Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

1 Introduction

In this paper, we first consider the existence of nontrivial weak solutions u∈H˙s⁢(ℝN) of the doubly critical equation of Choquard type with the Hardy–Sobolev–Maz’ya potential:

where x=(x′,x′′)∈ℝk×ℝN-k, 2≤k≤N-1, and 0≤θ<cN,k,s:=cN,s2⁢ak,s, with

also 𝒦α⁢(x)=1|x|N-α is a Riesz potential, 0<α,β<2⁢s<N, 2s,β*=2⁢(N-β)N-2⁢s is the critical Hardy–Sobolev exponent and 2s,α♯=N+αN-2⁢s is the fractional critical exponent for the Sobolev type embedding. We note that ak,s is the best constant of the following fractional Hardy–Maz’ya inequality:

The fractional Laplacian operator (-Δ)s is a nonlocal pseudo-differential operator, taking the form

where u∈Cc∞⁢(ℝN), P.V. stands for the Cauchy principle value and cN,s=22⁢s⁢π-N2⁢Γ⁢(N+2⁢s2)|Γ⁢(-s)|, see [11] and the references therein for the basics on the fractional Laplacian.

The weak solutions of (1.1) will be found in the space H˙s⁢(ℝN), which is defined as the completion of Cc∞⁢(ℝN) under the norm

By a weak solution u of the problem (-Δ)s⁢u=f in ℝN for f∈L2⁢NN-2⁢s⁢(ℝN), we mean that u∈H˙s⁢(ℝN) satisfies

Problems with two nonlinearities recently have been studied by several authors. For the case of local operators, such problems were considered in [2, 13, 18] for the Laplacian -Δ, the p-Laplacian -Δp and the bi-harmonic operator Δ2. Corresponding to the local case of such problems, the nonlocal fractional elliptic equations have been investigated widely in recent years. In [17], by using the harmonic extension [5] and concentration compactness principle [24, 25], Ghoussoub and Shakerian proved the existence of nontrivial weak solutions for a fractional Laplacian problem with critical Sobolev and Hardy–Sobolev terms. Then, by using the same methods, Chen [9] investigated such a problem with two critical Hardy–Sobolev terms. It is worth noting that the problems above are related to the Sobolev and Hardy–Sobolev type inequalities, see [10, 15, 32] and the references therein. In 2017, Yang and Wu [33] extended such problems to a broader scope. They studied the following fractional Choquard equation with critical nonlocal Hartree term and Hardy–Sobolev term:

where 𝒦α=|x|α-N and 0<α<2⁢s<N, 0≤γ<4s⁢Γ2⁢(N+2⁢s4)Γ2⁢(N-2⁢s4). The method they used is elementary, and they did not use the tools mentioned above, such as the extension methods or the concentration compactness principle, etc. Instead, they use an improved Sobolev inequality, which was proved in [29]. For more about nonlinear Choquard equations involving the singular Hardy potential and critical nonlinearities, see [14, 16, 28] and the references therein.

The problems mentioned above have a common characteristic, that is, they involve the singular Hardy potential and critical nonlinearities. Recently, as an application of the Hardy–Sobolev–Maz’ya inequality, a great attention has been focused on the study of fractional elliptic problems involving the Hardy–Sobolev–Maz’ya potential. In 2017, Cai, Chu and Lei [6] studied the existence of solutions for a fractional elliptic problem with the Hardy–Sobolev–Maz’ya potential and critical nonlinearities in a bounded domain. In 2019, Mallick [27] proved the following Hardy–Sobolev–Maz’ya type constant:

achieved by a nontrivial and nonnegative function in ℋ˙γs⁢(ℝN), where s∈(0,1), x′∈ℝk, 2≤k≤N-1, 0≤θ<2⁢s<N, 0≤γ<ak,s. (We refer to [12, 7], and many more references therein, for classical Hardy–Sobolev–Maz’ya inequalities.) As an application of this result, Mallick derived the existence of positive solutions for the Euler–Lagrange equation that related to the fractional Hardy–Sobolev–Maz’ya inequality in the entire space:

Furthermore, Mallick investigated the symmetry properties and a precise asymptotic behavior of the solutions (see also [26]).

Now, it is fairly natural to ask whether the nontrivial weak solutions exists for the fractional elliptic equations with Hardy–Sobolev–Maz’ya potential and doubly critical nonlinearities. To the best of our knowledge, there is not any such result in the literature on such problems. Hence, based on the results obtained by Mallick and inspired by the work of Yang and Wu, our first aim is to study the existence of nontrivial solutions for problem (1.1).

Solutions of problem (1.1) were found as critical points of a suitable functional, by the Mountain Pass Lemma without the (PS)c condition. In this case, the (PS)c condition only holds true for c in certain intervals related to the best Hardy–Sobolev–Maz’ya type constant. In the control of the Mountain-Pass level, the extremal function of the best Hardy–Sobolev–Maz’ya type constant plays an important role. The explicit form of this function is used in the process.

