SEMILINEAR ELLIPTIC PROBLEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN THE HALF-SPACE

. We perform an weighted Sobolev space approach to prove a Trudinger-Moser type inequality in the upper half-space. As applications, we derive some existence and multiplicity results for the problem under some technical condition on a , b and the the exponential nonlinearity f . The ideas can also be used to deal with Neumann boundary conditions.

1. Introduction and main results. In the paper [2], Alama-Tarantello considered the indefinite semilinear problem −∆u = µu + a(x)|u| p−2 u − b(x)|u| q−2 u, in Ω, u = 0, on ∂Ω, where Ω ⊂ R N is a bounded domain, 2 < p < q < 2 * := 2N/(N − 2) and a, b are nonnegative weight functions in L 1 (Ω). Under suitable conditions on the parameter µ, they proved existence, nonexistence and multiplicity of solutions according to the behavior of the competing terms a|u| p−2 u and b|u| q−2 u as determined by suitable integrability properties of the ratio a q /b p . After this, indefinite elliptic problem has been studied by many authors in bounded and unbounded domain. We specially quote the papers Rǎdulescu-Repovš [16] and Chabrowski [5], where it is considered the semilinear elliptic problem −∆u = λa(x)|u| p−2 u − b(x)|u| q−2 u, in Ω, u = 0, on ∂Ω, where 1 < p < 2 < q < 2 * and a, b ∈ L ∞ (Ω) are such that ess inf x∈Ω a(x) > 0 and ess inf x∈Ω b(x) > 0. The authors proved results of existence, nonexistence and multiplicity of solutions according the range of the parameter λ.
Roughly speaking, the growth condition (f 0 ) says that the problem has subcritical exponential growth with respect to some kind of Trudinger-Moser inequality. The main idea is looking for critical points of the associated energy functional with F (s) := s 0 f (t) dt and the domain of I is an appropriated Banach space E q . Notice that the L 2 -norm on the boundary does not appear in the definition of I. Actually, we shall prove that our working space E q is such that the first two terms in the above expression is exactly a half of the square of the norm, and therefore the functional behaves like that of the bounded domain case.
In our first result we prove the following: and q ≥ 2. If (a 1 ) there exist c 1 > 0 and β > 2 such that a(x) ≤ c 1 (1 + |x|) β , for all x ∈ R 2 + , then problem (P ) has a nonzero weak solution.
Condition (a 1 ) provides compactness for the Sobolev embedding of E q into some weighted Lebesgue spaces. This is important to recover compactness for the functional I, since we are dealing with a problem in the unbounded domain R 2 + . Another key point is the proof of an appropriated Trudinger-Moser inequality which guarantees that u → R 2 + a(x)F (u) is a well defined C 1 -functional. Actually, we present a general abstract framework to deal with problems with Robin (or Neumann) boundary condition in the upper half-space (see Section 2).
In our second result we suppose that a behaves like (1 + |x|) −2 . In this new setting, we argue as in [2] and recover compactness by assuming an integrability condition of a special function involving both a and h. More specifically, we prove the following: (h 1 ) the following holds then problem (P ) has a nonzero weak solution.
As it is natural, if we have symmetry we can obtain more and more solutions. So, in our last result we obtain multiple solutions for the problem: then problem (P ) has infinitely many weak solutions.
The study of elliptic problems involving Robin boundary condition has a wide literature, see [5,6,20] and references therein. Such kind of boundary conditions arise on some important biological models, see for example [8] for this particular aspect. Concerning the mathematical point of view, our main motivation comes from the papers [10,15], where the authors studied the quasilinear elliptic problem for an exterior domain Ω. Under certain conditions on the power p, q and the functions a, b, some results of existence, nonexistence and multiplicity are derived. It is worthwhile to mention that the Hardy type inequality (see [15,Lemma 1]) for 1 < p < N and some C 0 > 0 plays a fundamental role in their approaches. In order to prove this inequality it was fundamental to assume that there exist Here, we present a new Hardy type inequality (see Lemma 2.2) which allows us to consider h ≡ 1.
In the aforementioned papers the authors deal only with the Sobolev case N ≥ 3 (in the semilinear case i.e, p = 2). To the best of our knowledge, the borderline case N = 2 has not been considered, except when f is a pure power. Here, our main interest is to study problem (P ) with the nonlinearity f having exponential growth. One of the main difficult in our task relies on the fact that we can not use Schwarz Symmetrization as in [4,12,1], since is not possible to perform a reflection argument because the weight function a is not well defined in the whole space via the usual reflection. In order to overcome these difficulties we combine ideas of Kufner-Opic [13] with some arguments of Yang-Zhu [19] and the recent paper by DoÓ-Sani-Zhang [9].
