomposition.
Abstract
In this paper, we consider a class of nonhomogeneous fractional elliptic equations involving critical Hardy Sobolev exponents as follows where il C MN is a bounded domain, O < s < 1, A>Oisa parameter. We prove the existence of multiple solutions using the variational methods and the Nehari manifold decomposition.
2000 Mathematics Subject Classification: 34B15, 35B33, 58E30.
Key words: fractional Laplacian, critical Hardy-Sobolev exponent, variational methods.
1 Introduction
The paper deals with the following fractional Hardy-Sobolev equation with nonhomogeneous term being O < s < 1, where Q is a bounded domain in Rw, (TV >2s) containing the origin 0 in its interior, positive parameter, 0 < a < 2s, 2*s = n^2s is the fractional critical Hardy-Sobolev exponent. The fractional Laplacian (-A)s is defined by where
T is the Gamma function, ƒ is a given bounded measurable function. It has been seen that the fractional differential equations have better effects in many realistic applications than the classical ones. Qualitative theory and its ap-plications in physics, engineering, economics, biology, and ecology are extensively discussed and demonstrated in [5, 6, 8, 11, 12, 13] and the references therein. There have been by now a large number of papers concerning the existence, nonex-istence as well as qualitative properties of nontrivial solutions to critical elliptic problems of Hardy potential and fractional Laplace operator. For instance, Ben-nour and all in [1] handled the following singular equation
where Q is a bounded domain in R and N >5,Under sufficiënt conditions on the data, the existence and multiplicity of solutions was proven, via Ekland's variation principle and the Mountain Pass Lemma principle.
In the local setting case (s = 1) the problem (1) is reduced to the semilinear problem with Sobolev-Hardy exponents
This problem was further studied by Chen and Rocha [4], who based on varia-tional methods obtained the existence of four non-trivial solutions. Recently, the existence of nontrivial solutions for nonlinear fractional elliptic equations with Hardy's potential type have been studied by several authors. Wang and all [15] studied (2) with g(u) = \u\2*~2u + au, a >0 and discussed the infinitely many solutions. Daoues and all
[7] studied (2) with g(u) = X\u\q~2u + |M| ,,t- and obtained the existence and nonexistence of nonnegative distributional solutions.
In what follows, we state the main result for which we consider the following hypothesis
Where X is a Hilbert space defined as where H2s(RN) the usual fractional Sobolev space,
Thus, we write our results in the following theorem:
Theorem 1. Let 0 < /x < Jïl, 0 < A < Ai and ƒ is a bounded measurable function satisfying the condition (J), then (1) has at least two nontrivial solutions, ifO<2(3+(iI) + 2s-N.
This paper is organized as follows: in the forthcoming section, we give some preliminaries and technical lemmas used in our work. Section 3 is concerned by the proofs of our main results. In the following discussions, we shall denote various positive constants as c. ö(e*) means that |ö(e*)e~*| < c, as e ->0, and o(l) is an infinitesimal value, ->(respectively, -*) will denote strongly (respectively, weakly) convergence. We denote the norme of X~ (the dual of X) by ||.||-.
2 Notations and preliminary results
2.1 A functional framework for the nonlocal problems
The embedding is continuous for any r € [1; 2*] and compact for any r € [1; 2*). The space X is endowed with the norm defined as we find that the norm |.| is equivalent to the usual norm From the fractional Sobolev-Hardy inequality, for /x e [0;//s[, we can define the best constant of fractional Sobolev-Hardy
Ghoussoub and Shakerian [9] proved that there exists radial solutions U^{x) e HS(RN) positive, symmetrie, decreasing and solves satisfying lim U^{x) = 0 and U^ € C1(RAf \ {0}). Furthermore, U^ has the following properties where 5q and 5oo are positive constants and are zeros of the function
The author in [9] proved that A^tS is attained in Rw by the function
Due to the fractional Hardy inequality, the operator £ = (-A)s - /xr^pj is defined on X. Moreover, the following eigenvalue problem with Hardy potentials and singular coefficient where 0 < a < 2s, X € R, has the first eigenvalue Ai given by:
Definition 1. A functional I e C1(X, R) satisfies the Palais-Smaile condition at level c, ({PS)C for short), if any sequence (un) C X such that contains a strongly convergent subsequence.
Definition 2. We say that u e X is a weak solution of the problem (1) if for all
Lp e X, one has
2.2 Nehari manifold
The energy functional associated to (f) is given by the following expression:
We see that I is well defined in X and belongs to C1(X, R) and is not bounded. Moreover, all the critical points of I are precisely the solutions of (1). We define the Nehari manifold N associated with the functional by
It is usually effective to consider the existence of critical points in this smaller subset of the Sobolev space. We can split N for:
2.3 Some technical lemmas
Lemma 1. If jjl e]0;/I7[; ol >0 and 0 < X < X\, then
In particular
Proof. . The proof is similar to [2]. D
Lemma 2. Let ƒ / 0 satisfying the condition (J), then N° = 0 and c = c+.
