The Existence of Positive Solution for Semilinear Elliptic Equations with Multiple an Inverse Square Potential and Hardy-Sobolev Critical Exponents

Abstract

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 [8]. 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 [8] 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[9]).

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 [10]):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 [13] 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 [14].) 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. [[15], 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 [3], 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 Hence, up to a subsequence, we obtain that strongly in

Lemma 4. Under the assumptions of , , and , there is a nonnegative function , such that

Proof. Let us prove only for the following case , for the other case the proof is the same. We consider the following functions on the interval and Using the following formula, and using (17), (19), and (20), we have for sufficiently small. And since, for all , the function is a positive on , we have for sufficiently small.

3. Proof of Main Result 1

We verify that the functional satisfies the mountain pass geometry. To this end, we consider the energy level.where For any , by Hardy and Sobolev-Hardy inequality, (8) and (9) (take ), we get that Hence, there exists small enough such that Then for all with Let given in Lemma 4. Since , hence there exists such that and , by Lemma 4, we obtain Moreover, by the Mountain Pass Theorem [9] and Lemma 3, we obtain that is critical value of at point and thus is a solution of problem (1). Then the rest of the proof follows exactly the same lines as that in [3]. In order to find the positive solution of (1), we replace with defined as follows: where . Repeating the above arguments, we find a critical point of and by applying the maximum principle we obtain a positive solution. So, the proof of Theorem 1 is therefore completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.