Ground State Solution of Pohožaev Type for Quasilinear Schrödinger Equation Involving Critical Exponent in Orlicz Space

We study the following quasilinear Schrödinger equation involving critical exponent - Δ u + V ( x ) u - Δ ( u 2 ) u = A ( x ) | u | p - 1 u + λ B ( x ) u 3 N + 2 N - 2 , u ( x ) > 0 for x ∈ R N and u ( x ) → 0 as | x | → ∞ . By using a monotonicity trick and global compactness lemma, we prove the existence of positive ground state solutions of Pohožaev type. The nonlinear term | u | p - 1 u for the well-studied case p ∈ [ 3 , 3 N + 2 N - 2 ) , and the less-studied case p ∈ [ 2 , 3 ) , and for the latter case few existence results are available in the literature. Our results generalize partial previous works.


Introduction and Main Results
In this paper, we consider the following quasilinear Schrödinger equation where N ≥ 3, 22 * := 2 × 2 * = 4N N−2 , 1 < p < 22 * − 1, λ > 0. The solutions of Equation (1) are related to the existence of standing waves of the following quasilinear elliptic equations i∂ t z = −∆z + V(x)z − l(|z| 2 )z − k∆g(|z| 2 )g (|z| 2 )z, x ∈ R N , where V is a given potential, k ∈ R, l and g are real functions. Quasilinear Equation (2) has been derived as models of several physical phenomena (see e.g., [1][2][3] and the references therein). In recent years, extensive studies have been focused on the existence of solutions for quasilinear Schrödinger equations of the form One of the main difficulties of Equation (3) is that there is no suitable space on which the energy functional is well defined and belongs to C 1 -class except for N = 1 (see [4]). In [5], for pure power nonlinearities, Liu and Wang proved that Equation (3) has a ground state solution by using a change of variables and treating the new problem in an Orlicz space when 3 ≤ p < 22 * − 1 and the potential V(x) ∈ C(R N , R) satisfies V(x) ≥ a > 0 and for each M > 0, meas{x ∈ R N | V(x) ≤ M} < +∞.
To the best of our knowledge, there is no result in the literature on the existence of positive ground state solutions of Pohožaev type to the problem in Equation (1) with critical term. The first purpose of the present paper is to prove the existence of positive ground state solutions of Pohožaev type to the problem in Equation (1) with critical term. Since the approaches in [5,7,8,13], when applied to the monomial nonlinearity f (u) = |u| p−1 u, are only valid for p ∈ [3, 22 * − 1), we want to provide an argument which covers the case p ∈ [2, 3) and this is the second purpose of the present paper. Moreover, our argument does not depend on existence of the Nehari manifold.
Before state our main results, we make the following assumptions.
It is worth noting that the similar hypotheses on V(x) as above (V 1 ) and (V 2 ) are introduced in [14][15][16] and have physical meaning. Moreover, there are indeed many functions satisfying (V 1 ) and 1+|x| . Under conditions analogous to (A), (B), Zhao and Zhao [17] obtained the positive solutions of Schrödinger-Maxwell equations with the case p ∈ (2, 2 * ).
Our main result reads as follows. Theorem 1. Let V(x), A(x) abd B(x) be positive constants. If λ > 0 is sufficiently large, then the problem in Equation (1) has a positive ground state solution for N ≥ 3, 1 < p < 22 * − 1.

Remark 2.
The novelty of this works with respect to some recent results is that we treat the existence by using Pohožaev manifold method in an Orlicz space. The idea of Pohožaev manifold has been used in [8,12], where the authors studied problems with subcritical nonlinearity. It is worthy noting that their argument cannot be applied to our problem due to the presence of the critical term.
The rest of the paper is organized as follows. In Section 2, we state the variational framework of our problem and some preliminary results. The proof of Theorem 1 is contained in Section 3. Section 4 is devoted to establishing a global compactness lemma and proving Theorem 2.

