On the existence of positive solutions for a nonlinear elliptic class of equations in R 2 and R 3

We study the existence of positive solutions for an elliptic equation in R N for N = 2 , 3 which is related with the existence of standing (localized) waves and the existence of the ground state solutions for some physical model or systems in ﬂuid mechanics to describe the evolution of weakly nonlinear water waves. We use a variational approach and the well-known principle of concentration-compactness due to P. Lions to obtain the existence of this type of solutions, even in the case that the nonlinear term g is a non-homogeneous function or an operator deﬁned in H 1 ( R N ) with values in R .


Introduction
The first step in the study of some interesting physical problems in acoustics, in the context of gravity waves, in fluid mechanics, or optics is to establish the existence of special positive solutions as travelling and standing waves, as happens in the case of the generalized Schrödinger equation, the generalized Davey-Stewartson type systems, the Zakharov-Rubenchik (also known Benney-Roskes) system and the generalization the Zakharov-Rubenchik system. The main issue in previous models is that the existence of such special solutions is reduced to studying the existence of solutions for a single Schrödinger type nonlinear equation of the form where σ 1 , , σ, β, γ ∈ R, E 0 is a nonlocal linear operator defined via a Fourier multiplier, x ∈ R N for N = 2, 3, and ∆ = ∆ ⊥ + ∂ 2 z with ∆ ⊥ = ∂ 2 x in the case N = 2. In this general case, if we look for standing wave solutions for (1) of the form ψ(x, t) = e ict u(x), then u satisfies the equation something that also happens in the case of generalization of the KdV equation or the Gardner equation in R N for γ = 0. We note that this equation is elliptic in the case σ 1 > 0.
Regarding the existence or non existence of positive solutions for nonlinear models, there are plenty of result for different kind of problems in the elliptic and non elliptic case. For instance, J. Gidaglia and J.C Saut in [7] showed a non existence result of non trivial solutions for the nonlinear equation in the case j = ±1 with k = l for some (k, l), under the assumption that the nonsingular diagonal matrix diag(e 1 , · · ·, e N ) is not positive-definite (resp. negative definite) and that f is a continuous real-valued function such that |f (s)| ≤ C(1 + |s| p ), p ≤ 2 N − 2 , N = 2, 1 ≤ p < ∞, N = 2.
Moreover, the existence of a positive (radial) solution g ∈ C ∞ (R 2 ) ∩ H 1 (R 2 ) of the elliptic equation −g + ∆g + g 3 = 0 implies the existence of a nontrivial solutions of the nonelliptic equation −w + w xx − w yy + w 3 = 0.
Moreover, if we set u(x, y, t) = e it w(x, y), then u satisfies the equation On the other hand, for p ∈ 1, N +2 N −2 with N > 2 and b ∈ C(R N ) such that b(x) → b ∞ as |x| → ∞, J. Lions [9] established the solvability of the elliptic equation This result was extended by Bahri and Lions [2] to the case when b(x) ≥ b ∞ − Ce −δ|x| . In the case that b(x) < b ∞ and b Y is a convex combination of functions b(· − y) with y = (z 1 , · · · , z N ) and z i ∈ Z, then K. Tintarev in [15] showed the existence of a nontrivial solution of the elliptic equation In this paper we are interested in establishing a general existence result of positive solutions of the special elliptic equation in R N (N = 2, 3) cu − ∆u + g(u) = 0, c > 0 where g is either a function defined in R or an operator defined in H 1 (R N ) having some variational properties. In particular, we are very interested in this elliptic equation due to its relation with the existence of special solutions for some well known systems. For instance, we consider the existence of standing (localized) waves v(x, t) = e ict u(x) for the second order differential equation in R N iu t + ∆u − g(u) = 0, where g is a function or an operator defined such that g(u) =g(|u|)u. In this case, we find that u satisfies the elliptic equation (2). We note that equation (3) is related with the generalized Schrödinger equation in the case with the Davey-Stewartson type systems in the case with the Benney-Roskes/Zakharov-Rubenchik system in the case g(u) = a|u| 2 u + bE 2 (|u| 2 )u, where E j (j = 1, 2, 3) is a linear operator defined on H 1 (R N ) via a Fourier multiplier of the form We refer to the following works to get more details for these models: [4], [5], [6], [11], [12], [13].
