A study of biharmonic equation involving nonlocal terms and critical Sobolev exponent

In this paper, we investigate the existence of ground state solutions and non-existence of non-trivial weak solution of biharmonic equation with some nonlocal terms and critical Sobolev exponent. Firstly, we prove the non-existence by establishing Pohozaev type of identity. Next, we study the existence of ground state solutions by using the minimization method on the associated Nehari manifold.

The case when α = 0, (1.1) becomes the biharmonic Choquard equation In the last few decades, the Choquard equation has received a great attention and has been appeared in many different contexts and settings(see [1,3,11,16,17]).The following Choquard or nonlinear Schrödinger-Newton equation was first considered by Pekar [18] in 1954 for N = 3.In 1996, Penrose had used the equation (1.3) as a model in self-gravitating matter(see [19], [20]).The stationary Choquard equation arises in quantum theory and in the theory of Bose-Einstein condensation.
Equations involving biharmonic operator arise in many real life phenomena such as in biophysics, continuum mechanics, differential geometry and many more.For example, in the modeling of thin elastic plates, clamped plates and in the study of Paneitz-Branson equation and the Willmore equation(see [10]).As we could not apply maximum principle for the biharmonic operator which makes the problems involving biharmonic operator even more interesting from the mathematical point of view(see [4,12,21,22,23,24]) In the recent years, the biharmonic equations has received a considerable attention.In [7], Cao and Dai studied the following biharmonic equation with Hatree type nonlinearity ∆ 2 u = |x| −8 * |u| 2 |u| b , for all x ∈ R d , where 0 < b ≤ 1 and d ≥ 9.The authors applied the methods of moving plane and proved that the non-negative classical solutions are radially symmetric.The authors also studied the non-existence of non-trivial non-negative classical solutions in subcritical case 0 < b < 1.
Micheletti and Pistoiain [14] has investigated the following problem where Ω is smooth bounded domain in R N .The authors obtained the multiple non-trivial solutions by using the mountain pass theorem.Existence of infinitely many sign-changing solutions of the above problem had been studied by Zhao and Xu in [25] by the use of critical point theorem.
In this research article, we investigate the existence of ground state solutions and nonexistence of non-trivial weak solution to (1.1).Now, in the subsection below we provide some notations which we will using throught this paper, variational framework and main results.

Notations.
In this paper, we will be using the following notations.
is the Hilbert-Sobolev space endowed with the inner product and norm • L t (R N ) denotes the usual Lebesgue space in R N of order s ∈ [1, ∞] whose norm will be denoted by ||.|| t .

Note:
The embedding Variational Framework.
• We will require the following Hardy-Littlewood-Sobolev inequality ) and • We could easily notice that the equation (1.1) has variational structure.Now, let us define the energy functional (1.7) By using (1.5) and (1.6) together with the Hardy-Littlewood-Sobolev inequality (1.4) we get that the energy functional F α is well defined and . Also, solution of (1.1) is a critical point of the energy functional F α .

Main Results.
Non-existence (p < p * γ ) Firstly, we study the non-existence of non-trivial weak solution to (1.1).By weak solution of (1.1) one could understand that there exists u ∈ H 2 0 (R N ), u = 0 and ).Now, we present our main result on non-existence of non-trivial weak solution.
Next, we investigate the existence of ground state solutions for the equation (1.1).
Existence (p = p * γ ) Define the Nehari manifold associated with the energy functional F α by and the ground state solutions will be obtained as minimizers of We present the main result on ground state solutions.
Theorem 1.2.Assume that N ≥ 5, 2θ < γ, p θ > p * γ > 1 and α > 0. Further, if p θ satisfies (1.5), then the equation (1.1) has a ground state solution u ∈ H 2 0 (R N ).Now, in the Section 2 , we will be collecting some preliminary results and then it will be followed by Section 3 and 4 which consists of the proofs of our main results.