Preliminaries and Functional Setting
. We formally formulate the problem in Equation (1) in a variational structure as follows for u ∈ H 1 (R N ). From a variational point of view, J is not well defined in H 1 (R N ), which prevents us from applying variational methods directly. To overcome this difficulty, we employ an idea from Colin and Jeanjean [19]. First, we make a change of variables Thus, after the above change of variables, we can write the functional J(u) as Under the assumptions (V 1 ), (V 2 ), (A) and (B), I V is well defined and I V ∈ C 1 (E, R) on the Orlicz space ( [20]) and for any w ∈ E. Moreover, if v is a critical point for the functional I V , then v is a solution for the equation Therefore, u = f (v) is a solution of the problem in Equation (1) ( [19]).
Next, we prove a Pohožaev identity with respect to the problem in Equation (7), which plays a significant role in constructing a new manifold.
Proof. We only prove it formally. For any given positive constant R, B R = {x ∈ R N | |x| < R}. Let u i := ∂u ∂x i and n be the unit outer normal at ∂B R . By the divergence theorem, we have Next, by using div and the divergence theorem By Equations (9) and (10), one has Note that u is a solution of Equation (1); it follows from integration by parts that We get by Equations (7) and (12) that Next, we show that the right hand side of Equation (14) converges to 0 for at least one suitably chosen sequence R n → +∞. Since there exists a sequence R n → +∞ such that therefore, Φ(R) would not be in L 1 (0, +∞), which contradicts Equation (15), implying that i.e., The proof is finished.
In particular, if V(x), A(x), B(x) are positive constant V, A, B, the above-mentioned Pohožaev identity can be rewritten as follows Lemma 4. The functional I is not bounded from below on E.
Lemma 4 means that we can not obtain the boundedness of the (PS) sequence by usual method. We need to consider a constrained minimization on a suitable manifold.
To give the definition of such a manifold, we need the following lemma.
Then, h has a unique critical point which corresponds to its maximum.
Proof. For large enough λ > 0 such that a 4 λ − a 2 + a 3 > 0, consider derivatives of h : Note that h (t) → −∞ as t → +∞ and is positive for t > 0 small since N ≥ 3. Then, there exists t > 0 such that h (t) = 0. The uniqueness of the critical point of h follows from the fact that the equation has a unique positive solution Motivated by [8], we introduce the following Pohožave manifold where P(v) is defined by Equation (16).
Proof. For every v ∈ E\{0} and t > 0, keeping the definition of v t in mind. Denote By Lemma 5, we have that χ has a unique critical pointt > 0 corresponding to its maximum, i.e., which implies that P(vˆt) = 0 and vˆt ∈ M.
Lemma 7. The M is a natural C 1 manifold and every critical point of I| M is a critical point of I in H 1 (R N ).
Proof. By Lemma 6, it is easy to check that M = ∅. The proof consists of four steps. Step Note that, for any v ∈ M, using Lemma 1, Sobolev embedding inequality and choosing a number ρ > 0, then there exist r > 0, C 1 and C 2 > 0 such that for ρ small enough and λ > 0, so that M, ∂M ⊂ E\B ρ (0).
The equation P (v) = 0 can be written as and v satisfies the following Pohožaev identity We then obtain From above system, we have 2(N − 2)α = 0, then α = 0 since N ≥ 3, which is a contradiction. Thus, P (v) = 0 for any v ∈ M. This completes the proof of Step 2.
Step 3. Every critical point of I| M is a critical point of I in E.
If v is a critical point of I| M , i.e., v ∈ M and (I| M ) (v) = 0. Thanks to the Lagrange multiplier rule, there exists ρ ∈ R such that I (v) = ρP (v). We prove that ρ = 0. Firstly, in a weak sense, the equation and v satisfies the following Pohožaev identity Using notations α, β, γ and θ as in Step 3, we obtain that It is deduced from the above equations that If ρ = 0, then α = 0 since N ≥ 3, which is impossible. Therefore, ρ = 0 and I (u) = 0.
Proof. We use an idea from [22].
. It follows from the Hölder and Sobolev inequalities that Covering R N by a family of balls {B R (y i )} such that each point is contained in at most k such balls and summing up these inequalities over this family of balls we obtain Under the assumption of the lemma,