On the other hand, if we consider the existence of ground state solutions ( for the second order differential equation in R N we see that u satisfies the elliptic equation (2). In this case, the nonlinear model (4) with nonlinearity g(s) = as p + bs q could be though as a generalization of the KdV equation in R N (a = 0, q = 2), a generalization of the modified KdV and a generalization of the Gardner equation in R N (a = 0, b = 0, p = 2q).
The existence of solutions for the nonlinear elliptic model (2) is a consequence of the variational characterization of solutions and the well-known concentration-compactness principle by P. Lions, although the non-linear term g is not necessarily an homogeneous function or operator. The result is inspired in Cipolatti's approach in [4] related with the existence of standing waves for a Davey-Stewartson system.
Then, there exists λ > 0 and a positive solution u ∈ H 1 (R N ) of the equation for some ν ∈ R, then w = ρu for some ρ > 0. In particular, ν = λ.
This paper is organized as follows. In section 2, we include some preliminary results, state the main hypotheses on the operator g and provide a variational characterization of the ground state solutions for the general equation (3) and (4). In section 3, we prove the main results by using the variational characterization of ground state solutions and the Concentration-Compactness principle by P. Lions ([9, 10]). In section 4, we provide some non trivial examples, which include generalizations of KdV type model in R N , Davey-Stewartson type systems (see [4]), Benney-Roskes/Zakharov-Rubenchik system in spatial dimensions N = 2, 3 (see [1], [5], [11], [14]), the generalized Benney-Roskes/Zakharov-Rubenchik system in spatial dimensions N = 2, 3 ( [12], [13]). As far as our knowledge goes, the last result is new to the literature.
We assume that the nonlinear term g is such that the model has a Hamiltonian structure. In other words, we assume that there is an operator Ξ defined in H 1 (R N ) such that Ξ (φ) = g(φ) for φ ∈ H 1 (R N ). From previous assumption, we see directly that standing waves of the form v(x, t) = e ict u(x) for (3) for g(u) =g(|u|)u and travelling wave solutions in the x-direction of the form v(x, t) = u(x − ct, z) (x = x, z)) for (4) are characterized as critical points of the functional where the Hamiltonian H and the charge Q are defined respectively by Moreover, we have that F (u)(v) = 0 for any v ∈ H 1 (R N ) is equivalent to have solutions for the equation If we set the functionals T and V on H 1 (R N ) as we see that Before we go further, we note that any nontrivial solution u of the equation (2) satisfies the identities which imply that V (u) = 0 for N = 2 and V (u) < 0 for N = 3. These facts will be clever in the minimization argument for N = 2 and N = 3, as in the work by R. Cipolatti in [4]. Now, for each µ ∈ R we define the level set for V and the infimum j(µ) by As we mention above, we need to impose some natural conditions on the functional Ξ. Hereafter, we assume that (G3) There is m > 2 such that for λ > 0 and ψ ∈ H 1 (R N ), we have that G(λψ) = λ m F (λ, ψ), where F is a continuous From condition (G3), for a given ψ ∈ H 1 (R N ) we have that Remark 2.1 On the condition (G4). We want to point out that the condition (3.9) in R. Cipolatti's work [4] is trivially achieved in the case α < 2, but it unclear in the case α ≥ 2. It seems that the condition must be that with B 1 being defined as In order to assure that the condition (3.9) in Cipolatti's work holds, we introduce the condition (G4), which generates some simple restrictions on the set R ω,b in Cipolatti's work.
Under those conditions on G we are able to establish the following result. (ii) Let N = 2 and assume conditions (G1)-(G4), then there is I > 0 such that j(µ) = I for µ ∈ R.
