Multiple Solutions for Double Phase Problems with Hardy Type Potential

: In this paper, we are concerned with the singular elliptic problems driven by the double phase operator and and Dirichlet boundary conditions. In view of the variational approach, we establish the existence of at least one nontrivial solution and two distinct nontrivial solutions under some general assumptions on the nonlinearity f . Here we use Ricceri’s variational principle and Bonanno’s three critical points theorem in order to overcome the lack of compactness.


Introduction and Main Results
In the present paper, with the aid of variational methods, we establish the existence and multiplicity results for a class of singular elliptic problems, involving a double phase operator, subject to Dirichlet boundary conditions in a smooth bounded domain in R N . In the recent years, physical models containing a double phase operator have received extensive attention from scientists which is mainly due to applications as a models for describing a feature of strongly anisotropic materials and new examples of Lavrentiev's phenomenon (e.g., see Refs. [1][2][3][4]). A number of important results on the existence and multiplicity of nontrivial solutions for double phase problems, have been proved by Papageorgiou-Radulescu-Repovs [5,6], Perera-Squassina [7], Cencelj-Radulescu-Repovs [8], Radulescu [9], Zhang-Radulescu [10], Ge-Chen [11], Ge-Lv-Lu [12], Liu-Dai [13,14] and Colasuonno-Squassina [15]. For related regularity results dealing with minimizers of variational problems with double phase operator, we refer to the works of Baroni-Colombo-Mingione [16,17], De Filippis-Palatucci [18] and Esposito-Leonetti-Mingione [19].
on ∂Ω, (P 1,0 ) (P 1,0 ) has been studied more intensively in the last five years (see [11,13,14] and references therein), where the nonlinear term f being a Carathéodory function provided with suitable growth properties at zero and infinity, respectively. Using the variational method, the authors in [13] proved the existence and multiplicity of weak solutions of problem (P 1,0 ) when the nonlinear term has subcritical growth and satisfies the Ambrosetti-Rabinowitz condition. Then, Liu and Dai in [14] also proved the existence of at least three ground state solutions of (P 1,0 ) by applying a strong maximum principle for the double phase operator.
Recently, based on a direct sum decomposition of a space, Ge and Chen in [11] proved the existence of infinitely many solutions when the nonlinear term has a q − 1-superlinear growth at infinity and its primitive can be sign-changing. A similar treatment was recently performed by Hou-Ge-Zhang-Wang [20] via the Nehari manifold method. Replacing the strictly monotonicity condition (which was used in Refs. [13,14] to get ground state solution) by a weak version of Nehari type monotonicity condition, they discussed the existence of one ground state sign-changing solution by using the constraint variational method and quantitative deformation lemma. Following this, Zhang, Ge and Hou [21] established the existence of infinitely many positive solutions for the above problem under certain oscillatory conditions on the nonlinearity f at zero. For the case when µ = 0 and λ > 0, Ge, Lv and Lu in [12] obtained the existence of infinitely many solutions under the q-superliner condition and quasimonotonicity condition.
When µ = 0, the classical variational approach cannot be applied in our treatment due to the presence of the term µ |u| p−2 u |x| p . This is because the Hardy inequality only ensures that the embedding of the Sobolev space W 1,H 0 (Ω) into the weight Lebesgue space L p (Ω, |x| −p ) is continuous, but not compact. However, problems involving p-Laplacian operators have been discussed in several literatures, we refer to [22][23][24], in which the authors have used different techniques to prove the existence of solutions for problem (P λ,µ ) in the case a(x) ≡ 0. Motivated by the papers mentioned above, in this work we study the existence of solutions for problem (P λ,µ ) in which the function f is assumed to be subcritical growth condition. Our situation here is different from [11,13,14] in which the authors considered problem (P λ,µ ) in the case µ = 0 and f is q-superlinear at infinity.
The rest of this paper is organized as follows. In Section 2, we give some notation. We also include some useful results involving the Musielak-Orlicz-Sobolev space W 1,H 0 (Ω) in order to facilitate the reading of the paper. In Section 3, we establish the variational framework associated with problem (P λ,µ ), and we also establish some lemmas that will be used in the proofs of Theorems 1 and 2. We complete the proofs of Theorems 1 and 2 in Sections 4 and 5, respectively.

