High energy solutions for $p$-Kirchhoff elliptic problems with Hardy-Littlewood-Sobolev nonlinearity

This article deals with the study of the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left(\, \int\limits_{\mathbb{R}^N}|\nabla u|^p\right) (-\Delta_p) u + V(x)|u|^{p-2}u = \left(\, \int\limits_{\mathbb{R}^N}\frac{F(u)(y)}{|x-y|^{\mu}}\,dy \right) f(u), \;\;\text{in} \; \mathbb{R}^N, u>0, \;\; \text{in} \; \mathbb{R}^N, \end{array} \end{equation*} where $M$ models Kirchhoff-type nonlinear term of the form $M(t) = a + bt^{\theta-1}$, where $a, b>0$ are given constants; $1<p<N$, $\Delta_p = \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator; potential $V \in C^2(\mathbb{R}^N)$; $f$ is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for $\theta \in \left[1, \frac{2N-\mu}{N-p}\right) $ via the Poho\v{z}aev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem.


Introduction
The classical work by W. Ding and W. M. Ni in [11] and P. Rabinowitz in [26] opens a new stage in solving partial differential equations.The authors proved the existence of critical points using constrained minimization.The innovative idea was to establish that the mountain pass min-max level of the functional J, defined on the Hilbert space H and associated with the equation, is equal to the minimum of J restricted to the so-called Nehari manifold N = {u ∈ H| J ′ (u), u = 0}.Further, this technique has solved numerous problems, given the homogeneous and superlinear condition at infinity of the nonlinearity.
In contrast, if the nonlinear term is not homogeneous and superlinear, then we cannot employ the strategy of minimizing over the Nehari manifold.To overcome this, Jeanjean and Tanaka [17] defined the Pohožaev manifold and obtained solutions as minimizers of the functional over this manifold.Afterward, many authors used this technique to prove the existence of ground-state solutions to elliptic equations.In this article, we are interested in studying the equations for the p-Kirchhoff type operator with nonlocal non-homogeneous nonlinearity and without the superlinear condition at infinity.Elliptic equations of the Kirchhoff type have been studied extensively in the literature.Consider the following model problem where f (x, t) is subcritical and superlinear at t = 0.For this class of problems, extensive literature exists on the existence and multiplicity of results using variational methods.For instance, one can see [1,8,12,14,24,[27][28][29]31] and the references therein.In (1.1), Kirchhoff term is homogeneous with degree 4, so to obtain the geometric structure and the boundedness of Palais-Smale sequences for the energy functional, one usually assumes one of the following conditions (SL) 4-superlinear condition: (AR) Ambrosetti-Rabinowitz type condition: For all t ∈ R there exists λ > 4 such that f (x, t)t ≥ λF (x, t) > 0; (VC) Variant convex condition: f (x,t) |t| 3 is strictly increasing for t ∈ R\{0}.
Even though these conditions seem to be fundamental, they are restrictive in nature.For example, f (x, t) = |t| q−2 t, when 2 < q ≤ 4, the above conditions do not hold.So the elementary question arises as to whether one can obtain a ground-state solution without these conditions.
In 2014, Li and Ye [20] worked in this direction for the Laplacian and obtained a positive ground state solution of (1.1) on H 1 (R 3 ) for f (x, u) = |u| q−2 u and 3 < q ≤ 4 with some suitable assumptions on V .Authors used the minimizing argument on a new manifold called Nehari-Pohožaev manifold, defined as the set of all functions u ∈ H 1 (R 3 ) satisfying I ′ (u), u + Q(u) = 0, where I(u) is the energy functional and Q(u) := lim n→∞ I ′ (u), φ n (x • ∇u) is the Pohožaev functional for (1.1) and {φ n } are suitable test functions.Subsequently, Guo [13] generalized the results of [20] for a general nonlinearity f (u) with the following assumptions: Here, the author established that (1.1) admits a least energy solution using Jeanjean's [16] monotonicity approach on a Nehari-Pohožaev type manifold.However, in [32], Ye, with the superlinear condition at infinity, observed that the groundstate solution for (1.1) does not exist.Hence the author obtained a positive high energy solution by assuming suitable conditions for the potential function V .This was achieved by using the linking theorem on Nehari-Pohožaev type manifold.For more results, one can refer to [2,14,18,30].