Preliminary results
Lemma 2.1.([13, Lemma 1.1], [15, Lemma 2.3]) There exists a constant C 0 > 0 such that Lemma 2.2.([6, Proposition 4.7.12])Let 1 < t < ∞ and assume that (z n ) is a bounded sequence in L t (R N ) which converges to z almost everywhere.Then, z n converges weakly to ) which converges to z almost everywhere.Then, for every 1 ≤ q ≤ t, we have Proof.Let us fix ε > 0, then there exists a constant C(ε) > 0 such that for all c,d ∈ R, one could have Next, using (2.1), we get Further, using the Lebesgue Dominated Convergence theorem, we obtain Thus, we find that We finish the proof by letting ε → 0.
Proof.Let us take q = c = p θ , t = 2N 2N −θ in Lemma 2.3, then we get as n → ∞.By Lemma 2.2, we have Next, we use the Hardy-Littlewood-Sobolev inequality (1.4) to obtain On the other hand (2.6) Finally, passing to the limit in (2.6) and using (2.3)-(2.4), the result holds.Similarly, we prove the case when c = p * γ .
Lemma 2.5.Let us assume that N ≥ 5, θ ∈ (0, N ) and Proof.We will prove the lemma for c = p θ and the method will be similar to prove it for the the second case, that is, c = p * γ .Let us assume that h = h + − h − and v n = u n − u.We only require to prove the lemma for h ≥ 0. We use Lemma 2.3 with q = c = p θ and t = 2N 2N −θ together with (z n , z) = (u n , u) and (z n , z) = (u n h 1/c , uh 1/c ) respectively in order to obtain Now, we use the Hardy-Littlewood-Sobolev inequality to get (2.7) Also, Lemma 2.2 yields Then, the combination of (2.7) and (2.8) will give us (2.9) Next, we use the Hardy-Littlewood-Sobolev inequality together with Hölder's inequality in order to find (2.10) Furthermore, by Lemma 2.2 we have v Therefore, using (2.10) we obtain On the other hand, we notice that (2.12) Finally, by passing to the limit in (2.12) together with (2.9) and (2.11) we get the desired result.

Proof of Theorem 1.1
In order to prove this theorem, we establish the following Pohozaev type of identity.Proposition 3.1.Assume that u ∈ H 2 0 (R N ) is a solution of (1.1).Then, we have Proof.Let us define the cut-off function by where are smooth functions.The functions φ κ (x) and ψ δ (x) satisfy the following properties: • suppφ = (1, ∞) and suppψ = (−∞, 2).
Also, u is a smooth function away from the origin (see [10]) and (x.∇u)ϕ κ,δ ∈ C 3 c (R N ).Next, we multiply the equation (1.1) by (x.∇u)ϕ κ,δ and we obtain (3.2) Following the similar approach as in [5], we get On the other hand, we have

Completition of Proof of Theorem 1.1
Since, u is a solution of (1.1), we also have Now, we use the Hardy-Littlewood-Sobolev inequality together with the fact that the embeddings The above inequality is imposssible as 2N −θ−p θ (N −4) > 0 and on the other hand p < p * γ .This concludes our proof.Now, we look into the ground state solutions to (1.1) in the following section.
4 Proof of Theorem 1.2 Our proof will be relying on the analysis of the Palais-Smale sequences for F α | Nα .We will follow the ideas from [8,9] in order to prove that any Palais-Smale sequence of F α | Nα is either converging strongly to its weak limit or differs from it by a finite number of sequences, which are the translated solutions of (1.2).We will be using several nonlocal Brezis-Lieb results which we have presented in Section 2. Assume α > 0. For u, v ∈ H 2 0 (R N ) we have Also, we have for some s > 0.
Since p θ > p * γ > 1, this gives us that the equation F ′ α (su), su = 0 has a unique positive solution s = s(u), also known as the projection of u on N α .Now, we will discuss the main properties of the Nehari manifold N α : Proof.(i) One could notice that We conclude the proof by taking c (ii) We will be using the Hardy-Littlewood-Sobolev inequality and the fact that the embeddings Hence, there exists some constant C 0 > 0 such that Next, we use the fact that F α | Nα is coercive together with(4.1) in order to obtain . Now, for any u ∈ N α we use (4.1) to get Now, let us assume that u ∈ N λ is a critical point of F α in N α .Using the Lagrange multiplier theorem, we obtain that there exists λ ∈ R such that F ′ α (u) = λG ′ (u).Hence, we get that F ′ α (u), u = λ G ′ (u), u .Next, because G ′ (u), u < 0, which yields λ = 0 and further, we get F ′ α (u) = 0.
Lemma 4.3.Assume that the sequence (u n ) is a (P S) sequence for F α | Nα .Then (u n ) is a (P S) sequence for F α .
Proof.Let us assume that (u n ) ⊂ N α be a (P S) sequence for F α | Nα .Since, , for some λ n ∈ R, which further gives us Next, by the use of (4.2), we deduce that λ n → 0 and this yields F ′ λ (u n ) → 0.

