Positive Radial Solutions for a Class of Semilinear Elliptic Problems Involving Critical

In this paper, we investigate the solvability of a class of semilinear elliptic equations which are perturbation of the problems involving critical Hardy-Sobolev exponent and Hardy singular terms. The existence of at least a positive radial solution is established for a class of semilinear elliptic problems involving critical Hardy-Sobolev exponent and Hardy terms. The main tools are variational method, critical point theory and some analysis techniques.


Introduction and Main Results
In this paper, we are concerned with the existence of positive radial solutions for the following semilinear elliptic problem with Hardy-Sobolev exponent and Hardy singular terms: : is radial , denotes the space of the functions ( )  , endowed with scalar product and norm, respectively, given by that coincides with the completion of ( ) with respect to the L 2 -norm of the gradient.By Hardy inequality [1], we easily derive that the norm is equivalent to the usual norm: Clearly, ( ) is a closed subset of ( ) Hilbert space.By the symmetric criticality principle, in view of [2], we know that the positive radial solutions of problem (1.1) correspond to the nonzero critical points of the functional ( ) The reason why we investigate (1.1) is the presence of the Hardy-Sobolev exponent, the unbounded domain N  and the so-called inverse square potential in the linear part, which cause the loss of compactness of embedding 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.There are abundant results about this class of problems.
For example, by using the concentration compactness principle, the strong maximum principle and the Mountain Pass lemma, Li et al.In the present paper, we investigate the existence of positive radial solutions of problem (1.1).There are several difficulties in facing this problem by means of variational methods.In addition to the lack of compactness, there are more intrinsic obstructions involving the nature of its critical points.Based on a suitable use of an abstract perturbation method in critical point theory discussed in [5] [13] [14], we show that the semilinear elliptic problem with Hardy-Sobolev exponent and Hardy singular terms has at least a positive radial solution.
In this paper, we assume that h satisfies one of the following conditions: , ( ) The main results read as follows.
Then for δ small, problem (1.1) has a positive radial solution u δ .
Remark 3 It is easy to check that the following function ( ) h r satisfies the conditions of Theorem 3 for all 3 N ≥ and 0 2 s < < , ( ) We can also give the following example for 3 N = and 1 s = , and by a direct computation, we have ( ) Theorem 5 Let h satisfy (H'), and suppose that ( ) Remark 5 It is easy to check that the following function ( ) h r satisfies the conditions of Theorem 5, ( ) sin 2 .

h r r =
This paper is organized as follows.After a first section we devoted to studying the unperturbed problem The main results are proved in Section 3. In the following discussion, we denote various positive constants as C or ( ) This idea is essentially introduced in [5] [13]., ; , 0, .

The
It is well-known that the nontrivial solutions of problem (2.1) are equivalent to the nonzero critical points of the energy functional Obviously, the energy functional ( ) solves the equation (2.1) and satisfies The unperturbed problem is invariant under the transformation that transforms the function ( ) u r in the function . The purpose of this section is to show the following lemmas.Lemma 2.1.For all 0 ε > , ( ) Proof.We will prove the lemma by taking 1 ε = , hence z U ε = .The case of a general 0 ε > will follow immediately.It is always true that ( )  .We will show the converse, i.e., that if ( ) , where D ε denotes the derivatives with respect to the parameter ε .We look for solutions ( ) and then a first solution is given by . If we look for a second independent solution of the form ( ) ( ) ( ) u r c r w r = , we will check that u is not in ( ) and because w is a solution, we have where C is a constant.This implies ( ) , ; , where , ; Passing to a subsequence we may assume that n u u  in ( )  Now, we give the abstract perturbation method, which is crucial in our proof of the main results of this paper.
Lemma 2.3.[13] (Abstract Perturbation Method) Let E be a Hilbert space and let ( ) Suppose that 0 f satisfies: 1) 0 f has a finite dimensional manifold of critical points Z, let ( ) for all z Z ∈ ; 2) for all z Z ∈ , ( ) D f z is a Fredholm operator with index zero; 3) for all z Z ∈ , ( ) Hereafter we denote by Γ the functional Z G .
Let 0 f satisfy ( 1)-( 3) above and suppose that there exists a critical point z Z ∈ of Γ such that one of the following conditions hold: 1) z is nondegenerated; 2) z is a proper local minimum or maximum; 3) z is isolated and the local topological degree of ′ Γ at z , ( ) still prove that f ε has a critical point near 0 Z .The set 0 Z could also consist of local minima and the same for maxima.

