Existence and concentration of positive solutions for a critical p & q equation

: We show existence and concentration results for a class of p & q critical problems given by where u ∈ W 1, p ( R N ) ∩ W 1, q ( R N ), ϵ > 0 is a small parameter, 1 < p ≤ q < N , N ≥ 2 and q * = Nq /( N − q ). The potential V is positive and f is a superlinear function of C 1 class. We use Mountain Pass Theorem and the penalization arguments introduced by Del Pino & Felmer’s associated to Lions’ Concentration and Compactness Principle in order to overcome the lack of compactness.


Introduction
In this paper we are concerned with a class of problems, named p&q problems type. In the last years the main interest in this general class of problems has been due to the fact that they arise from applications in physics and related sciences, such as biophysics, plasma physics and chemical reaction, as it can be seen for example in [20], [23] and [35]. In addition, such a class of problems encompasses a large class of problems, as can be seen in [4], [15] and [17].
More precisely, we show existence and concentration results of positive solutions for the critical problem given by ⎧ ⎪ ⎨ ⎪ ⎩ −div a ϵ p |∇u| p ϵ p |∇u| p−2 ∇u + V(z)b |u| p |u| p−2 u = f (u) + |u| q * −2 u in R N , where ϵ > 0, N ≥ 2, 1 < p ≤ q < N and q * = Nq/(N − q). The hypotheses on the function a are the following: (a 1 )the function a is of class C 1 and there exist constants k 1 , k 2 ≥ 0 such that k 1 t p + t q ≤ a(t p )t p ≤ k 2 t p + t q , for all t > 0; (a 2 )the mapping t → a(t p ) t q−p is nonincreasing for t > 0; (a 3 )if 1 < p < 2 ≤ N the mapping t → a(t) is nondecreasing for t > 0. If 2 ≤ p < N the mapping t → a(t p )t p−2 is nondecreasing for t > 0. As a direct consequence of (a 2 ) we obtain that the map a and its derivative a ′ satisfy a ′ (t)t ≤ (q − p) p a(t) for all t > 0. (1.1) Now if we define the function h(t) = a(t)t− q p A(t), using (1.1) we can prove that the function h is nonincreasing. Then, there exists a positive real constant γ ≥ q p such that 1 γ a(t)t ≤ A(t), for all t ≥ 0. (1. 2) The hypotheses on the function b are the following: (b 1 )The function b is of class C 1 and there exist constants k 3 , k 4 ≥ 0 such that is nondecreasing for t > 0. If 2 ≤ p < N the mapping t → b(t p )t p−2 is nondecreasing for t > 0.
We need to put some hypotheses on the potential V ∈ C(R N ). (V 1 )There is V 0 > 0, such that 0 < V 0 ≤ V(z) for all z ∈ R N .
(V 2 )There exists a bounded domain Ω ⊂ R N such that In order to illustrate the degree of generality of the kind of problems studied here, with adequate hypotheses on the functions a and b, in the following we present more some examples of problems which are also interesting from the mathematical point of view and have a wide range of applications in physics and related sciences.
The Problem 1 comes from a general reaction-diffusion system: u t = div(Du∇u) + g(x, u), where Du := [|∇u| p−2 +|∇u| q−2 ]. In such applications, the function u describes a concentration, the term div(Du∇u) corresponds to the diffusion with a diffusion coefficient Du and g(·, u) is the reaction and relates to source and loss processes. Usually, in chemical and biological applications, the reaction term g(·, u) is a polynomial of u with variable coefficients.
In this case we are studying problem and it is related to the main result showed in [3] in the case q = 2. In [19] the author have studied the case 1 < q < N.
and b(t) = 1. In this case we are studying problem In this case we are studying problem −ϵ p Δp u − ϵ q Δq u − div ϵ p |∇u| p−2 ∇u The main result is the following: Then there are ϵ 0 > 0 and λ * > 1 such that (Pϵ) has a positive solution wϵ ∈ W 1,p (R N ) ∩ W 1,q (R N ), for every ϵ ∈ (0, ϵ 0 ) and for every λ > λ * . In addition, if Pϵ is the maximum point of wϵ, then Moreover, there are positive constants C and α such that for all ϵ ∈ (0, ϵ 0 ) and for all z ∈ R N .
In a seminal paper [31], Rabinowitz used his famous Mountain Pass Theorem(joint with Ambrosetti) [5] and showed the existence of solution for a Nonlinear Schrödinger Equation given by where V is a continuous potential satisfying (V 1 ) and In [31], Rabinowitz used the force of the parameter ϵ and the geometry of the potential V in order to overcome the lack of compactness of Sobolev's embedding to obtain the positive solution. In [33], Wang showed that the solution found by Rabinowitz concentrates around a local minimum of the potential V, when ϵ converges to zero. Wang also noted that the concentration of any family of solutions with energy uniformly bounded can only occur in a critical point of V. In [12], Del Pino and Felmer weakened the hypothesis (R 1 ) of Rabinowitz and created a method that is known as Del Pino and Felmer's penalization method.
As can be seen in [4], [15] and [17], p&q problems are generalizations of (R). However, as can seen below, we show that the arguments found in [12], [31] and [33] cannot be used directly. But before that, we are going to report some results on p&q problems type. There are interesting papers on such class of problems. We start with some problems in a bounded domain. For example, in [15] the author shows the existence and multiplicity of solutions for a critical p&q problem considering nonlinearity of type concave and convex. The critical case with discontinuous nonlinearities has studied in [16]. Now we comment some results in R N . Existence results was studied in [11] and [17]. In [2] the authors studied concentration results in Orlicz-Sobolev spaces with subcritical nonlinearity and the potential satisfying the local condition introduced by Del pino and Felmer [12]. In [4], it was showed the existence and concentration results with subcritical nonlinearity and the potential satisfying the global condition introduced by Rabinowitz [31]( see also [33]).
The present work is strongly influenced by the articles above. Below we list what we believe that are the main contributions of our paper.
(3) Since the operator is not homogeneous, some estimates are different and more delicate than some estimates that can be found in [12] and [31] . For example, see Lemma 3.4, Proposition 5.1, Lemma 5.7 and all the Lemmas of Section 7. (4) In order to overcome the lack of compactness provoked by the critical growth, it is very common to use the Talenti's function (see [32]) to have some control on the minimax level, as can be seen in [10, Lemma 1.1]. The lack of homogenity of the p&q operator does not allow to use this argument. We overcome this difficulty using the solution of a problem in a bounded domain, as can be seen in Lemma 3.5.
The interest in the study of nonlinear partial differential equations with p&q operator or fractional p&q operator has increased because many applications arising in mathematical physics may be stated with an operator in this form. We cite the papers [6], [7], [8], [9], [18], [21], [22], [26], [27], [28], [29], [30] and their references. Several techniques have been developed or applied in their study, such as variational methods, fixed point theory, lower and upper solutions, global branching, and the theory of multivalued mappings.
This paper is organized as follows. In Section 2, we define an auxiliary problem using the penalization argument introduced by Del Pino and Felmer [12]. The existence of solution for the auxiliary problem was showed in Section 3. In order to show the concentration result, in Section 4 we studied the autonomous problem. The concentration result was showed in Section 5. In Section 6 we showed that the solutions of the auxiliary problem are solutions of the original problem. In Section 7 we showed the exponential decay of these solutions. To conclude the paper, we showed in an appendix the existence of a solution to a problem in a bounded domain that was important to overcome the lack of compactness.

