Hamiltonian elliptic system involving nonlinearities with supercritical exponential growth

: In this paper, we deal with the existence of nontrivial solutions to the following class of strongly coupled Hamiltonian systems:


Introduction
In the literature, the existence of nontrivial solutions for strongly coupled Hamiltonian systems has been extensively studied by many authors [3, 6, 15-18, 20, 25, 29, 34, 38-41]. A Hamiltonian system is a mathematical expression of the following form: x ∈ Ω, −∆v = H u (x, u, v), x ∈ Ω, (1.1) where Ω is a smooth domain in R N , N ≥ 2, and H(x, u, v) is a nonlinear function.
In the case N ≥ 3. A classical model for H is given by H(x, u, v) = |u| p+1 /(p + 1) + |v| q+1 /(q + 1) and the maximal growth for the exponents p and q are related to the curve [17,20,29]: If the couple (p, q) lies on (1.2), some features of noncompactness arises, this motivates one to name (1.2) as the critical hyperbola, and we say that the nonlinearities H v = |v| q−1 v and H u = |u| p−1 u possess critical growth; alternatively, if the couple (p, q) is below (1.2) the growth of the nonlinearities are denominated subcritical. We want to point out that the critical hyperbola results from the borderline between existence and nonexistence of solutions for (1.1) (see [6]). In the case when N = 2, the critical hyperbola is not defined. Notice that, if Ω is a bounded domain in R N , the Sobolev embeddings state W 1,2 0 (Ω) ⊂ L q (Ω) for all 1 ≤ q ≤ 2 * = 2N/(N − 2) for N ≥ 3. In dimension N = 2, one has 2 * = +∞ and W 1,2 0 (Ω) L ∞ (Ω). Therefore, H u and H v may have any arbitrary polynomial growth. It was shown independently by Yudovich [44] , Pohožaev [32], and Trudinger [43] that the growth is of exponential type. More precisely, e αu 2 ∈ L 1 (Ω) for all u ∈ H 1 0 (Ω) and α > 0. Furthermore, Moser [30] proved the existence of a positive constant C = C(α, Ω) such that α ≤ 4π, = +∞, α > 4π. (1.3) From now on, the estimate of the type (1.3) will be referred to as the Trudinger-Moser inequality. These inequalities have been extended in many directions (see [3,10,12,14,23,25,28,31,36,42] among others). The above results motivate us to say that the function f has subcritical exponential growth if lim s→+∞ f (s) e αs 2 = 0, for all α > 0, and critical exponential growth if there exists α 0 > 0 such that α > α 0 , +∞, α < α 0 . (1.4) Nonlinear equations considering nonlinearities involving subcritical and critical exponential growth were treated by Adimurthi [1], Adimurthi-Yadava [2], de Figueiredo, Miyagaki, and Ruf [19] (see also [10,12,24,31,35]). We recall that a nonlinear equation in a domain Ω ⊂ R N with N ≥ 3 a classical assumption on the nonlinearity is given by | f (s)| ≤ c(1 + |s| q−1 ) , with 1 < q ≤ 2 * = 2N/(N − 2) (see [5,7,8,11,26,27] among others). If there exist positive constants k and s 0 such that g 1 (s) ≤ g 2 (ks) for s ≥ s 0 , we shall write g 1 (s) ≺ g 2 (s). Additionally, we shall say that g 1 and g 2 are equivalent and write g 1 (s) ∼ g 2 (s) if g 1 (s) ≺ g 2 (s) and g 2 (s) ≺ g 1 (s). Therefore, f possesses critical exponential growth if and only if f (s) ∼ e |s| 2 . The existence of a nontrivial solution of the system (1.1) under H v ∼ e v 2 and H u ∼ e u 2 and considering H 1 0 (Ω) × H 1 0 (Ω) as the setting space was proved by de Figueiredo, doÓ, and Ruf [18]. Now, we recall some facts about Lorentz-Sobolev spaces. Let 1 < r < +∞, 1 ≤ s < +∞ and Ω subset of R N , the Lorentz space L r,s (Ω) is the collection of all measurable and finite almost everywhere functions on Ω such that φ r,s < +∞, where where φ * denotes the spherically symmetric decreasing rearrangement of φ. In addition, if Ω is an open bounded domain in R N , the Lorentz-Sobolev space W 1 0 L r,s (Ω) is defined to be the closure of the compactly supported smooth functions on Ω, with respect to the quasinorm u W 1 0 L r,s := ∇u r,s . Brezis and Wainger [9] proposed the following Trudinger-Moser inequality