Ground State of Equation (1) with Constant Coefficient
In this section, we study the existence of positive ground state solutions of Pohožaev type to Equation (1) with constant coefficient. Proof. Inspired by [8], we divide the proof into three steps.
Step 1. Let {v n } ⊂ M be a sequence such that I(v n ) → inf M I. We claim that {v n } is bounded.
Step 2. Since {v n } is bounded in E, passing to a subsequence, we may assume v n v in E, v n v in L s (R N ) for 2 ≤ s ≤ 22 * . We prove that v ∈ M and v n → v in E. Thus, I| M attains its minimum at v. By Lemma 2, we get that Using the Ekeland's Variational Principle in Ekeland [23], we can assume that I(v n ) → inf M I and Arguing by a contradiction, supposing that which is a contradiction. Then, R N (|∇v n | 2 + V f 2 (v n )) = lim inf n→∞ R N (|∇v n | 2 + V f 2 (v n )) and P(v) = lim inf n→∞ P(v n ) = 0. Therefore, v ∈ M and v n → v in E.
Step 3. We now show that I (v) = 0. Thanks to the Lagrange multiplier rule, there exists τ ∈ R so that I (v) = τP (v) = 0. As in the proof of Step 4 in Lemma 7, we can prove that τ = 0. Thus, Proof of Theorem 1. For N ≥ 3 and large enough λ > 0, it is deduced from Lemma 10 that there exists v ∈ M such that I(v) = inf I| M and I (v) = 0. Then, v is a nontrivial critical point of I| M . Hence, by Lemma 7, the v is a nontrivial ground state solution of (7) Furthermore, it is easy to see that |u| is also a ground state solution of Equation (1) since the functional I(v) and P(v) are even. Therefore, we may assume that such a ground state solution does not change sign, i.e. u ≥ 0. The strong maximum principle and standard arguments [24] imply that u(x) > 0 for all x ∈ R N and the proof is completed.