Variational framework and an auxiliary problem
To prove Theorem 1.1, we will work with the problem below, which is equivalent to (Pϵ) by change variable z = ϵx, which is given by where ϵ > 0, N ≥ 2 and 1 < p ≤ q < N.
In order to obtain solutions of ( P ϵ), consider the following subspace of W 1,p (R N ) W 1,q (R N ) given by which is a Banach space when endowed with the norm where Since the approach is variational, consider the energy functional associated J ϵ : Wϵ → R given by where u + = max{u, 0}. By standard arguments, one can prove that Jϵ ∈ C 1 (Wϵ , R). As we are interested in nonnegative solutions we can assume that f (s) = 0 for s ≤ 0.
Let β be a positive number satisfying β > max pγθ q(θ − pγ) , V 0 pγ q , 1 , where θ was given in (f 3 ) and V 0 appeared in (V 1 ). From (f 4 ), there exists η > 0 such that f (η) + η q * −1 η q−1 = V 0 β . Then, using the above numbers, we define the function of C 1 class given by We now define the function and the auxiliary problem where χ Ω is the characteristic function of the set Ω. It is easy to check that (f 1 ) − (f 4 ) imply that g is a Carathéodory function and for x ∈ R N , the function s → g(ϵx, s) is of class C 1 and satisfies the following conditions, uniformly for x ∈ R N : The function is also a positive solution of (Pϵ).