Now we proceed to establish (iii) for N = 3. We set As in previous case, from the Gagliardo-Nirenberg-Sobolev inequality, the condition (G1) with 0 < r j < 4 for 1 ≤ j ≤ k, and that V (ψ) = −1, we conclude that I > 0. In fact, assume that I = 0. So, choose a sequence (ψ n ) n ⊂ H 1 (R 3 ) such that V (ψ n ) = −1 and ∇ψ n 2 → 0, as n → ∞. If for some subsequence (ψ n k ) k ⊂ (ψ n ) n we have that ψ n k 2 → 0, as k → ∞, then we have from the Gagliardo-Nirenberg-Sobolev inequality that ψ n k rj +2 → 0, as k → ∞, since So, we reach the contradiction In other words, ψ n 2 ≥ L for some L > 0 and n ∈ N. From this and the Gagliardo-Nirenberg-Sobolev inequality, we conclude for some 0 < r j < 2 that which again is a contradiction. So, we have that I > 0. On the other hand, for λ > 0 we have that We see now that the minimization problem associated with j has an equivalent formulation.
is equivalent to the minimization problem Proof. LetĨ Clearly, we have thatĨ 0 ≤ I 0 . Let ψ ∈ H 1 be such that ψ = 0 and V (ψ) < 0. We set the function h ψ (λ) = V (λψ). We know that h ψ (1) < 0 and that h ψ (0) = 0, so there is 0 < λ < 1 such that h ψ (λ) = V (λψ) = 0. Then, we conclude that Following the same arguments, we also have that Theorem 2.3 For N = 3 and µ < 0, the minimization problem is equivalent to the minimization problem We split the proof in the cases N = 2 and N = 3. In the case N = 2 , we use conditions (G1)-(G4), that the embedding H 1 (U ) ⊂ L q (U ) is compact, for 1 ≤ q < ∞ and U ⊂ R 2 bounded, the concentration-compactness and the variational characterization j.
(ii) ϕ ∈ X if and only if ϕ solves the minimization problem Proof. Let j(µ) be defined in (5). We want to show that the problem (12) has a solution. In fact, let (ϕ n ) n ⊂ Σ 0 such Moreover, due to the fact that (φ n ) n ⊂ Σ 0 is a minimizing sequence for I 0 , we have that ∇φ n 2 is also a bounded Now, we set the measure ν n with density ρ(φ n ) with respect to the Lebesgue measure given by So, we have that We now apply the Lions' Concentration-Compactness Principle (see [9]- [10]). First, we see that vanishing is not possible.
In fact, from the Sobolev inequality we have for any open box J in R 2 and for any r j > 0 that On the other hand, from V (φ n ) = 0 and condition (G1), we conclude that which gives us a contradiction, since φ n rj +2 → 0, as n → ∞ for r j > 0, in the case that we had vanishing.
If we assume Dichotomy, then there is 0 < γ < σ 0 such that for a given > 0, there exist R 0 > 0, a sequence (y n ) n ⊂ R 2 , R n ↑ +∞, and a bounded sequence (φ i n ) n ⊂ H 1 (R 2 ) for i = 1, 2 (all depending on ) such that The first remark (passing to a subsequence) is that In particular, we have that φ k n = 0 for k = 1, 2. Now, from (15), and using that j(µ) = I 0 for any µ ∈ R, we have that implying that I 0 ≥ 2I 0 − C 2 , but this is a contradiction if > 0 is small enough. In other words, we have ruled out Dichotomy, meaning that we have Compactness. From this fact, there is a sequence (y n ) n ⊂ R 2 such that for a given > 0, there exists R 0 ≥ 1 such that where A(n) = R 2 \ B R0 (y n ) with B R0 (y n ) being the open ball of radius R 0 around y n . If we setφ n (x) = φ n (x − y n ), then we have thatφ n φ 0 in H 1 (R 2 ). Moreover, we also have that ∂ lφn → ∂ lφ0 a. e. in L 2 (R 2 ) for l = 0, 1, 2 and φ n →φ 0 a. e. in L 2 (R 2 ) for p ≥ 2. From the compactness condition (17), we have for n large enough that, On the other hand, from the Sobolev inequality, we conclude for q ≥ 2 that Now, from Fatuo's Lemma for q ≥ 2 we have that where we are using that the embedding H 1 (B R0 (0)) → L q (B R0 (0)) for 1 ≤ q < ∞ is compact. From this, we have that φ n →φ 0 in L q (R 2 ) for 1 ≤ q < ∞, since we have weak convergence (φ n φ 0 in L q (R 2 )) and the convergence of the norms ( φ n L q (R 2 ) → φ 0 L q (R 2 ) ). From this fact, we conclude that meaning thatφ 0 ∈ Σ 0 . Moreover, again from Fatou's lemma and the compactness of the embedding H 1 (B R0 (0)) → L q (B R0 (0)) for 2 ≤ q < ∞, we also have that which implies that Now, we will see that X = ∅. In fact, let ψ ∈ Σ 0 be a solution of the minimization problem (8). Then, there is a Lagrange multiplier λ such that for any where the pairing ·, · is between the spaces H −1 (R 2 ) and H 1 (R 2 ). In other words, ψ satisfies the equation We see directly that λ = 0. Now, we claim that λ > 0. In fact, let v ∈ H 1 (R 2 ) such that δV (ψ), v < 0 and take t ∈ R.