On the other hand, problems that involve two nonlocal terms have been studied by Lü [22] with the potential V (x) = 1 + λg(x).More precisely, he considered the problem where M (t) = a + bt, q ∈ (2, 3 + α), λ > 0 is a parameter and g(x) is a non-negative steep potential well function.For λ large enough, Lü proved the existence of ground state solutions using the Nehari manifold and the concentration-compactness principle.Since the associated functional is not bounded below, the assertions used here can not be carried forward for the case of q ∈ (1 + α/3, 2].But using Nehari-Pohožaev manifold with some strong assumptions on the potential V (x), Chen & Liu [9], obtained a ground state solution in R 3 for the complete range, q ∈ (1 + α/3, 3 + α).For more recent results on ground state solutions, refer to [10,33].
Inspired by the works described above, in this paper, we study the existence of high energy solutions for a class of p-Laplace equations with a general Kirchoff terms M (t) = a + bt θ−1 and the nonlinearity that does not satisfy (SL), (AR), and (VC) assumptions.Precisely, we study the following class of Kirchhoff problems with the general Choquard term: where N > max{2, p} with 1 < p < N , a, b, θ are positive parameters and 0 < µ < N .Here, ∆ p is the p-Laplacian operator, defined as ∆ p u = div(|∇u| p−2 ∇u).The function f ∈ C(R, R) is such that f (t) ≡ 0 for t ≤ 0 and F is the primitive of f .The general function f satisfies the following growth conditions: f is a monotonic function and for some positive constant C and q ∈ p * ,µ , p * µ |f (t)t| ≤ C|t| p * ,µ + C|t| q ; where p * ,µ = p(2N −µ)

2N
is the lower critical exponent and p * µ = p(2N −µ) 2(N −p) is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.Moreover, we assume that the potential function for all x ∈ R N with the strict inequality on a subset of positive Lebesgue measure; where H represents the Hessian matrix of the function V .
This paper presents a new approach to show the existence results to (1.2) for all N ≥ 3. We established the existence of high energy solutions to (1.2) for a larger class of nonlinearity f .Most of the literature mentioned above for Kirchhoff problems is for the case when N = 3, as the case N ≥ 3 is more intricate.The techniques and ideas presented in [32] cannot be extended to higher dimensions using the Nehari-Pohožaev type manifold.This is primarily because the super quadratic growth in the Kirchhoff operator dominates the nonlinearity.We study the problem with the Kirchhoff term M (t) = a + bt θ−1 , for all values of θ ∈ 1, 2N −µ N −p .We remark that we incorporated the case θ = 2, that is, M (t) = a + bt when µ < min{N, 2p}.To successfully establish the existence of a solution to (1.2), we need first to study the following limit problem We study the existence of a solution to this limit problem using the concentration compactness arguments of Lions and the radial symmetry nature of the solution.Proving the radial symmetry nature of the solutions for this class of equations is an independent topic of interest.In the Laplacian case ( p = 2, M = 1), the radial nature of the solution was established long back by Li & Ni [19] for general ground state solutions, under the condition f ′ (s) < 0. However, the same approach cannot be carried forward for the p-Laplacian case (p = 2, M = 1).In [3], the radial symmetry of solutions for a class of p-Laplacian equations was proved under the assumption that solution v has only one critical point in R N .Subsequently, many researchers obtained radial symmetry result by using the apriori estimates on the solutions.But in our case, we establish the radial nature of the solution by aptly using the non-local nature of nonlinearity.Our arguments are based on the Euler-Lagrange equation satisfied by polarizations and equality cases in polarization inequalities.We also establish some inequalities in the frame of polarization which we didn't find explicitly in the literature.We dedicate Section 3 and Section 4 to establish that the ground state solution of (1.3) is radially decreasing.We develop a unified approach in this paper to study both (1.2) and (1.3).In the next section, we give some preliminary framework of the problem and state the main results of the article.

Preliminaries and Main result
This section of the article is intended to provide the variational setting.Refer to [21,34] for the complete and rigid details.Further in this section, we state the main results of the current article with a short sketch of the proof.The functional space associated to this problem is with the norm, It is easy to see from the condition ( which infers that the norm is well-defined. Proposition 2.1.(Hardy-Littlewood-Sobolev inequality) Let t, r > 1 and 0 < µ < N Then there exists a sharp constant C(t, r, µ, N ) independent of f , h such that By Hardy-Littlewood-Sobolev inequality and condition (F 2 ), there exists constant The energy functionals associated with the problems (1.2) and ( 1.