Existence of ground state for problem (P ϵ aux )
Hereafter, let us denote by Iϵ : Wϵ → R the functional given by We denote by N ε the Nehari manifold of Iε, that is, and define the number bε by setting Using (f 1 ), (f 2 ) and (g 2 ) we have: for every ξ > 0 there exists C ξ such that Then, by definition of g and (3.2), there is r ε > 0 such that The main result in this section is: Moreover, we would like to highlight that in section 5, more precisely in Lemma 5.5, we are going to show that if Pϵ ϵ is the maximum point of uϵ then lim In order to use the Mountain Pass Theorem [5], we define the Palais-Smale compactness condition. We say that a sequence (u n) ⊂ Wϵ is a Palais-Smale sequence at level c for the functional Iϵ if If every Palais-Smale sequence of I ϵ has a strong convergent subsequence, then one says that Iϵ satisfies the Palais-Smale condition ((PS) for short).
Lemma 3.2. The functional I ϵ : Wϵ → R satisfies the following conditions (i) There are α, ρ > 0 such that Proof. Using (a 1 ), (b 1 ) and (3.2) we obtain By Sobolev embeddings, choosing ξ > 0 appropriate and taking u < 1 there are positive constants Then the item (i) follows. Now we show that the item (ii) holds. Consider a positive function w ∈ C ∞ 0 (Ωϵ), t > 0 and using (a 1 ), (b 1 ), (f 3 ) and Sobolev embedding, we have This proves the second item.
Hence, there exists a Palais-Smale sequence (un) ⊂ Wϵ at level cϵ. Using (a 2 ), (b 2 ) and (f 4 ), it is possible to prove that where b ϵ was defined in (3.1). In order to prove the Palais-Smale condition, we need to prove the next lemma.
Proof. Since (un) is a (PS) d sequence for functional I d , then using (1.1), (1.3), (g 3 ) i and (g 3 ) ii we have that Then, arguing as the [4, Lemma 2.3] , we can concluded that (un) is bounded in Wϵ.
Passing to the limit in the last estimate, we get for some R sufficiently large and for some fixed ξ > 0.
In the next result we show that the functional Iϵ satisfies the Palais-Smale condition for some levels. For this work we are denoting by S the best Sobolev constant for the embedding of D 1,q (R N ) into L q * (R N ), that is, the largest positive constant S such that Lemma 3.4. The functional I ϵ satisfies the Palais-Smale condition at any level Using the same kind of ideias contained [4, Lemma 2.3], we may conclude that u is a critical point of I ϵ. From Lemma 3.3 and for each ξ > 0 given there exists R > 0 such that This inequality, (a 1 ), (b 1 ), (f 1 ), (f 2 ), (g 2 ) and the Sobolev embeddings imply, for n large enough, there exists a positive constant C 1 such that On the other hand, taking R large enough, we suppose that Therefore, by (3.6) and (3.7), We claim that Indeed, we have, in view of the definition of g, Then we obtain an at most countable index set Γ, sequences (x i ) ⊂ R N and (μ i ), (ν i ) ⊂ (0, ∞), such that for all i ∈ Γ, where δx i is the Dirac mass at x i ∈ R N . Thus it is sufficient to show that {x i } i∈Γ ∩ Ωϵ = ∅. Then, we suppose by contradiction that x i ∈ Ωϵ for some i ∈ Γ. Consider R > 0 and the function ψ R := ψ( where R > 0 will be chosen in such way that the support of ψ is contained in Ω ϵ. Then, as (ψ R un) is bounded and I ′ ϵ (un)ψ R un = on(1), Note that, using (a 1 ), (b 1 ) and that the function f has subcritical growth, we have Therefore, by (a 1 ) again, Since ψ R has compact support and letting n → ∞ in the above expression, we see that From this inequality and (3.11) one easily sees that S N/q ≤ ν i . As β > pγθ q (θ − pγ) and S N/q ≤ ν i we have, by previous arguments, Hence, taking the limit and using (3.11), we get which does not make sense. Thus we obtain the convergence (3.10). Therefore Finally, we prove that, up to a subsequence, u n → u in Wϵ. Since I ′ ϵ (un)un = on(1), I ′ ϵ (u) = 0, (3.12) and Fatou's Lemma we have Then, using (a 1 ) and (b 1 ), we obtain un − u = on (1), that is, the sequence (un) converges strongly to u.
For each fixed ϵ > 0, let us consider the following problem −k 2 Δp u − Δq u + V(k 4 |u| p−2 u + |u| q−2 u) = |u| τ−2 u in Ωϵ , u ∈ W 1,q 0 (Ωϵ), where τ is the constant which appears in the hypothesis (f 5 ) and V := max x∈Ωϵ V(x) is a positive constant. We have associated to problem (Pτ) the functional and the associated Nehari manifold From Appendix there exists w τ ∈ W 1,q 0 (Ωϵ) such that Since λ is the parameter which appears in the hypothesis (f 5 ) we have the following result.

Proof of the Theorem 3.1
Proof. The proof is a consequence of Lemma 3.2, Lemma 3.4 and Lemma 3.5.