Preliminaries
In order to study problem (P λ,µ ), we need some basic concepts on space W 1,H 0 (Ω) which are called Musielak-Orlicz-Sobolev space. Based on this reason, we first recall some properties on Musielak-Orlicz spaces. A comprehensive presentation of the theory of such spaces can be found in [15,[25][26][27].
Denote by N(Ω) the set of all generalized N-function. For 1 < p < q and 0 ≤ a(·) ∈ L 1 (Ω), we define It is already clear that H ∈ N(Ω) is a locally integrable and In addition, we introduce the Musielak-Orlicz-Sobolev space W 1,H (Ω) is defined by which is equipped with the Luxemburg u given by The space W 1,H 0 (Ω) is defined to be the u −closure of the compactly supported elements of W 1,H (Ω), that is, .
Further, we recall Hardy's inequality, which states that where S H := N−p p p (see [28]). By Proposition 2(1), it follows that From now on, in the paper we denote by E the space W 1,H 0 (Ω). In order to study the problem (P λ,µ ), we consider the function J : E → R defined by It is standard to check that J ∈ C 1 (E, R) and double phase operator −div(|∇u| p−2 ∇u + a(x)|∇u| q−2 ∇u) is the derivative operator of J in the weak sense. Set L = J , then for any u, v ∈ E. Her ·, · is the duality pairing between E and its dual space E * . Then, we have the following important properties.

Proposition 3 ([13]
, Proposition 3.1). Let L be as above. Then, the following properties hold: (1) L : E → E * is a bounded, continuous and strictly monotone operator; (2) L : E → E * is a mapping of type (S) + , that is, if u n u in E and lim sup

Variational Setting and Some Preliminary Lemmas
To prove our theorems, we recall the variational setting corresponding to the problem (P λ,µ ). Now we introduce the Euler Lagrange functional ϕ λ,µ : E → R associated with problem (P λ,µ ) defined by where and Ψ(u) = Ω F(x, u(x))dx. Thus, using Proposition 1 and (3), it is easy to see that, if Now, in [13,23], it is shown that Φ µ (u) is a Gâteaux differentiable functional in E, and its Gâteaux derivative is the functional Φ µ (u) ∈ E * , given by is weakly lower semi-continuous and coercive. Moreover, similar to the proof of ( [13], proof of Theorem 3.1(ii)), we also can deduce that On the other hand, standard arguments show that Ψ is a well defined and continuously Gâteaux differentiable functional whose Gâteaux derivative Definition 1. Fixing the real parameters µ and λ, we say that u ∈ E is a weak solution of (P λ,µ ) if Therefore, the critical points of ϕ λ,µ are exactly the weak solutions of (P λ,µ ). Next, we give some important lemmas which will play important roles to prove our main results. First of all, let us recall the following the Ricceri's variational principle, which we use in the proof of Theorem 1. Lemma 1. Let X be a reflexive real Banach space, and let G, H : X → R be two Gâteaux differentiable functionals, such that G is strongly continuous, sequentially weakly lower semicontinuous and coercive. Further, assume that H is sequentially weakly upper semi-continuous.
This result is a refinement of the variational principle of Ricceri, see the quoted paper [29]. For the proof of Theorem 2, we need some definitions and results.
has a strongly convergent subsequence.
To prove Theorem 2, we will use the following Bonanno's three critical points theorem.
(T 2 ) There exists u 1 ∈ X and ρ > 0 such that , with τ > 1, and assume that, for every λ ∈ [0, γ], the functional G − λH is sequentially weakly lower semi-continuous and satisfies the Palais-Smale condition, and Then, there is an open interval Λ ⊆ [0, γ] and a number σ > 0, such that for each λ ∈ Λ, the equation G (u) − λH (u) = 0 admits at least three solutions in X, having a norm of less than σ.

The Proof of Theorem 1
In this section, we will prove Theorem 1. Firstly, we show that ϕ λ,µ possesses a nontrivial global minimum point in E.

Lemma 3. For every
Proof. Using inequalities (3), we obtain that for any u ∈ E, From this and 0 ≤ µ < pS H q , we conclude that This means that Φ µ (u) is coercive and this ends the proof.
Proof. Let {u n } n≥1 be a sequence that converges weakly to u in E. In view of (3), we can deduce that { |u n (x)| |x| } n≥1 is bounded in L p (Ω), so that the sequence has a weak limit, and, since u n u in E, u n → u in L p (Ω), and u n (x) → u(x) a.e. in Ω, it holds that Assume that 0 is the weak limit of the sequence, because if we denote by v n = u n − u, Brezis-Lieb lemma [31] yields for all ξ ∈ C c (Ω) where C c (Ω) is the space of those functions on R that are indefinitely differentiable and have compact support contained in Ω.
Note that the sequence {u n } n≥1 is bounded in E, while the sequence { |u n (x)| |x| } n≥1 is bounded in L p (Ω), so that the weak * limits of the sequences in the measure space exist. Due to P.L. Lions (see [32,33]), we have |u n (x)| p |x| p dx → ν and |∇u n | p + a(x)|∇u n | q dx → χ, in the * -weak convergence of measures. Given any ξ ∈ C c (Ω), using the functions ξu n in the Hardy inequality, we have The left-hand side member of (9) goes to Ω |ξ(x)| p dν as n goes to +∞. On the other hand, the right-hand side member is estimated as follows Using the fact that ξ has compact support, and the Rellich theorem, we see that this bound goes to 0 as n goes to +∞. Hence, passing to the limit in (9), we find for all ξ ∈ C c (Ω) Choosing ξ, such that 0 / ∈ supp(ξ), we have that |u n (x)| p dx → 0, since the function { |ξ(x)| |x| } belongs to L p (supp(ξ)), u n 0 in E and the embedding E → L p (Ω) is compact. The above information implies that ν is a measure concentrated in 0 and is absolutely continuous with respect to a Dirac mass (since χ contains Dirac masses). Hence, it holds that Fixing a function ξ ∈ C c (Ω), such that Then, by (10), one easily deduces Since ε is arbitrary, we have 0 ≤ ν 0 ≤ χ 0 S H . Then from (11), we deduce that and this ends the proof. Now, we are ready to prove Theorem 1.
These above facts enable us to apply Lemma 1 in order to find that there exists Moreover, by f (x, 0) = 0 in Ω, it follows that u λ is not trivial, that is, u λ = 0. Therefore, for any µ ∈ [0, pS H q ), there exists a λ * µ > 0 such that for any λ ∈ (0, λ * µ ) problem (P µ,λ ) admits at least one non-trivial weak solution u λ ∈ E.