Observe that both the functionals J, and We also define the following Pohozaev manifolds associated with problem (1.2) and (1.3) Here, P (u) and P ∞ (u) are the Pohožaev identities associated with problem (1.2) and (1.3) respectively, (see [23,Theorem 3])

2) and
With these preliminaries, we state our main results Theorem 2.1.Assume that f and V satisfy 3) has a positive ground state solution that is radially symmetric.
Theorem 2.2.Assume that f and V satisfy (F 1 ) − (F 2 ) and (V 1 ) − (V 4 ) along with where u is the ground state solution of the functional J ∞ obtained in Theorem 2.1.Then there exists at least one positive solution to problem (1.2), which is a high energy solution.
A standard methodology to obtain solutions using variational techniques consists of looking for minimizers of the functional.Here, using the (SL) and (AR) conditions, one can prove the boundedness of the minimizing sequences, which will subsequently give us solutions.But in the above-mentioned results, due to the absence of conditions (SL) or (AR), verifying the mountain-pass geometry and obtaining the boundedness of the Palais-Smale sequence for the functionals J and J ∞ is a tedious job.Moreover, the lack of conditions (VC) and (f 3 ) − (f 4 ) makes the classical method of the Nehari manifold inappropriate.Taking into account all these points, we observed that J ∞ satisfies the mountain-pass geometry of the functional with the scaling function u t defined as for u ∈ W 1,p (R N )\{0} and t > 0. Keeping this in mind, we take minimum over the Pohožaev manifold P ∞ .Using the assumptions on f and V ∞ , we conclude that (1) P ∞ is a natural constraint, in the sense that J ∞ is coercive and bounded below on Further, to construct a bounded Palais-Smale sequence for J ∞ , we employ Jeanjean's [15] technique and prove the existence of a non-trivial critical point of functional J ∞ by the concentration-compactness lemma.This motivates us to use the same type of approach for the existence of a solution to (1.2).That is, we look for the solution in the Pohožaev manifold P and with the same type of scaling, one can establish that the functional J satisfies the mountain-pass geometry.However, we encounter that the functional J couldn't attain the mountain pass level, thus indicating the absence of a ground state solution for equation (1.2).To overcome this obstacle, we establish that J satisfies the Palais-Smale condition above the min-max level, leading to solutions with higher energy.To address the issue of compactness, we employ the splitting lemma (refer to Lemma 5.9).Additionally, we utilize the linking theorem in conjunction with a Barycenter mapping to achieve the goal in Theorem 2.2.To successfully apply the machinery of Barycenter mapping, we need the radial nature of ground state solution to (1.3).The salient feature of the approach is the existence of the radial solution to (1.3).To the best of our knowledge, there has been no attempt till now on the existence of high energy solutions to a generalized class of Kirchhoff-Choquard equation driven by the p-Laplacian operator.
Remark 2.2.There are a number of functions that satisfy Throughout the paper, we make use of the following notations: • u ± := max{±u, 0} and (H * , • * ) denotes the dual space of (H, • ).
• The letters C, and C i denote various positive constants possibly different in different places.
We organize the rest of the paper as follows.In Section 3, we give some technical lemmas that will help prove our Theorem 2.1.In Section 4, we study the limit problem and prove it has a positive radial solution.In section 5, we give some background material for Theorem 2.2, and in the last section, we present its proof.

Technical lemmas
In this section, we will first collect some variational frameworks and results that form the background material.Then we prove the limiting case problem (1.3), using a couple of critical results.
First, we look at the Mountain-pass geometry of the functional J ∞ .
Proof.(i) From (2.1) and Sobolev embedding theorem, there exist constants C 1 , C 2 > 0 such that for each u ∈ W 1,p (R N ), Then it follows that It is easy to see that there exists α > 0 such that J ∞ (u) > α, for u ≤ ρ, provided that ρ is sufficiently small.(ii) For any u ∈ W 1,p (R N )\{0} and t > 0 we define u t (x) := u( x t ).
Thus for sufficiently large t, we have e = u t ∈ W 1,p (R N ), with e > ρ, such that J ∞ (e) < 0. This completes the proof.
