Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.


Introduction and Main Results
Consider the following semilinear elliptic equations with Dirichlet boundary value conditions: in Ω, where Ω is a smooth bounded domain in R N ðN ≥ 3Þ, 0 < s < 2, 2 * ðsÞ = 2ðN − sÞ/N − 2 is the Hardy-Sobolev critical exponent, 2 * = 2 * ð0Þ = 2N/ðN − 2Þ is the Sobolev critical exponent, μ < μ = Δ ðN − 2Þ 2 /4, and γ ∈ ð0, 1Þ. The energy functional associated with problem (1) is defined by for any u ∈ H 1 0 ðΩÞ. In general, a function u is called a weak solution of problem (1) if u ∈ H 1 0 ðΩÞ and uðxÞ > 0 for all x ∈ Ω; it holds The paper by Crandall et al. [1] is the starting point on semilinear problem with singular nonlinearity. There is a large literature on singular nonlinearities (see [2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein). In particular, the following Dirichlet problem has been shown in [2], in which the authors proved that problem (4) possesses at least one solution for μ > 0 small enough, and there exists no solution when μ is large. Chabrowski in [15] considered the Neumann boundary problems with singular superlinear nonlinearities by approximation and variational methods. When the superlinear term is subcritical, he obtained two solutions, a mountain-pass solution and a local minimizer solution. And, in the critical case, he obtained a local minimizer solution and proved that there is no moutain-pass solution.
Up to our knowledge, the literature does not contain any result on the existence of positive solutions to the problem (1) with the nonlinearity containing singularity and Hardy-Sobolev exponents. Motivated by reasons above, the aim of this paper is to show the existence of positive solutions of problem (1). We study problem (1) and obtain at least two solutions via the Nehari method. It is well-known that the singular term leads to the nondifferentiability of the functional I on H 1 0 ðΩÞ, so I does not belong to C 1 ðH 1 0 ðΩÞ, RÞ. In order to get the existence of multiple positive solutions of problem (1), we use the Nehari method and differentiate the two solutions by their different Nehari-type sets.
The main result can be described as follows.
The paper is organized as follows: in Section 2, we give some preliminaries; in Section 3, we prove Theorem 1. This idea is essentially introduced in 20]. Throughout this paper, we make use of the following notations: (i) The norm in H 1 0 ðΩÞ is denoted by By Hardy inequality [28], we easily derive that the norm is equivalent to the usual norm: (ii) D 1,2 ðR N ÞðN ≥ 3Þ denotes the space of the functions u ∈ L 2 * ðR N Þ such that |∇u| ∈ L 2 ðR N Þ, endowed with scalar product and norm, respectively, given by that coincides with the completion of C ∞ 0 ðR N Þ with respect to the L 2 -norm of the gradient. By Hardy inequality [28], we easily derive that the norm is equivalent to the usual norm: in D 1,2 ðR N Þ.
(iii) The norm in L p ðΩÞ is denoted by j·j p (iv) C, C 0 , C 1 , C 2 , ⋯ denote positive constants

Preliminaries
In this section, we will study the unperturbed problem It is well-known that the nontrivial solutions of problem (9) are equivalent to the nonzero critical points of the energy functional Obviously, the energy functional I 0 ðuÞ is well-defined and is of C 2 with derivatives given by For all ε > 0, it is well-known that the function solves equation (9) and satisfies

Journal of Function Spaces
Moreover z ε ðxÞ is the extremal function of the minimization problem In view of [27,29], we have the following exact local behavior of the solutions of (1).

Proof of Theorem 1
In this section, we will prove Theorem 1. The proof of Theorem 1 is based on solving the minimization problem (18) and the minimization problem We divide the proof into two steps. In the first step, we prove that if there is w ∈ N + λ such that d + ðλÞ = I λ ðwÞ and there is v ∈ N − λ such that d − ðλÞ = I λ ðvÞ, then w and v are two positive weak solutions of (1). In the second step, we prove that the minima d + ðλÞ in (18) and d − ðλÞ in (23) are achieved, respectively.

Lemma 5.
For each φ ∈ H 1 0 ðΩÞ and φ ≥ 0, we have the following: There is η 0 such that Proof. The proof follows exactly the scheme in the proof of Lemma 3 in [31].
Proof. We only prove (24) since the proof of (25) is similar. Let φ ≥ 0 and ε > 0. By (i) of Lemma 5 and simple computations, we have that Since the right hand side of the inequality has a finite limit value as ε ⟶ 0, by direct calculations, we conclude ε −1 ððw + εφÞ 1−γ − w 1−γ Þ increases monotonically as ε ⟶ 0 and The Fatou lemma yields w −γ φ ∈ L 1 ðΩÞ and we get (24). Since w, v ≥ 0 and w, v≡0, by the strong maximum principle, it follows that Lemma 7. We have that w and v are positive weak solutions of (1).
Step 2. The minima d + ðλÞ and d − ðλÞ are achieved. The proof is exactly the same as [32]. We omit the details here. 4 Journal of Function Spaces We point out that v ε and the exact local behavior of w (see Lemma 2.) play essential roles. From Lemma 2., we have So there is m > 0 such that wðxÞ ≥ m for x ∈ supp w \ f0g.

Lemma 8.
Under the assumptions of Theorem 1,