Proof. . The lemma is proved in the same way as in [14]. D
Let the cut-off function tp(x) = <f(\x\)>
Proposition 1. Suppose that N >2s, /jé [0;/I7[. Then
Proof. . For the estimates (1), (2) and (4) one can see in [16], we only verify (3)
Lemma 3. Let ƒ / 0 satisfies (J). For every u e X, u / 0 there exists a unique t+ = t+(u) >0 such that t+u e N~. In particular:
Lemma 4. Let ƒ / 0 satisfies (J). For e >0 and a differentiable function t = t(w) >0, 10 e X \ {0}, \\w\\ < e, satisfying the following: applying the implicit function theorem at the point (1,0) we can get the result of this lemma. D
3 Proof of Theorem 1
The current section contains two subsections. We consider 0 < A < Ai and 0 < /x < /IJ.
3.1 Existence of solution in 3\f+
Using Ekeland's variational principle, we prove the existence of a solution in N+.
Proposition 2. Let f be a function satisfying (J). Then c = inf I(u) is achieved at a point uq e N+ which is a critical point and even a local minimum for I.
Proof. . We start by showing that / is bounded from below in N. Indeed for u € N we have
Applying the Ekeland's variational principle to the minimization problem (1), we can get a minimizing sequence {un)n C N+ satisfying :
Next, we shall prove that ||/'(«n)|| ->0 as n ->+00. Suppose that ||/'(«n)|| >0 for n be large enough. By Applying Lemma 4 with
u = un and w = ^(iij/fa )ii)i er >0, we can find some tn(a) = ^(iij/fa )ii) sucn that
Dividing by a and passing to the limit as a goes to zero we derive that:
The proof will be completed once we have shown that |t^(0)| uniformly bounded with respect to n. From Lemma 4 and the estimate (6), we get:
C\ is a suitable constant.
Hence we must prove that \T(un) - (2*s - 1)J\un\2sdx\ is bounded away from zero.
n Arguing by contradiction, assume that for a subsequence still called (un), we have
According to (6) and (7), there exists a constant C2 >0 such that ƒ \un\2*sdx >C2.
n In addition, from (7) and by the fact that un e N, we get
This together with (J) imply that
which is clearly impossible. In conclusion,
Let uq e X be the weak limit in X of (un)n.
From (5) we derive that Jfuo >0, and from (8) that
i.e uq is a weak solution for (1). In fact, uq e N and
So, we deduce that un -)• uo strongly in X and I(-uo) = c = inf I(u). Moreover, uo S ^N"+- So uo is a local minimum for i". D
3.2 Existence of solution in 3\f~
In this subsection, to prove the existense of a solution in N~, we shall find the range of c~ where / verifies the (PS)C- condition.
Lemma 5. Let (un)n be any sequence of X satisfying the following conditions:
Solutions for a nonlocal elliptic equation 63
Then (un)n has a strongly convergent subsequence. Proof. . From (a) and (b), we have
I{un) = c + o(l), and then By using Hölder inequality, we get
From (3), (9) and (10) then, we have for all e positive
So T(un) is uniformly bounded. For a subsequence of (un)n, we can get auel such that
So, from (6), we obtain that
{I'(u),w) = 0, Vwel.
Then u is a weak solution for (1). In particular u / 0, -u e N and /(«) >c. We have
un ^ u in X and L2s (Q),
un ^ v, in L2(Q, |a;|_2,s),
Un^-u in L2(Q, la;!""2-5),
Let -ura = u + vn- So,0 in X. As in Brezis-Lieb Lemma (see [3]), we conclude that
Without loss of generality, as n ->+00 we may assume that
By (11), we deduce that l = 0 and un ->• u strongly in X as n ->+00. Assume that u = 0 in Q, from (I'(un),un) = o(l), we have this contradicts c < -^(A^)^. Therefore -u / 0 and m is a nontrival solution of problem (1).
Lemma 6. Let f ^ 0 be a function satisfying (J) £/ien for all 0 < X < X\, there exists v e X such that
Proof. . For t < 0, we consider the functions and
n the function ~g attains it's maximum. By Proposition 1, we can get that
On the other hand, usmg the denmtions of g and u&temppound;, we get g(t) = l{tu&temppound;) < 4||-uJ|2, for alli >0 and 0 < A < Ai.
Combining this with Proposition 1, let e e]0; 1[, then there exists to g]0; 1[ independent of e >0 such that
Hence, for all 0 < A < Ai and, by (17), we have
As 0 < a < 2/?+(/x) + 2s - N.
Combining this with (17) and (18), for any 0 < A < Ai, we can choose e small enough such that By taking v = u&temppound;. From Lemma 3, the definition of c and (16), for any 0 < A < Ai, we see that there exists t~ >0 such that fve N~ and Proposition 3. Suppose that ƒ verifies the conditions of Lemma 6. Then I has a minimizer u e N~ such that c~ = I{u). Moreover, u is a solution of problem Proof. Ii O < A < Ai, then, by Lemma 5 and Lemma 6, there exists a (Pb)cjv sequence (un) C N~ G X for I with c~ G (0; -fjA^s). Since 7 is bounded on CNT , we see that (-u„) is bounded in X. From Lemma 5, there exist a subsequence still denoted by (un) and a nonzero solution u G X of (1). such that un ^ u strongly inX.
Now, we first prove that u G 3\T-. Arguing by contradiction, we assume u G N+.
Then, by Lemma 3, there exists a unique t~ such that t~u G 3\T-. It follows that
This is a contradiction. Consequently, u G N~.
Proof of Theorem 1. By Proposition 2, 3, we obtain that the problem (?) has two positive solutions -uo and -u such that uq g ?Nf+, u G 3\T-. Since N+ n 3\T- = 0, this implies that uq and u are distinct. This completes the proof. D
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