By the classical Mountain pass theorem, we have the minimax value defined as where By Lemma 3.1, there exists a (P S) . Thus, we have established the existence of the Palais-Smale sequence.Now, to ensure that the Pohožaev identity acting on the Palais-Smale sequence approaches zero, we will use Jeanjean's technique [15].Lemma 3.2.There exists a sequence {u n } in W 1,p (R N ) such that, as n → ∞, Proof.Define the map φ : and the corresponding minimax level Since {φ • γ : γ ∈ Γ∞ } ⊂ Γ ∞ and Γ ∞ × {0} ⊂ Γ∞ , the mountain pass levels of the functional For each n ∈ N, by (3.1), there exists Setting gn (t) := (g n (t), 0) for t ∈ [0, 1], we see that gn ∈ Γ∞ and using (3.3), we get max By Ekeland's variational principle, there exists a sequence min It is easy to verify that for every (w, s) Employing (3.5) and (3.4) with (w, s) = (0, 1), we reach the conclusion that For any z ∈ W 1,p (R N ), set w = φ(z, −t n ) = z(e tn x) and s = 0 in (3.5), we get . Hence, we have got a sequence {u n } in W 1,p (R N ) that satisfies (3.2).Further, the boundedness of (P S) c∞ sequence {u n } follows from where C > 0. This completes the proof.
Lemma 3.3.For any u ∈ W 1,p (R N )\{0}, there exists a unique t > 0 such that u t ∈ P ∞ , where Proof.For any u ∈ W 1,p (R N )\{0} and t > 0, we define Observe that for t small enough, h ∞ (t) > 0 and h ∞ (t) → −∞ as t → ∞, since 2N − µ > max{N − p, (N − p)θ, N }.Thus, h ∞ has at least one critical point, say at t > 0. Further, taking the derivative of h ∞ , we obtain that t d dt h ∞ (t) = P ∞ (u t ), which implies u t ∈ P ∞ .Now we claim that t is unique.Let us suppose that t 1 and t 2 are two critical points of h ∞ , such that 0 < t 1 < t 2 .Depending upon the range of θ, we consider the following cases: This is not true as 0 < t 1 < t 2 .
Case II: N N − p < θ < 2N − µ N − p Following the same argument as in Case I, we divide by t (N −p)θ i to get a contradiction.This completes the proof. Set which is well defined, since for any u ∈ P ∞ , Lemma 3.4.For c ∞ as defined in (3.1) and p ∞ as in ( * ), we have p ∞ = c ∞ .

Radial ground states of limiting problem
This section is devoted to the proof of Theorem 2.1.First, we will revisit the polarization technique and derive some necessary lemma to prove the radial symmetry of a given ground state solution.Then, we deploy the concentration-compactness principle to obtain the existence of a positive radial groundstate solution.
Let H ⊂ R N be a closed half-space and that σ H denotes the reflection concerning its boundary In particular, we have Proof.As F is a non-decreasing function, we get that (4.3)We divide the proof into four cases: From the definition of polarization and (4.3), we deduce that Case II: When u(x) ≤ u(x H ) and u(y) ≤ u(y H ) One can prove the desired result using the arguments as in Case I.
Case III: When u(x) ≤ u(x H ) and u(y Employing the definition of polarization, (4.3) and (4.2), we have Furthermore, by direct calculation and the assumption of the case, we see that Thus, we get the desired claim from (4.4) and above inequality.
Case IV: When u(x H ) ≤ u(x) and u(y) ≤ u(y H ) Similar to Case III.Hence the results hold.
with equality if and only if either Lemma 4.5.Let u ∈ W 1,p (R N ) be a positive ground state solution to problem (1.3).Then u is radially symmetric to a point.
Proof.Using the strategy of Tanaka and Jeanjean [17], we can easily lift the ground state solution to the path h 1)) < 0. Now applying the polarization on the path h, we can achieve that u H is also a critical point (for a more details, we refer to [23]).Consequently, applying Lemma 4.1, we deduce that Employing (4.1) and Lemma (4.3), we get either , we get the desired result.