Then, by applying the same strategy of (1.1), and as an application of the results obtained by Su and Chen [30], we continue to study another Choquard equation involving critical nonlinearities and the local p-Laplace operator:

where N≥3, Δp⁢u:=div⁢(|∇⁡u|p-2⁢∇⁡u) is the p-Laplace operator, μ∈[0,(N-pp)p), p∈(1,N), λ∈(0,N), p∗=N⁢pN-p and pλ∗=p2⋅2⁢N-λN-p is the Hardy–Littlewood–Sobolev upper critical exponent; the Riesz potential 𝒦λ⁢(x) here is taking the form 𝒦λ⁢(x)=|x|-λ. The Sobolev space W1,p⁢(ℝN) is defined as the completion of Cc∞⁢(ℝN) with respect to the norm

In [30], Su and Chen investigated the following minimizing problem:

where N≥3, λ∈(0,N), p∈(1,N) and pλ*=p2⋅2⁢N-λN-p. By using a refinement of the Hardy–Littlewood–Sobolev inequality, they proved that Λμ,λ is achieved in ℝN by a radially symmetric decreasing and nonnegative function. Moreover, they gave an estimation of the extremal functions; we refer the reader to [30] for details.

This paper is organized as follows. In Section 2, we establish the appropriate space which is applicable to the study of problems (1.1) and (1.2), and we give some preliminary results and the main results of this paper. In Section 3, we investigate the attainability of the minimization problem related to problem (1.1), and then we give the proof of the existence of a solution to problem (1.1) in Section 4. Finally, we give the proof of the existence result for problem (1.2).

2 Preliminaries and Statements of the Main Results

Equations (1.1) and (1.2) are related to some specific functional embedding and geometric inequalities.

Towards problem (1.1), for fixed 2≤k≤N-1, 0≤θ<cN,k,s, we define ℋ˙θs⁢(ℝN) as the completion of Cc0,1⁢(ℝk∖{0}×ℝN-k) under the following norm:

and it is evident that ℋ˙θs⁢(ℝN)=H˙s⁢(ℝN) for 0≤θ<cN,k,s.

Thus, in the present paper, we work in the space H˙s⁢(ℝN) for problem (1.1). Note that if 0≤θ<cN,k,s, it follows from the Hardy–Maz’ya inequality that

is comparable to the norm ∥u∥H˙s⁢(ℝN)2, since the following inequalities hold:

for all u∈H˙s⁢(ℝN), where θ+=max⁡{θ,0} and θ-=max⁡{-θ,0}. Next, we work in the H˙s⁢(ℝN) for problem (1.1) endowed with the norm (2.1).

For problem (1.2), the standard Hardy inequality asserts that W1,p⁢(ℝN) is embedded in the weighted space Lp⁢(ℝN,|x|-p) and this embedding is continuous. More precisely,

for all u∈W1,p⁢(ℝN). Moreover, the constant μ1 is optimal. If μ<μ1, it follows from the Hardy inequality (2.2) that

is well defined on W1,p⁢(ℝN). By using the same arguments mentioned above, we know that ∥⋅∥W is an equivalent norm in W1,p⁢(ℝN). Thus, we work in W1,p⁢(ℝN) for problem (1.2) and we take the equivalent norm (2.3).

Solutions of problems (1.1) and (1.2) are equivalent to a nontrivial critical point of the following functionals, respectively:

where u∈H˙s⁢(ℝN), and

where u∈W1,p⁢(ℝN).

Now, we introduce the following improved version of Sobolev inequality, which can be founded in [29, 27]. We say that a measurable function u:ℝN→ℝ belongs to the Morrey space ∥u∥ℒr,ϖ⁢(ℝN), with r∈[1,∞) and ϖ∈(0,N], if only and if

Mallick [27] extended the improved version of fractional Sobolev inequality that originated in [29], we state it below and give a proof for the convenience of the readers. This can be viewed as a generalization of the classical Caffarelli–Kohn–Nirenberg inequalities (see [4] and also [21]).

By the Hardy–Littlewood–Sobolev inequality, we have

The exponent 2s,α♯ is critical in the sense that it is the limit exponent for the Sobolev type inequality. Combining (2.4) and (2.8), we have the following inequality:

Similarly, combining Lemma 2.2 and Lemma 2.3, Su and Chen [30] derived the following result:

where u∈W1,p⁢(ℝN).

For 0≤θ<cN,k,s, we define the following best fractional Hardy–Sobolev–Maz’ya constant:

We may easily see, as in [27] with minor changes, that the infimum is achieved in H˙s⁢(ℝN).

We first consider problem (1.1). Since solutions of (1.1) can be founded as a nontrivial critical point of the functional I0, the Nehari manifold related to I0 is given by

And a minimizer of the minimization problem

is a solution of problem (1.1). In order to establish the existence of solutions for problem (1.1) for 0≤θ<cN,k,s, we investigate the minimization problem

We will show that Sθ,α is achieved by a nontrivial and nonnegative function in H˙s⁢(ℝN) and this fact allow us to establish the existence of solution for problem (1.1).

Based on the results in [30] that the sharp constants Λμ,λ and

are achieved, by using similar arguments as in the proof of Theorem 2.5, we obtain the following result.

Throughout this paper, we assume that the projection of a point x∈ℝN to ℝk (2≤k≤N-1) and ℝN-k is denoted by x′ and x′′, respectively; BRk⁢(z) denotes the k-dimensional ball of radius R>0 centered at z∈ℝk, o⁢(1) is a generic infinitesimal value, and we always denote positive constants with C for convenience.

3 The Proof of Theorem 2.4