Then we see directly that If we assume that λ < 0 and take t small enough but negative, then form the continuity of T and V , we conclude that T (ψ + tv) < T (ψ) with V (ψ + tv) < 0, which contradicts the second characterization of the minimization problem (Theorem (2.2)). So, we see that ψ λ (x) = ψ(y) with x = √ λy satisfies the travelling wave equation (2) and ψ λ ∈ X.
In the case N = 3 , we use the embedding H 1 (U ) ⊂ L q (U ) is compact, for 2 < q < 6 and U ⊂ R 3 bounded, the concentration-compactness, the variational characterization j, the conditions (G1)-(G3), and the additional condition on G, whenever supp (φ) ∩ supp (ψ) = ∅, and c) For any bounded sequence (ψ n , φ n ) n ⊂ H 1 (R 3 ) × H 1 (R 3 ), the sequence (G 0 (ψ n , φ n )) n converges. (ii) There is µ 0 < 0 such that ϕ ∈ X if and only if ϕ solves the minimization problem Proof. Let j(µ) be defined in (5). We want to show that the problem (21) has a solution. In fact, let (φ n ) n ⊂ Σ µ0 such that We claim that the sequence ( φ n 2 ) n is bounded. If not, assume that for some subsequence, we have that φ n l 2 → ∞, as l → ∞. So, from the Gagliardo-Nirenberg-Sobolev inequality and that the sequence ( ∇φ n 2 ) n is bounded, we conclude that which is a contradiction, meaning that the sequence (ϕ n ) n is bounded in such that, for a subsequence (denoted the same), if necessary, we have that φ n φ 0 (weakly) in H 1 (R 3 ) Now, we set the measure ν n with density ρ(φ n ) with respect to the Lebesgue measure given by So, we have that σ 0 ≥ 2 β j(µ 0 ) = −µ 1 3 0 I. We now apply the Lions' Concentration-Compactness Principle (see [9]- [10]). First,we see that vanishing is not possible. In fact, as in previous result, covering R 3 with a sequence of open boxes J k in such a way that J k ∩ J m = ∅, then we see that which implies, in the case we have vanishing, that φ n rj +2 → 0, as n → ∞ for 0 < r j < 2. On the other hand, from V (φ n ) = µ 0 and condition (G2), we conclude that , which gives us a contradiction.