We finally mention that, as a byproduct of the abstract framework introduced here, we are also able to deal with Neumann boundary conditions. More specifically, if we assume (a 0 ) − (a 1 ) and that the nonlinearity f satisfy the assumptions in Theorems 1.1 and 1.3, with some minor modifications we can obtain the same kind of results for the problem The paper is organized as follows: in the next section we prove a Hardy type inequality, we introduce the variational setting to deal with (P ) and also prove a Trudinger-Moser type inequality. In Section 3, as an application of the previous results, we obtain existence and multiplicity of solutions for problem (P ).
2. Variational framework. Throughout the paper we denote by C ∞ 0 (R 2 ) the set all functions with derivatives of any order and with compact support. If (X, · X ) is any normed vector space, R > 0 and a ∈ X, then B R (a) := {y ∈ X : y − a X < R} and B c R (a) := X \ B R (a). When a = 0, we write only B R and B c R , respectively. The points x ∈ R 2 will be written as x = (x 1 , x 2 ), with x 1 , x 2 ∈ R and B + R := B R (0) ∩ R 2 + . Finally, we denote by C, C 1 , C 2 , . . . , positive constants (possibly different).
Let C ∞ δ (R 2 + ) be the space of C ∞ 0 (R 2 )−functions restricted to R 2 + and define the weighted Sobolev space E as the completion of C ∞ δ (R 2 + ) with respect to the norm Given a positive function ω ∈ L 1 loc (R 2 + ) and p ≥ 1, we denote by L p ω the weighted Lebesgue space In our first results we establish some embedding results from E into weighted Lebesgue spaces. We start with following: Proposition 1. The weighted Sobolev embedding E → L p (1+x2) −2 is continuous for any 2 ≤ p < ∞. The same holds for the Sobolev trace embedding E → L p (∂R 2 + ). Proof. First notice that, for any 1 ≤ m < N , we can apply the Gagliardo-Nirenberg-Sobolev inequality and a suitable reflection argument to obtain a constant C = C(m, N ) > 0 such that where m * := N m/(N − m). Picking m = 1 and N = 2, we obtain where we have used (1 + x 2 ) −1 ≤ 1, for x 2 > 0. Using Young's inequality in the last term above we obtain In particular, since 2 < 3 < 4, one has E → L 3 (1+x2) −2 . Thus, applying (2.1) with v = (1 + x 2 ) −2 |u| 3 and repeating the argument, we get Iterating this process with k = 4, 5, . . . , one has E → L 2k (1+x2) −2 . Hence, given p ≥ 2, it is sufficient to choose k ≥ 2 such that 2 < p < 2k and use interpolation as above to conclude that the embedding E → L p (1+x2) −2 is continuous. Now we will prove the trace embedding. In order to do that we compute, for p ≥ 2 fixed, Integrating, using Hölder's inequality together and (1 + Since 2(p − 1) ≥ 2, we obtain from the first part of the proof that which completes the proof of Proposition 1.

Remark 1.
The above embeddings fail if p = ∞. In fact, considering the function In what follows we show that (1 + x 2 ) −2 can de replaced by a large class of weights. Essentially, it is important the decay rate of the weight at infinity. If it is suitable, we are also able to obtain compactness properties. This will be useful for our applications in the next section.
Then the Sobolev embedding E → L p ω is continuous for any 2 ≤ p < ∞. Moreover, the embedding is compact if β > 2.
Proof. The first statement is a direct consequence of the inequality ( for any k ∈ N. On the other hand, since the restriction operator Since ε > 0 is arbitrary, the above expression and (2.2) imply that, along a subsequence, u k → 0 strongly in L p (1+|x|) −β . It follows from (a 1 ) that u k → 0 strongly in L p ω and the lemma is proved. A fundamental result in the context of this paper regards on a weighted Hardytype inequality. This is the subject of next lemma (see [14] for a similar result in dimension N ≥ 3).

Lemma 2.2. The following inequality holds
Proof. If u ∈ C ∞ 0 (R 2 ) and t ∈ R, we have Since the normal unit vector pointing out of ∂R 2 + is (0, −1), we can use the Divergence Theorem to get Combining the above expression and (2.3), we obtain It follows from (2.4) that B 2 − 4AC ≤ 0, and therefore The above inequality and  defines on E a norm which is equivalent to · .
Proof. If u ∈ C ∞ 0 (R 2 ), it follows from Lemma 2.2 that u 2 ≤ 5 u 2 E . On the other hand, since (1 + x 2 ) −1 < 1, if x 2 ≥ 0, we have that Integrating this inequality we obtain Therefore, and the proof is complete.
Since E is embedded into Lebesgue spaces of arbitrary order, it is natural to ask if we can obtain embeddings from E into Orlicz space. In order to give a positive answer to this question we consider, from now on, the following weight According to Lemma 2.1, the embedding E → L p b is continuous for any 2 ≤ p < ∞. Moreover, we have the following Trudinger-Moser type result: for any 0 < α ≤ α 0 .