version on Lorentz-Sobolev spaces: If Ω be a bounded domain in R 2 and s > 1, then e α|u| s s−1 belongs to L 1 (Ω) for all u ∈ W 1 0 L 2,s (Ω) and α > 0. Furthermore, Alvino [4] proved the following refinement of (1.3), there exists a positive constant C = C(Ω, s, α) such that Ruf [34] showed that, if the setting space of the system (1.1) is given by the product space W 1 0 L 2,q (Ω) × W 1 0 L 2,p (Ω), the maximal growth of the nonlinearities can be considered like H u ∼ e |u| p and H v ∼ e |v| q with p, q > 1 satisfying 1 In analogy to (1.2), the curve (1.6) is called exponential critical hyperbola. The existence of solutions of the system (1.8) for p = q = 2 has been treated in many works [3,18,38,40,41] among others, and the case where (p, q) lies on the exponential critical hyperbola given by (1.6) was studied in [15,25,39]. Trudinger-Moser type inequalities for radial Sobolev spaces with logarithmic weights were considered by Calanchi and Ruf [12] . Denote by H 1 0, rad (B 1 , w), the subspace of the radially symmetric functions in the closure of C ∞ 0 (B 1 ) with respect to the norm Calanchi and Ruf [12] found that The above results represent an increase in the maximal growth of the exponential type. For λ = 1, the weight given by (1.7) allows us to consider double exponential growth, see [12,13,37] for more details. In this paper, we deal with the existence of solutions to the following Hamiltonian system: where w is given by (1.7) and B 1 denotes the unit open ball center at the origin in R 2 . In order to use variational methods, we consider an associated functional defined on the space H 1 0, rad (B 1 , w) × H 1 0, rad (B 1 , w), which allows us to have nonlinearities of the form f (u) ∼ e |u| 2/(1−γ) and g(v) ∼ e |v| 2/(1−γ) . We assume the following conditions on the nonlinerities f and g: ( uniformly on x ∈ B 1 . (H 5 ) There exist constants p > 2 and C p > 0 such that f (x, s) ≥ C p s p−1 and g(s) ≥ C p s p−1 , for all s ≥ 0, , which is a Hilbert space endowed with the inner product Additionally, we denote the dual space of E with its usual norm by E * . We say that (u, v) ∈ E is a weak solution of (1.8) if Under the assumption on f and g, we establish the Euler-Lagrange functional J : E → R defined by for all (u, v) ∈ E. Furthermore, using standard arguments [21] , J ∈ C 1 (E, R) and, for all (u, v), In particular, (u, v) ∈ E is a nontrivial weak solution of the system (1.8) if only if (u, v) ∈ E is a nontrivial critical point of the functional J. Next, we present our existence result for the system (1.8). First, observe that if γ = 0, then the system (1.8) is reduced to −∆u = g(x, v) and −∆v = f (x, u) with Dirichlet conditions, and the growth of the functions are given by f (x, u) ∼ e |u| 2 and g(x, v) ∼ e |v| 2 uniformly on x ∈ B 1 , whose existence of nontrivial weak solutions was found in [18]. In the case for γ > 0, the nonlinearities under the assumption (H 4 ) behaves like f (x, u) ∼ e |u| p and g(x, v) ∼ e |v| q uniformly on x ∈ B 1 , where 1/p+1/q < 1 and p = q, that is, the pair (p, q) lies in the diagonal direction above the exponential critical hyperbola. Therefore, our result treats the Hamiltonian system (1.8) involving nonlinearities with supercritical exponential growth. Consequently, our result complements the works which study nonlinearities f (x, u) ∼ e |u| p and g(x, v) ∼ e |v| q for values where (p, q) lies under and on the the curve (1.6), that is, for nonlinerities that possess subcritical and critical exponential growth, respectively [15,18,25,[38][39][40][41].
The paper is organized as follows: Section 2 contains some preliminaries results and properties our setting space. In Section 3, we show that the Euler-Lagrange energy functional possesses the geometry of the linking theorem. In Section 4, it is established the finite-dimensional approximation and estimated the minimax level of the functional. Finally, in Section 5, we present the proof of Theorem 1.1.