If we assume Dichotomy, then there is 0 < γ < σ 0 such that for a given > 0, there exist R 0 > 0, a sequence (y n ) n ⊂ R 3 , R n ↑ +∞, and a bounded sequence (φ i n ) n ⊂ H 1 (R 3 ) for i = 1, 2 (all depending on ) such that The first remark (passing to a subsequence) is that Passing to a subsequence, if necessary, we may assume for i = 1, 2 that In particular, form (22), we have that From the fact that V ∈ C 1 (H 1 (R 3 )), we easily see that using that From the hypothesis (G5), we have that Now, if we setφ i n (x) = φ i n (x − y n ), then we note where χ A denotes the characteristic function on the set A and B n = B Rn (0). On the other hand, the embedding ) for 2 < q < 6. Moreover, for any ψ ∈ H 1 (R 3 ) we have that G 0 (ψ,φ 1 n ) → G 0 (ψ, φ 1 0 ) and also that χ B c n G 0 (φ 1 n ,φ 2 n ) → 0 almost everywhere, then the Lebesgue convergence theorem implies that From the estimates (23) and (24), we conclude for n large enough that where lim →0 + δ( ) = 0. So, taking limit as n → ∞, we see that Assume that lim →0 + µ 1 ( ) ≥ 0, then we conclude that Now, we note that If for a subsequence of (φ n ) n (denoted the same) we have that lim n→∞ T (φ n ) = 0, then we conclude that φ 1 n rj +2 → 0, as n → ∞, which implies that lim n→∞ V (φ 1 n ) = 0. In other words, there is M 1 > 0 such that T (φ n ) ≥ T (φ 1 n ) > M 1 with M 1 independent of and n. Moreover, which implies from Lemma (2.1), after taking → 0, that but this is a contradiction since 0 > µ 0 ≥ µ 2 . So, we may assume that µ i ( ) ≤ 0 for i = 1, 2. Then, from (15), we have but this contradicts the fact that the function f (t) = t 1 3 is strictly concave for t ∈ R + , since we have that So, we have ruled out Dichotomy. Using the compactness property as in the case N = 2, we conclude that there is a minimizer ϕ 0 ∈ H 1 (R 3 ) for j(µ 0 ), T (ϕ 0 ) = 2j(µ). Now, we will see that X = ∅. In fact, let ψ ∈ Σ µ0 be a solution of the minimization problem (10). Then, there is a Lagrange multiplier λ such that for any v ∈ H 1 (R 2 ) where the pairing ·, · is between the spaces H −1 (R 3 ) − H 1 (R 3 ). In other words, ψ satisfies the equation We see directly that λ = 0. Now, we claim that λ > 0. In fact, let v ∈ H 1 (R 3 ) such that δV (ψ), v < 0 and take t ∈ R.
Then we see directly that If we assume that λ < 0 and take t small enough but negative, then form the continuity of T and V , we conclude that T (ψ + tv) < T (ψ) with V (ψ + tv) < 0, which contradicts the second characterization of the minimization problem (Theorem (2.2)). So, we see that ψ λ0 (x) = ψ(y) with x = 3 √ λ 0 y satisfies the travelling wave equation (2) and ψ λ0 ∈ X with λ 0 = λ 3 2 .
In this section we provide some examples of operators g satisfying the conditions imposed on G. Before we go further, we introduce some notation to be considered. Hereafter, we set

Generalizations of the KdV and the Gardner equation in R N
We consider the dispersive model of the form u t + ∆u x − a|u| p−1 u − b|u| q−1 u x = 0. For this model, the x-travelling equation is given by -cu+ ∆u − a|u| p−1 u − b|u| q−1 u = 0. In this particular case, we have that and that G 1 is given by We will establish the existence of travelling waves by imposing some restrictions on the parameters a and p. We start defining a 0 (p) = for p > q. , We also define the sets A c,q,b = {(p, a) : 0 < p < p * , a < a 0 (p)}, and B c,q,b = {(p, a) : 0 < p < p * , a 1 (p) < a < a 0 (p)}, for c > b and p > q in the case a 1 (p) < a 0 (p).
We are going to verify the conditions (G1)-(G5) for the function G 1 . First note that meaning that the condition (G1) holds. Now, note that the condition (G3) holds. In fact, Now, we verify the condition (G2) holds for (p, a) ∈ A c,q,b .
Now, if we consider φ R,s = sχ B R , where χ B R denotes the characteristic function on the ball B R of radius R denoted, then we have that ||φ R,s || r r = |B R |s r for r ≥ 1 and that which implies that V ((φ R,s0 ) λ ) = |B R |g λ (s 0 ) < 0. By a density argument, we have that there is ϕ 0 ∈ H 1 (R N ) such that V (ϕ 0 ) < 0.
