Research Article | Open Access
The Existence of Positive Solution for Semilinear Elliptic Equations with Multiple an Inverse Square Potential and Hardy-Sobolev Critical Exponents
Via the concentration compactness principle, delicate energy estimates, the strong maximum principle, and the Mountain Pass lemma, the existence of positive solutions for a nonlinear PDE with multi-singular inverse square potentials and critical Sobolev-Hardy exponent is proved. This result extends several recent results on the topic.
1. Introduction and Main Result
Let be smooth open bounded with In this paper, we study the existence of solutions to the following nonlocal problem:for arbitrary such that , , , , if and , is the critical Sobolev-Hardy exponent .
We suppose the following: , for every There is an , , such that
and where is given in The function is a positive bounded on , for every Furthermore,
The reason why we investigate (1) is the presence of the Hardy-Sobolev exponent and the so-called inverse square potential in the linear part, which cause the loss of compactness of embedding , and ).
Hence, we face a type of triple loss of compactness whose interacting with each other will result in some new difficulties. In last two decades, loss of compactness leads to many interesting existence and nonexistence phenomena for elliptic equations. Many important results on the singular problems with Hardy-Sobolev critical exponents (the case that and were obtained such as the existence and multiplicity of solutions in these works and these results give us very good insight into the problem; see, for example, [1–7] and references therein. In the present paper, we use a variational method to deal with problem (1) with general form and generalize the results in . As to our knowledge, there are no results on the existence of non-trivial solutions for (1). It is therefore significant for us to study the problem (1) deeply. However, because of the singularities caused by the terms , our problem becomes more complicated to deal with than  and therefore we have to face more difficulties. Despite the multiple terms of hardy and the coefficients of the critical nonlinearity, but we will see how, they will play an important role in the search for the bubble whose energy is below the level of local compactness (PS). The existence result is obtained via constructing a minimax level within this range and the Mountain Pass Lemma due to A. Ambrosetti and P.H. Rabinowitz (see also).
Our main result is the following.
Theorem 1. Assume that conditions , , hold. Then problem (1) has at least one positive solution.
The paper is organized as follows: in Section 2, preliminary results about Palais-Smale condition for in a suitable interval and construct some auxiliary functions and estimate their norms. In Section 3, fill the conditions of Mountain Pass Theorem and we establish our result.
2. Preliminary Results
Throughout this paper, , represent all kinds of positive constants. We denote the standard norm of the Sobolev space by is a ball centered at with radius denotes and denotes as We will look for solutions of (1) by finding critical points of the functional : given byfor all The function is said to be a solution of problem (1) if satisfies for all
Problem (1) is well defined by the both inequalities, Sobolev-Hardy inequalities which is essentially due to Caffarelli, Kohn, and Nirenberg (see ):where , and the Hardy inequality (see [11, 12]), that is a special case () of the above Sobolev-Hardy inequality.By (8) and (9), for , , and we can define the best Sobolev-Hardy constant:In the case where , then ; note is the best constant in the Sobolev inequality, i.e.,The best Sobolev-Hardy constant is achieved only when by a family of functions:Let where and , (see  for details). Moreover,We consider such that and define a cut function such that , , for and for Setso that Then we have the following estimates.
Lemma 2. For any and ,For , we have
Proof. It is easy to get the following results (17),(18) (see .) We show (19) and (20) and for the proof (19). By using (18) and assumption we haveNow we show (20). Let and We haveForwe know thatand since we deduct thatBy using (23) and (26), we obtainwhere are constant.
For the second integral,For , we have Since , this implies that Also, and Then, if , we haveAnd, if , we haveIf , this implies that since , we deduct that We deduct thatSo, if , we haveLet , and ; we have ThenSo, for and taking (27),(39), and (41) in (22), we get For the case Using (29) we haveThen if we getBy (44), (45), and (46), we derive (20) for
Let be a Banach space and be the dual space of The functional is said to satisfy the Palais–Smale condition at level ( in short), if any sequence satisfying , strongly in as contains a subsequence converging in to a critical point of the functional
In our case, and
Lemma 3. The functional satisfies condition for any
Proof. Suppose is a sequence for with Then,First, we show that is bounded in Let for all So, and By Hardy and Sobolev-Hardy inequality we haveTherefore, up to a subsequence, we may assume that Then is a weak solution of problem (1). We may suppose that Using the concentration-compactness principle due to Lions (cf. [, Lemma 1.2]), we obtain an atmost countable set , a set of different points , real numbers , and for such thatand since we have where is the Dirac mass at
Let such that for any , Choose a smooth cut-off function centered at the point satisfying , for , for and Since is bounded, , that is,Moreover, we haveArguing as in , we can prove thatFrom (55)-(56), let and in expression (54), we obtainBy the definition of , we deduce thatCombining (57) with (58), we get which implies that Arguing by contradiction, let us suppose that there exist such thatThus,Letting , we get so, by (61), we obtain which contradicts the assumption that