Preliminaries
Let H 1 0,rad (B 1 , w) be the subspace of the radially symmetric functions in the closure of C ∞ 0 (B 1 ) with respect to the norm where w(x) = log 1/|x| γ and 0 ≤ γ < 1.
We note that H 1 0, rad (B, w) is a Hilbert space endowed with inner product Now, we state a compactness result.
Using the change of variable |x| = e −s , we obtain Therefore, there exists C > 0 such that Thus, H 1 0, rad (B 1 , w) → W 1,1 0 (B 1 ) continuosly, which implies the continuous and compact embedding [12]) Let w be the weight given by (1.7). Then, Lemma 2.4. Let (u n ) be a sequence in H 1 0, rad (B 1 , w) such that u n 0 in H 1 0, rad (B 1 , w) and u n = 1 for every n ∈ N. Then, for every 0 < α < α * γ , there exists a subsequence, still denoted by (u n ) such that Proof. Choosing > 0 such that α + < α * γ . We have the following limits: Thus, we can find C > 0 such that Using the Hölder inequality with r > 1 such that r(α + ) < α * γ and Proposition 2.3, we get By the weakly convergence of u n 0 in H 1 0, rad (B 1 , w) and Lemma 2.2 for a subsequence, we have

Linking geometry
This section is devoted to prove that the functional J possesses the geometry of the linking theorem. We start setting the following subspaces: In particular, we have Using the Cauchy-Schwarz inequality and (2.2), we obtain provided that u ≤ ρ 0 for some ρ 0 > 0 such that 4α 0 ρ 2 2−γ 0 < α * γ . A similar inequality holds for the function G. By Lemma 2.2, we obtain c 1 > 0 and c 2 > 0 such that Therefore, taking > 0 sufficiently small, we can choose ρ > 0 sufficiently small and σ > 0 such that J(z) ≥ σ, for all z ∈ ∂B ρ ∩ E + .

Finite-dimensional approximation
Observe that the leading part of the functional J is strongly indefinite, that is, J can assume positives and negatives values on infinite-dimensional subspaces of E. Therefore, we can not use the linking theorem. To deal with this inconvenience, we follow the arguments developed by de Figueiredo, doÓ, and Ruf [16], that is, we use a finite dimensional approximation.
Let e = u p ∈ H 1 0,rad (B 1 , w) be a nonnengative function with u p p = 1 where S p is attained. We consider {e i } i∈N a Hilbert basis of e ⊥ and setting Setting the following class of functions: where Q n = Q e ∩ H n , and set c n = inf γ∈Γ n max z∈Q n J(γ(z)). Now, let J n be the restriction of J to the finite-dimensional space H n . Moreover, Lemmas 3.1 and 3.2 are still valid for J n . Additionally, it follows from [16] that for ρ given by Lemma 3.1. Moreover, Lemma 3.1 and (4.2), implies that c n ≥ σ > 0, for all n ≥ 1.
Using the fact that the identity map I n : Q n → H n belongs to Γ n and the fact that F and G are nonnegative functions, we obtain for each z = r(e, e) + (u, −u) ∈ Q n . Hence, c n ≤ R 2 1 , for all n ≥ 1. Next, this proposition follows from the linking theorem for J n (see [33]). where σ and R 1 > 0 are given by Lemmas 3.1 and 3.2, respectively, and for each (φ, ψ) ∈ H n .
Then, the sequence (u n , v n ) is bounded in E.
combined with the fact that |J(u n , v n )| ≤ d, we obtain From (H 2 ), we get Similarly for the function g, there exists M g > 0 such that From (4.10) and (4.11) in (4.9), we obtain We define V n = v n v n and U n = u n u n .
We finally obtain (u n , v n ≤ c + n (u n , v n ) which implies that (u n , v n ) ≤ c, for every n ∈ N, for some positive constant c.
Lemma 4.4. Assuming the conditions (H 1 )-(H 5 ), are hold. Let (u n , v n ) be a sequence in E and (u, v) ∈ E such that (u n , v n ) (u, v) weakly in E, J(u n , v n ) → c and J (u n , v n ) E * → 0. Then, Proof. From Lemma 2.2, we can suppose that u n converges to u in L 1 (B 1 ). By Proposition 2.3, and the assumptions (H 1 ) and (H 4 ), we imply that f (x, u n ) ∈ L 1 (B 1 ). Moreover, using J (u n , v n )(u n , v n ) = o n (1), we can find c > 0 such that According to [19, Lemma 2.10], we obtain the limit (i) . On the other hand, from (i), we obtain Therefore, there exists p ∈ L 1 (B 1 ) such that From (H 1 ) and (H 3 ), we obtain Using (4.22) and (4.23), we have Therefore, F(x, u n ) → F(x, u) in L 1 (B 1 ), which follows from Lebesgue's dominated convergence theorem.
Let recall that for p > 2, u p ∈ E denotes the nonnegative function such that u p p = 1 and Lemma 4.5. Suppose that f and g satisfy (H 1 ) − (H 5 ). Then, the following inequality holds: , v ∈ E and u p is given by (4.25). Then, Using condition (H 5 ), we have Since u p = S p and u p p = 1, we obtain Since the function λ(t) = t 2 S 2 p − 2C p t p p achieves its maximum on t 0 = S 2/(p−2) p C 1/(p−2) p and using the estimate Remark 4.6. By Lemma 4.5, there exists δ > 0 such that for every n ∈ N.