Assume now that p > q and a > 0. In this case, we have that there is a unique λ 0 > 0 given by (26). If we had λ 0 = 1, On the other hand, we also have for 2 < q < p that which implies that If we assume for example that c > b, and choose a > 0 such that then we conclude that λ 0 = 1. Now, if we had λ 0 < 1 and a > 0 satisfies (27), then from (26) we see that meaning that the function w(p) = pλ p 0 is an increasing function, but this happens, only if 1 + p ln(λ 0 ) > 0, which requires λ 0 > 1, but we are assuming that λ 0 < 1. In other words, under the assumption that a > 0 satisfies (27), we have necessarily that λ 0 > 1, and so, we see directly that h ψ (λ) < 0 for λ > 1, but close to 1 + . Finally, the case p = q and a < b follows trivially.
Finally, we note that condition (G5) is trivially obtained. Now, we consider the model u t + ∆u x − a|u| p−1 u − bu q x = 0, which the x-travelling equation is given by -cu+∆u − a|u| p−1 u − bu q = 0. So, we have that and that G 2 is given by In order to obtain similar results as in previous case, we need to adjust some issues. For instance, we take q > 2, b > 0 and (p, a) ∈ C c,q,b , where C c,q,b = {(p, a) : 0 < p < p * , a 21 (p) < a < a 22 (p)}, and for p > q.
, First, we have that G 2 is given by As in the case above, the conditions (G1) and (G3) hold using the same argument.
We now verify the condition (G2) holds for (p, a) ∈ A c,b,q with p = q.
Now, in the case a R N ψ q+2 dx > 0, we have that there is a unique λ 0 > 0 such that Moreover, we also have thath if we assume a(q − p) > 0. So, supposing that a > 0, then we have that q > p and that R N ψ q+2 dx > 0. In this case, we conclude that From this fact, we conclude that q > p, and also that λ 0 < 1. So, from this analysis, we have that h ψ (λ) < 0 for λ > 1, but close to 1 + . Now, assume that a < 0, then we have that p > q and that R N ψ q+2 dx < 0. In this case, we conclude that because the condition V (ψ) = 0 implies that b(p + 2) a(q + 2) Moreover, we also have thath ψ (λ 0 ) = c ψ 2 2 + aλ p 0 q(p+2) (q − p) ψ p+2 p+2 > 0 and also that λ 0 < 1. So, from this analysis, we have that h ψ (λ) < 0 for λ > 1, but close to 1 + , for a < 0 and p > q.
Finally as in previous case, condition (G5) for N = 2 follows trivially.
We note that using the same type of arguments as above, we also verify conditions (G1)-(G5) in the case g 3 (t) = at 2p+1 − bt p+1 = a|t| 2p t − bt p+1 , for appropriates p and a.

Davey-Stewartson type systems
We consider the standing wave equation for the Davey-Stewartson system in R N iu t + ∆u + bE(|u| 2 )u − a|u| p u = 0, where p > 0, b = b 1 b 2 > 0, a ∈ R, and E is a (non local) linear operator defined via the Fourier transform F by In this case, standing wave equation associated with the Davey-Stewardson system is reduced to find solutions u of the Schrödinge like model, -c u +∆u = a|u| p u − bE(|u| 2 )u, u ∈ H 1 (R N ) \ {0}. From this model, we have that We will establish the existence of travelling waves by imposing some restrictions on the parameters a and p. We start defining a 0 (p) = , We also define the sets A c,b = {(p, a) : 0 < p < p * , a < a 0 (p)}, and B c,b = {(p, a) : 0 < p < p * , a 1 (p) < a < a 0 (p)}, for c > b and p > 2 in the case a 1 (p) < a 0 (p).
If we assume for example that c > b, and choose a > 0 such that then we conclude that λ 0 = 1. So, as in the first case, under the assumption that a > 0 satisfies (29), we have necessarily that λ 0 > 1, and so, we see directly that h ψ (λ) < 0 for λ > 1, but close to 1 + . Finally, the case p = 2 and a < b follows trivially as in the first case. Now, we verify the condition (G5) only for N = 3 (the case N = 2 does not require this argument) for the operator G 5 (u) = b 4 E(|u| 2 )|u| 2 . Let ψ, φ ∈ H 1 (R 3 ) be such that supp (ψ) ∩ supp (φ) = ∅, then we have that where G 0 (ψ, φ) = 2E(|ψ| 2 )|φ| 2 . Due to the nature of the operator E, we have that