To prove our main proposition we will combine the ideas of Kufner-Opic [13] and Yang-Zhu [19]. First we recall a local estimate concerning the Trudinger-Moser inequality.
Lemma 2.4. For any R > 0, there exists a constant A 0 > 0 such that for any y ∈ R 2 and v ∈ W 1,2 0 (B R (y)) with ∇v L 2 (B R (y)) ≤ 1 we have Proof. For the proof, we refer the reader to [ Our strategy for proving Proposition 2 consists in consider, for any u ∈ E, its extension to the whole R 2 defined bȳ x 2 < 0.
We now take the control of the integral outside a ball.
In order to estimate the integral ofū in A σ 3r , we fix 1 < r < σ and follows as in [13] we introduce the set of indices and we obtain Now notice that 2 3 |x k | ≤ |x| ≤ 4 3 |x k |, ∀ x ∈ U k , and therefore 1 (2.8) For any k ∈ K r,σ fixed, in view of (2.8) we get Now, consider a cut-off function ϕ k ∈ C ∞ 0 (U k ) such that 0 ≤ ϕ k ≤ 1 in U k , ϕ k ≡ 1 in U 1/2 k and ∇ϕ k L ∞ (U k ) ≤ C/|x k | for some constant C > 0. Note that ϕ kū ∈ H 1 0 (U k ) and, by (2.8), Since x k ∈ A σ r , we have that |x k | > r > 1. This and the above estimate imply that Thus, applying Lemma 2.4 with B R (y) = U k and v = v k , we get for any 0 < α ≤ α 2 := 4π/(10C 3 ) and hence This, together with (2.7), (2.9) and the fact that s 2 /(1 + cs) 2 ≤ 1/c 2 for any c, s > 0 imply that where we have used (2.6) in the last inequality. It follows from (2.5) that

ELLIPTIC PROBLEMS WITH EXPONENTIAL GROWTH ON THE HALF-SPACE 4947
Taking the limit as σ → +∞ and arguing as before we get for any 0 < α ≤ α 2 := 4π/(10C 3 ). The lemma is proved.
We are now ready to prove the main result of this subsection.
Proof of Proposition 2. For any u ∈ E, we have that Picking r > 1 and setting R := 3r, after splitting the last integral above into B R and B c R , we can use Lemmas 2.5 and 2.6 to get whenever u E ≤ 1 and 0 < α < α 0 := min{α 1 , α 2 }.

3.
Applications. In this section we apply our embedding theorems. We consider the problem (P ) and notice that, since h does not belong to any Lebesgue space, we need to introduce a suitable subspace of E in order to use variational methods. So, we consider Since one of the terms of the energy functional associated to (P ) has the form a(x)F (u) and the nonlinearity f can behave like e s at infinity, we need the following Trudinger-Moser type inequality: Suppose ( a 1 ). Then, for any u ∈ E and α > 0, we have that (e αu 2 − 1) ∈ L 1 a . Moreover, there exists α 0 > 0 such that for any 0 < α ≤ α 0 .
Proof. The result is a direct consequence of Proposition 2 and the inequality a(x) ≤ c 1 (1 + |x|) 2 , where c 1 > 0 comes from (a 1 ).
Using the above lemma, (f 0 ) − (f 1 ) and Proposition 1, we can prove that the functional I : E q → R defined by is well defined, I ∈ C 1 (E q , R) and for any ϕ ∈ E q . Thus, weak solutions of (P ) are exactly the critical points of I. We shall prove our existence result as an application of the Mountain Pass Theorem. So, the first step is to prove the following: (ii) there exists e ∈ E q with e E q > ρ such that I(e) < 0.
In order to prove (ii) we notice that, by (f 2 ), there exists C 7 , C 8 > 0 such that So, given a nonzero function ϕ ∈ E q with support in the compact set Ω, we obtain Hence I(tϕ) → −∞, as t → +∞, and it is sufficient to set e := t 0 ϕ, with t 0 > 0 large enough. This finishes the proof of the lemma.
In the next steps we shall prove a compactness property for the functional I. First, a compact embedding result for the space E q : Lemma 3.2. Suppose ( a 1 ) and (h 1 ). Then the weighted Sobolev embedding E q → L 2 a is compact.