Proof of the Theorem 1.1
Let (u n , v n ) ∈ H n be the sequence given by Proposition 4.1. From Lemma 4.3, this sequence is bounded in E. Thus, up to a subsequence, we can assume that (u, v) ∈ E such that (u n , v n ) (u, v) weakly in E, for some (u, v) ∈ E. Taking (0, ψ) and (φ, 0) in (4.7), where φ and ψ belongs to C ∞ 0, rad (B 1 )∩ H n . Therefore, Taking the limit in (5.1) and (5.2) as n → ∞, by Lemma 4.4 and the density C ∞ 0, rad (B 1 ) ∩ n∈N H n in H 1 0, rad (B 1 , ω), we obtain Therefore, (u, v) ∈ E is a critical point of J. Now, we prove that (u, v) is nontrivial. Since the system (1.8) is strongly coupled, if we assume that u ≡ 0 we get that v ≡ 0. Therefore, by Lemma 2.2, up to a subsequence, we have u n → 0 and v n → 0 in L p (B 1 ), for all p ≥ 1 (5.5) and u n → 0 and v n → 0 almost everywhere in R 2 .
If we suppose that u n is not bounded below by a positive constant, we can get a subsequence of (u n ) such that u n → 0. Therefore, w(x)∇u n ∇v n dx → 0. (5.6) Considering the pairs of functions (φ, ψ) = (u n , 0) and (φ, ψ) = (0, v n ) in (4.7), we have Using Lemma 4.4 and (5.5), we have Thus, by the above limits, we get that J(u n , v n ) tends to zero which contradicts (4.5); consequently, u n is bounded below by a positive constant, in particular, we can assume that u n 0 for all n ∈ N. Now, taking (φ, ψ) = (0, u n ) in (4.7), we get We assume that max{α 0 , β 0 } = α 0 . Then, we can write Applying Lemma 4.2 with ), where δ > 0 is given by Remark 4.6. By (H 4 ) and the assumption α 0 ≥ β 0 , we can find C > 0 such that Replacing the above inequality in (5.13), we get Since v n → 0 almost everywhere in B 1 and g is bounded in B 1 \Y n for all n ∈ N (being independent of n), by the dominated convergence theorem, we get Since J n (u n , v n )(u n , v n ) = 0, we get

Conclusions
In this work, we apply variational methods to find a nontrivial solution for a Hamiltonian systems where the nonlinearities possess maximal growth related to Trudinger-Moser type inequalities. To the best of our knowledge, this is the first result to demonstrate the existence of nontrivial solutions for a Hamiltonian involving supercritical exponential growth in the sense of the exponential critical hyperbola in the literature. According to our definition of logarithmic weight, we restricted the domain to the unit ball. It is of interest to further our results to solutions for Hamiltonian systems involving supercritical exponential growth on the whole space R 2 .

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