A result on compactness
Let us define the energy functional R : Consider the corresponding Nehari manifold for R by And we have, In such as case, there exists a solution u ∈ H 2 0 (R N ) of (1.1) such that, on replacing the sequence (u n ) with the subsequence, we have either of the following alternatives: (i) u n → u strongly in H 2 0 (R N ); or (ii) u n ⇀ u weakly in H 2 0 (R N ).Also, there exists a positive integer l ≥ 1 and l nontrivial weak solutions to (1.2), that is, l functions u 1 , u 2 , . . ., u l ∈ H 2 0 (R N ) and l sequences of points (q n,1 ), (q n,2 ), . . ., (q n,l ) ⊂ R N such that the following conditions hold: Proof.As we know that (u n ) ∈ H 2 0 (R N ) is a bounded sequence, then there exists u ∈ H 2 0 (R N ) such that, up to a subsequence, one could have We use (4.3) together with Lemma 2.5 and get which further yields that, u ∈ H 2 0 (R N ) is a solution of (1.1).Next, if u n → u strongly in H 2 0 (R N ) then (i) holds.Suppose that (u n ) ∈ H 2 0 (R N ) does not converge strongly to u and define e n,1 = u n − u.In this case (e n,1 ) converges weakly (not strongly) to zero in H 2 0 (R N ) and Next, we use Lemma 2.4 in order to obtain By combining (4.4) and (4.5) we get Further, by the use of Lemma 2.4 we have ), e n,1 + o(1) = R ′ (e n,1 ), e n,1 + o(1).
On the other hand, we also have Hence, we could find q n,1 ∈ R N such that Therefore, for any sequence (e n,1 (• + q n,1 )), there exists u 1 ∈ H 2 0 (R N ) such that, up to a subsequence, one could have e n,1 (• + q n,1 ) → u 1 a.e. in R N .Now, we pass to the limit in (4.9) and get which yields, u 1 ≡ 0. As (e n,1 ) ⇀ 0 weakly in H 2 0 (R N ), one could obtain that (q n,1 ) is unbounded.Next, passing to a subsequence, we get that |q n,1 | → ∞.Further, using (4.8), we have R ′ (u 1 ) = 0, which implies that u 1 is a nontrivial solution of (1.2).Let us define e n,2 (x) = e n,1 (x) − u 1 (x − q n,1 ).
Then, in the same manner as before, we have + o(1).
By using Lemma 2.4 we get .
By (4.6), we get Again, we use the same approach as above and get Next, if (e n,2 ) → 0 strongly, then by taking l = 1 in the Lemma 4.4 we could finish our proof.
Let us assume that e n,2 ⇀ 0 weakly (not strongly) in H 2 0 (R N ) and one could iterate the process and in l number of steps one could find a set of sequences (q n,j ) ⊂ R N , 1 ≤ j ≤ l with |q n,j | → ∞ and |q n,i − q n,j | → ∞ as n → ∞, i = j and l nontrivial solutions u 1 , u 2 , . . ., u l ∈ H 2 0 (R N ) of (1.2) such that, by letting e n,j (x) := e n,j−1 (x) − u j−1 (x − q n,j−1 ) , 2 ≤ j ≤ l, we obtain e n,j (x + q n,j ) ⇀ u j weakly in H 2 0 (R N ) and R(u j ) + R(y n,l ) + o(1).
Since, F α (u n ) is bounded and R(u j ) ≥ b R , by iterating the process a finite number of times, we get the desired result.Proof.Suppose that (u n ) is a (P S) d sequence of F α in N α .Next, using the Lemma 4.4 we obtain R(u j ) ≥ b R .Then, upto a subsequence u n → u strongly in H 2 0 (R N ), which further yields that u is a solution of (1.1).
To complete the proof of Theorem 1.2, we require the following result.Proof.Suppose that ground state solution of (1.2) is denoted by T ∈ H 2 0 (R N ) and such solution exists (see [2] and references therein).Assume that sT is the projection of T on N α , that is, s = s(T ) > 0 is the unique real number such that sT ∈ N α .As, T ∈ N R and sT ∈ N α , we get −γ * |u| p |u| p−2 u(x.∇u)ϕ κ,δ dx = − 2N − γ 2p R N |x| −γ * |u| p |u| p .(3.5) Next, passing to limit in (3.2) and using (3.3), (3.4) and (3.5), we get the result.
is bounded from below by a positive constant.

Lemma 4 . 5 .
If any sequence is a (P S) d sequence of F α | Nα , then it is relatively compact for any d ∈ (0, b R ) .