Proof of Theorem 2.1: Let {u n } be a (P S) c∞ sequence for J ∞ , by Lemma 3.2, we have |u n | p .Now we divide the proof into the following parts: Case II: If δ > 0 Then there exists {y n } ⊂ R N such that Then there exists u ∈ W 1,p (R N ) such that up to a sub-sequence and by Fatou's Lemma, we conclude that Let us assume that Employing (4.5), we deduce that J ′ A,∞ (u) = 0, also u satisfies the Pohožaev identity associated with J A,∞ (4.7) Putting together (4.6) and (4.7), we get Hence by Lemma 3.3, there exists t 0 < 1 such that u t 0 ∈ P ∞ .Using this information, we get which is not possible.So our assumption (4.6) is not true, that is which further implies that J ′ ∞ (u) = 0 and u ∈ P ∞ .Following the same argument as in (4.8) for t 0 = 1, we get J ∞ (u) = c ∞ .Thus u is a non-trivial solution of (1.3).Moreover, we claim that u is a ground state solution.
where the last inequality holds by the definition of that is, u is a non-negative.Using the similar Moser's iteration as in [6], we get u ∈ L ∞ (R N ).Now employing the standard elliptic regularity and maximum principle (see [25]), we can achieve that u is a positive solution.At this moment using Lemma 4.5 we have the required.and t i > 0, we deduce that (N − p)a p which gives us a contradiction as 0 < t 1 < t 2 .
Case II: (N V (tx) + ∇V (tx) • tx) |u| p dx, using (V 3 ) and (V 4 ), we get that ψ is a decreasing function.By similar reasoning as in Case I, we have which is not true.Consequently, from both cases, we infer the uniqueness.
(iii) From (i), we see that there exists a ρ small enough such that P (u) > 0 for 0 < u ≤ ρ.
Proof.First, we will show that Γ is non-empty.Indeed by Lemma 5.1(i), there exists ρ > 0 small enough, such that J(u) ≥ α > 0 for 0 < u ≤ ρ.Additionally for any For fixed t > 0 and |τ | → ∞, we see from Hence for t and |τ | large enough, we have J(u t,τ ) < 0, and thus Γ is non-empty and c is well defined.
To show that c ≤ c ∞ , let ǫ > 0 be arbitrary, and take γ ∈ Γ ∞ such that max This gives c ≤ c ∞ and hence the equality is proved. Set it is well defined, by Lemma 5.1(i).J(γ(t)).
Proof.The proof is similar to the proof of Lemma 3.4.
Lemma 5.5.c is not a critical value of the functional J.
Proof.Let us suppose that c is a critical value of the functional J; that is, there exists As a consequence of Lemma 5.3 and 5.4, we see that J(u) = p and u t ∈ P ∞ , for some unique t ∈ (0, 1).Further employing assumption (V 2 ), Lemma 3.4 and Lemma 5.2, we obtain which is not possible.
Lemma 5.6.The following holds Proof.(i) We claim that P ′ (u) = 0, for all u ∈ P. Indeed, suppose for the sake of contradiction, there exists u ∈ P such that P ′ (u) = 0, namely, u is a solution to the problem (5.2) As a consequence, u satisfies the Pohožaev identity related to problem (5.2), that is, 3) In view of (5.3) and P (u) = 0, we get Using hypothesis (V 2 ) − (V 4 ), we obtain that (ii) Let us suppose that u ∈ P is a critical point of the functional J | P .Then there exists a Lagrange multiplier λ ∈ R such that J ′ (u) + λP ′ (u) = 0. (5.4) We will show that λ = 0. From (5.4), we see that u weakly satisfies the following problem: Consequently, u satisfies the following Pohožaev identity: P 2 (u) = P (u) + λP 1 (u) = 0, ( where P 1 (u) is defined in (5.3).Using the fact that P (u) = 0 in (5.5), we infer that From (V 2 ),(V 3 ) and (V 4 ), we see that it is only possible when λ = 0.
Lemma 5.7.For any u ∈ P ∞ , (i) there exists a unique t > 1, such that u t = u x t ∈ P.
(ii) For any y ∈ R N , there exists a unique t y > 1, such that u y,ty = u x − y t y ∈ P and (5.8) Proof.The proof of (i), (ii) is similar to that of [9,Proposition 3.1].It is enough to prove (5.8).Let {u n } ⊂ P be a minimizing sequence for p, then by Ekeland's variational principle, there exists a sequence {v n } ⊂ P such that By Lemma 5.6 (ii), Lemma 5.3 and Lemma 5.8, we conclude that {v n } is a bounded (P S) c .Then either (i) or (ii) holds, but by Lemma 5.5, as c is not the critical value of the functional J, so (i) cannot hold.As {v n } is a bounded (P S) c sequence, thus there exists a function v ∈ W 1,p (R N ) such that v n ⇀ v weakly in W 1,p (R N ).We make the following cases: Case I: When v = 0