Remark 1.
It is important to point out that accurate estimation of parameters λ * µ are very important in Theorem 1. In order to obtain such an estimation, let us fix µ ∈ [0, pS H q ). It is easy to compute directly that where τ 1 , τ 2 , τ 3 are positive constants and satisfy the following equations

Remark 2.
We observe that, if f is a (r − 1)-sublinear growth at infinity, with r ∈ (1, p), then for all λ > 0, Theorem 1 shows that problem (P λ,µ ) admits at least one nonzero solution. We explicitly observe that, in this case, the existence of at least one nontrivial solution can be obtained by classical direct methods.

The Proof of Theorem 2
In the section, we use Lemma 2 to prove Theorem 2, so we will first prove these lemmas.
Proof. Recall that the embedding W 1,H 0 (Ω) → L r (Ω) is compact. So, from (h 1 ) it follows that Ψ has sequentially weak continuity. As a result of Lemma 4, we obtain that Φ µ is sequentially weakly lower semi-continuous for every µ ∈ [0, pS H q ). Hence, ϕ λ,µ is sequentially weakly lower semi-continuous on E. This completes the proof. Lemma 6. Suppose that the assumption (h 3 ) is satisfied. Then, for every µ ∈ [0, pS H q ) and λ ∈ R, the functional ϕ λ,µ is coercive and satisfies the (PS) condition.
Next, we shall prove that Indeed, combining assumption (h 1 ) and Proposition 2(2), we calculate and Noting that Thus, in view of (27), (29)-(31), we see that (28) holds. Finally, since Φ µ is of type (S) + , we obtain u n → u 0 in E. The proof of Lemma 6 is complete.
In the following, we verify the condition (T 2 ). Due to (h 4 ), there exists a s 0 ∈ R such that F(x, s 0 ) > 0 for all x ∈ Ω. Additionally, choose R 0 > 0 in such a way that R 0 < dist(0, ∂Ω). For τ ∈ (0, 1) define where B R (0) = {x ∈ R N : |x| < R}. Moreover, denoting by w N the volume of the N-dimensional unit ball, one has Owing to assumption (h 4 ), we deduce that As τ → 1 − , the first term on the right hand side of the above inequality tends to the positive constant w N R N 0 F(x, s 0 ), and the second term goes to zero. We thus pick up some τ 0 and u τ 0 such that Ψ(u τ 0 ) > 0. Thus, in view of (5), we see that Again from Lemma 7, we may choose ρ 0 > 0 such that By choosing u 1 = η τ 0 , the condition (T 2 ) of Lemma 2 is verified. Define γ = γ µ =: In view of Lemmas 5 and 6, we testify all the conditions in Lemma 2. Hence, there exist an open interval Λ µ ⊂ [0, γ µ ] and a number σ µ > 0, such that, for any λ ∈ Λ µ , the equation Φ µ (u) = λΨ (u) has at least three solutions in E having E-norm less than σ µ . Because one of them may be the trivial solution (since f (x, 0) = 0, see (h 2 )), so the problem (P λ,µ ) still has at least two distinct nontrivial solutions.

Conclusions
In this paper, we have discussed the double phase problems with Hardy type potential.
Due to the presence of the term µ |u| p−2 u |x| p , the embedding of the Sobolev space W 1,H 0 (Ω) into the weight Lebesgue space L p (Ω, |x| −p ) is continuous but not compact, so the classical variational approach are not applicable. In view of this difficulty, few papers turn their attention to the existence of solutions of problem (P µ,λ ). In order to overcome this difficulty, In the present paper, we use the Ricceri's variational principle to obtain the existence of at least one nontrivial solution for problem (P µ,λ ), formulated in the paper as Theorem 1. Moreover, we use Bonann's three critical points theorem to obtain the existence of at least two nontrivial solutions for problem (P µ,λ ), formulated in the paper as Theorem 2. The main results in this paper extend and complement the previous research results.