Proof. Let (u k ) ⊂ E q be such that u k 0 weakly in E q . The Sobolev embedding provides C > 0 be such that u k 2 L q h ≤ C. Hence, using Hölder's inequality with exponents q/2 and q/(q − 2) we obtain, for any R > 0, Given ε > 0, we can use the above expression together with the integrability condition of (h 1 ) to choose R > 0 large in such way that for any k ∈ N. On the other hand, as in the proof of Lemma 2.1, Since ε > 0 is arbitrary, the result is proved.
and Ω ⊂ R 2 + its compact support. Since u k W 1,2 (Ω) ≤ u k ≤ C, we may assume that, for any p ≥ 1, u n → u strongly in L p (Ω) and u n (x) → u(x) for a.e. x ∈ Ω. Using (f 0 ) and (f 1 ) we can check that f (u k ) ∈ L 1 (Ω) for any k ∈ N. Moreover, since I (u k ), u k → 0, as k → +∞, we obtain C 1 > 0 such that where o k (1) stands for a quantity approaching zero as k → +∞. It follows from [7, Lemma 2.1] that f (u k ) → f (u), as k → +∞, strongly in L 1 (Ω), and therefore The strong convergence in L q−1 (Ω) provides ψ ∈ L q−1 (Ω) such that |u k (x)| ≤ ψ(x) for a.e. x ∈ Ω. Thus, The right-hand side above belongs to L 1 (Ω) and therefore we infer from the Lebesgue Theorem that This, (3.3) and the weak convergence of (u k ) imply that By density we conclude that I (u) = 0.
We recall that I satisfies the Palais-Smale condition at level c ∈ R if any sequence (u k ) ∈ E q such that I(u k ) → c and I (u k ) → 0 has a convergent subsequence. In the next result we show that this is true in our setting. Proposition 5. Suppose (a 1 ) and q ≥ 2. Then I satisfies the Palais-Smale condition at any level c ∈ R. The same holds if we drop (a 1 ) by both the conditions ( a 1 ) and (h 1 ).
Proof. Let (u k ) ⊂ E q be such that I(u k ) → c and I (u k ) → 0. We claim that (u k ) has a bounded subsequence. Indeed, computing I(u k ) − (1/µ) I (u k ), u k and using (f 2 ), we get Suppose, by contradiction, that u k E q → ∞ as k → ∞. Since 2 ≤ q < µ, the above expression implies that The first equality above combined with the fact that shows that Consequently, u k 2 L q h → +∞ and we can use q ≥ 2 and (3.4) to conclude that which does not make sense. Hence, (u k ) has a bounded subsequence. For simplicity, we write only (u k ) to denote this subsequence. Suppose first that (a 1 ) holds. Then, in view of Lemma 2.1, we may assume that, for some u ∈ E q ,      u k u weakly in E q , u k (x) → u(x) for a.e. x ∈ R 2 + , u k → u strongly in L 2 a . By the last lemma we have that I (u) = 0, and therefore where We claim that B(k) = o k (1). In fact, we have a(x)|f (x, u)||u k − u|dx.
From the above remarks and (3.5) we infer that o k (1) = u k − u 2 E + A(k). We now recall that (|s 1 | q−2 s 1 − |s 2 | q−2 s 2 )(s 1 − s 2 ) ≥ C 4 |s 1 − s 2 | q , for all s 1 , s 2 ∈ R and some C 4 = C 4 (q) (see [17, inequality (2.2)]). This and the above equality imply that and therefore u k → u strongly in E q . For the case that ( a 1 ) and (h 1 ) hold we can argue along the same lines but using Lemma 3.2 instead of Lemma 2.1.
We are ready to prove our existence results.
Proof of Theorems 1.1 and 1.2. By the last proposition the functional I satisfies the Palais-Smale condition. Hence, it follows from Proposition 4 and the classical Mountain Pass Theorem that there exists u 0 ∈ E q \ {0} such that I (u 0 ) = 0. This function is a nonzero weak solution of (P ).
In order to prove our multiplicity result we shall use the following version of the symmetric Mountain Pass Theorem.
Theorem 3.4. Let E be a real infinite-dimensional Banach space and I ∈ C 1 (E, R) an even functional satisfying the Palais-Smale condition at any level and the following hypotheses: (I 1 ) I(0) = 0 and there are constants ρ, C > 0 such that I | ∂Bρ(0) ≥ C; (I 2 ) for any finite dimensionalX ⊂ X,X ∩ {u ∈ X : I(u) ≥ 0} is bounded. Then I has an unbounded sequence of critical values.
We finish the paper proving Theorem 1.3.
Proof of Theorem 1.3. Since f is odd we have that I is even and I(0) = 0. By Proposition 4(i) the functional verifies (I 1 ). Moreover, by Proposition 5, it also satisfies Palais-Smale.
LetẼ ⊂ E q be a finite dimensional subspace and (u k ) ⊂Ẽ be such that I(u k ) ≥ 0. Using (f 3 ) and the equivalence of norms inẼ, we obtain ELLIPTIC PROBLEMS WITH EXPONENTIAL GROWTH ON THE HALF-SPACE   4953 for some constants C 1 , C 2 , C 3 > 0. Since µ 0 > q ≥ 2, the above expression implies that (u k ) is bounded. Hence, (I 2 ) holds and Theorem 3.4 provides infinitely many critical points for I.