Ground State of Equation (1) with Nonconstant Coefficient
In this section, we investigate Equation (1) in the case that V(x), A(x) and B(x) are nonconstant. A starting point is the following lemma. Lemma 11. ( [25]) Let (X, · ) be a Banach space and T ∈ R + be an interval. Consider a family of C 1 functionals on X of the form Φ δ (u) = C(u) − δD(u), for all δ ∈ T, with D(u) ≥ 0 and either C(u) → +∞ or D(u) → +∞, as u → ∞. Assume that there are two points v 1 , v 2 ∈ X such that Then, for almost every δ ∈ T, there is a bounded (PS) c δ sequences in X.
For δ ∈ [ 1 2 , 1], we consider the functional I V,δ : E → R defined by where We also need to consider the associated limit problem (QS) ∞ It is clear that (QS) ∞ is the Euler-Lagrange equations of the functional The following lemma ensures that I V,δ has the mountain pass geometry with the corresponding mountain pass level denoted by c V,δ . Proof.
Taking v = v t for t large, this shows at once that I V,δ (v) ≤ I ∞,δ (v) < 0.
Lemma 12 means that, if I V,δ (v) satisfies the assumptions of Lemma 11 with X = E and Φ δ = I V,δ , we then obtain immediately, for a.e. δ ∈ [ 1 2 , 1], there exists a bounded sequence {u n } ⊂ E such that Introduce the following manifold According to Section 3, M ∞,δ (v) has some similar properties to those of the manifold M, such as containing all the nontrivial critical points of I ∞,δ (v). Lemma 14. If N ≥ 3 and δ ∈ [ 1 2 , 1], m ∞,δ is obtained at some v ∞,δ ∈ M ∞,δ . Moreover, Proof. The proof is similar to that of Theorem 1, and is omitted here.
Step 2. We prove that I V,δ (v 0 ) ≥ 0. From (V 2 ) and N ≥ 3, we deduce that Step 3. Set w 1 n = v n − v 0 , then we get w 1 n 0 in E. Let us define Vanishing: If µ = 0, then it follows from Lemma 8 that in L s (R N ) for s ∈ (2, 22 * ). By I V,δ (v 0 ) = 0 and Fatou's Lemma, we have which means that w 1 n → 0. Non-vanishing: If µ > 0, we can find a sequence {y 1 n } ⊂ R N such that wherew 1 n = w 1 n (· + y 1 n ). Note that w 1 n = w 1 n (· + y 1 n ) , we see that {w 1 n } is bounded. Going if necessary to a subsequence, we have a v 1 ∈ E such thatw 1 we see that v 1 = 0. Moreover, w 1 n 0 in E implies that |y 1 n | → +∞. Next, we prove that I ∞,δ (v 1 ) = 0. Similar to the proof of Step 1, for any fixed ϕ ∈ C ∞ 0 (R N ), it suffices to show that I ∞,δ (w 1 n ), ϕ → 0. By (V 1 ), (A), (B) and |y 1 n | → +∞, as n → ∞, we have that Since w 1 n 0 in E, one has that as n → ∞. Thus, using Equations (34)-(37), one has I ∞,δ (w 1 n ), ϕ → 0. Therefore, I ∞,δ (v 1 ) = 0. In the following, we prove that and Firstly, we claim that the relation below holds: We have by ( f 2 ) and ( f 3 ) of Lemma 1 that Thus, { f (w 1 n )} is bounded in E and f (w 1 n ) ∈ L l (R N ). Because of the local compactness of the Sobolev embedding theorem, we have, up to a subsequence, f (w 1 n ) → f (v 0 ) almost everywhere on R N . Then, the conclusion follows from the Brrézis-Lieb Lemma. This implies that Equation (40) holds. Using similar arguments above, for any ϕ ∈ C ∞ 0 (R N ), we also obtain In addition, by Lemma 9, we have Now, from Equations (40) and (43), we know that Equation (38) holds. We deduce from Equations (20) and (22) that It is deduced from Equations (40)-(44) that Equation (39) holds.

Similar to the proof in
Step 2 of Lemma 16, we obtain that I ∞,δ (v 1 ) ≥ 0. Then, we get from Equation (30) that Repeating the same type of arguments explored in Step 3, set If vanishing occurs, then w 2 n → 0 in E. Thus, Lemma 16 holds with j = 1. If w 2 n is non vanishing, then there exists a sequence {y 2 n } and v 2 ∈ E such thatw 2 n = w 2 n (· + y 2 n ) v 2 in E and I ∞,δ (v 2 ) = 0.
V(x) f 2 (v n ),b n := As in the proof of Step 2 of Lemma 16, we can see that every critical point of I V has nonnegative energy. Thus, 0 ≤ m V ≤ I V (v 0 ) < c V,1 < +∞. Let {v n } be a sequence of nontrivial critical points of I V satisfying I V (v n ) → m V . Since I V (v n ) is bounded, using the similar arguments as Equation (49), we can conclude that {v n } is bounded (PS) m V sequence of I V . Similar arguments in Lemma 17, there exists a positive and nontrivial v * ∈ E such that I V (v * ) = m V , which implies that u * = f (v * ) is a ground state solution for Equation (1). By strong maximum principle, u * = f (v * ) is a positive ground state solution for Equation (1). The proof is complete.

Discussion
Our results generalize partial results in Xu and Chen [8] and Zhao and Zhao [16]. The case of p ∈ [1, 2) is still unknown, which can be a problem for further study.