Proof of the Theorems
We will now solve the bifurcation equation.In order to do this, let us define the reduced functional, see [14], : Hence we are led to study the finite-dimensional functional The functional ( ) Proof.From the definition of ( ) ε Γ and U, we have ) where N s α < − .It is easy to get the first integral in the right hand side; next we show the second integral: In fact, Proof of Theorem 1. Firstly, we claim that ( ) Γ is not identically equal to 0. To prove this claim we will use Fourier analysis.We introduce some notation that will be used in the following discussion.If [ ) M is nothing but the Mellin transform.The associated convolution is defined by With this notation we can write our r Γ in the form ( ) ( ) ( )   ( )

Conclusion
We study a class of semilinear elliptic problems involving critical Hardy-Sobolev exponent and Hardy terms, and obtain positive radial solutions for these problems via an abstract perturbation method in critical point theory.
Extensions of nonradial solutions for these problems are being investigated by the author.Results will be submitted for publication in the near future.

[ 3 ]
had obtained the existence of positive solutions for singular elliptic equations with mixed Dirichlet-Neumann boundary conditions involving Sobolev-Hardy critical exponents and Hardy terms.Bouchekif and Messirdi [4] obtained the existence of positive solution to the elliptic problem involving two different critical Hardy-Sobolev exponents at the same pole by variational methods and concentration compactness principle.Lan and Tang [5] have obtained some existence results of (1.1) with 0 µ = via an abstract perturbation method in critical point theory.There are some other sufficient conditions, we refer the Y.-Y.Lan interested readers to ([6]-[18]) and the references therein.

Remark 1 >Theorem 3
It is easy to check that the following function ( ) .Then for δ small, problem (1.1) has a positive radial solution u δ .Remark 2 It is easy to check that the following function ( ) Assume that (H) holds, and suppose

Theorem 4
Suppose that assumption (H') holds, and satisfies the condition that ( ) ( ) problem (1.1) has a positive radial solution u δ , provided 1 δ  .Remark 4 It is easy to check that the following function ( ) h r satisfies the conditions of Theorem 4, Case 0 δ = In this section, we will study the unperturbed problem Y.-Y.Lan DOI: 10.4236/jamp.2017.5111802209 Journal of Applied Mathematics and Physics = − , where I is an identical operator.By the fact that I T λ − is a Fredholm operator with index zero, where 0 λ ≠ and T is compact, we obtain that Fredholm operator with index zero.This completes the proof of Lemma.

→
Here we will prove the existence result by showing that problem (1.1) has a positive radial solution provided that h satisfies some integrability conditions.Before giving the proof of the main results, we need the following lemma.Y.-Y.Lan DOI: 10.4236/jamp.2017.5111802213 Journal of Applied Mathematics and Physics Lemma 3.1.If (H) holds, then as ε → +∞ .

→>
real analytic and so has a discrete number of zeros.By continuity it follows that [ ] 0 M g ≡ .Then g and hence h are identically equal to 0. We arrive at a contradiction.This proves the claim.Since as ε → +∞ , and 0 r Γ ≡ / , it follows that r Γ has a maximum or a minimum at some 0 ε > .By a straight application of Lemma 2.3 jointly with Remark 2.4, the result follows. Proof of Theorem 2. Using Lemma 3.1, we have ) or a (global) minimum (if ( ) 0 0 h < ), at some 0 ε > .This allows us to use the abstract results, yielding a radial solution of problem (1.1), for δ small. Proof of Theorem 3. It suffices to remark that 0 h ≤ (resp.( ) 0 0 h ≥ ) and, once more r Γ has a (global) maximum (resp.a (global) minimum ) at some 0 ε > .In the rest of the section we will give the proof of Theorem 4 and Theorem 5.
Mathematics and PhysicsFirst we give the following Lemma.Hypothesis (H') allows us to use the following Riemann-Lebesgue convergence result.

∫
On the other hand, the remainder integral over the interval 0 r R ≤ < tends to 0 as ε → ∞ because of hypothesis (H') and the Riemann-Lebesgue lemma. Proof of Theorem 4. Using Lemma 3.3, we have

>
) or a (global) minimum (if ( ) 0 0 h < ), at some 0 ε > .This allows us to use the abstract results, yielding a radial solution of problem (1.1), for δ small. Proof of Theorem 5.It suffices to repeat the arguments used to prove Theorem 1 using Lemma 3.1 instead of Lemma 3.3.