The Autonomous Problem
In order to prove the concentration result, we consider the following problem which the functional associated I 0 is given by and the corresponding Nehari manifold is given by We also define c 0 = inf N0 I 0 .
Using the same arguments of the prove of Lemma 3.5, we conclude that The next result allows to show that problem (P 0 ) has a solution that reaches c 0 . Hence, from (f 1 ) − (f 3 ), Since we also have (g 3 ) and that I ′ ϵn (un)un = on(1), we get We claim that l > 0. Indeed, if the claim is not true then, by (a 1 ) and (b 1 ), we have c 0 = 0 which is a contradiction. Therefore By definition of the constant S, we have Using (4.3), (4.4) and that c 0 > 0, we obtain c 0 ≥ 1 θ − 1 q * S N/q which is a contradiction with (4.1).
We are going to show that the problem (P 0 ) has a solution that reaches the level c 0 .

Lemma 4.2. (A Compactness Lemma) Let (u n)
⊂ N 0 be a sequence satisfying I 0 (un) → c 0 . Then there exists a sequence (ỹn) ⊂ R N such that, up to a subsequence, v n(x) = un(x + yn) converges strongly in W 1,p (R N ) ∩ W 1,q (R N ). In particular, there exists a minimizer for c 0 .
Proof. Applying Ekeland's Variational Principle (see Theorem 8.5 in [34]), we may suppose that (u n) is a (PS)c 0 for I 0 . Since (un) is bounded in W 1,p (R N ) ∩ W 1,q (R N ) we can assume, up to subsequences, that un u in W 1,p (R N ) ∩ W 1,q (R N ).
Using arguments found in [  By (4.5), (a 1 ), (b 1 ) and Lebesgue's theorem we conclude that un → u in W 1,p (R N ) ∩ W 1,q (R N ). Consequently, I 0 (u) = c 0 and the sequence ( yn) is the null sequence.
If u ≡ 0, then in that case we cannot have un → u strongly in W 1,p where v n := un(x + yn). Therefore, (vn) is also a (PS)c 0 sequence for I 0 and v ≢ 0. It follows from the above arguments that, up to a subsequence, (v n) converges strongly in W 1,p (R N ) ∩ W 1,q (R N ) and the proof is complete.

Concentration results
In this section we prove some technical results in order to show the concentration result. Proof. Since V satisfies (V 1 ) and c 0 > 0, we repeat the same arguments in Lemma 4.1 to conclude that there exist positive constants R and β and a sequence ( which implies that I 0 (ṽn) → c 0 , as n → +∞. From boundedness of (v n) and (5.2), we obtain that (tn) is bounded. As a consequence, the sequence (ṽn) is also bounded in W 1,p (R N ) ∩ W 1,q (R N ) which implies, up to a subsequence,ṽn ṽ weakly in W 1,p (R N ) ∩ W 1,q (R N ).
Note that we can assume that t n → t 0 > 0. Then, this limit implies thatṽ ≢ 0. From Lemma 4.2, we conclude thatṽ n →ṽ in W 1,p (R N ) ∩ W 1,q (R N ) and this implies that vn → v in W 1,p To conclude the proof of this proposition, we consider y n := ϵnỹn. Our goal is to show that (yn) has a subsequence, still denoted by (y n ), satisfying y n → y for y ∈ Ω. First of all, we claim that (y n ) is bounded. Indeed, suppose that there exists a subsequence, still denote by (y n), verifying |yn| → ∞. From (a 1 ), (b 1 ) and (V 1 ) we have Fix R > 0 such that B R (0) ⊃ Ω and let X B R (0) be the characteristic function of B R (0). Since X B R (0) (ϵx + yn) = on (1) for all x ∈ B R (0) and vn → v in W 1,p (R N ) ∩ W 1,q (R N ), then R N X B R (0) (ϵx + yn)g(ϵx + yn , vn)vn dx = on (1).
By definition of f we obtain that It follows that v n → 0 in W 1,p (R N ) ∩ W 1,q (R N ), obtain this way a contradiction because c 0 > 0. Hence (y n) is bounded and, up to a subsequence, Arguing as above, if y ∉ Ω we will obtain again vn → 0 in W 1,p (R N ) ∩ W 1,q (R N ), and then y ∈ Ω. Now if V(y) = V 0 , we have y ∉ ∂Ω and consequently y ∈ Ω. Suppose by contradiction that V(y) > V 0 . Then, we have Using the fact that v n → v in W 1,p (R N ) ∩ W 1,q (R N ), from Fatou's Lemma we obtain Since u n ∈ Nϵ n , this implies that Proof. Note that v n is a solution of problem −div a |∇vn| p |∇vn| p−2 ∇vn where yn = ϵnỹn. Adapting some arguments explored in [4, Lemma 5.5], we have that the sequence (vn) is bounded in L ∞ (R N ) and there exist R > 0 and n 0 ∈ N such that vn L ∞ (R N \B R (0) < ξ , for all n ≥ n 0 .
Hence from continuity of V it follows that We claim that V(y) = V 0 . Indeed, suppose by contradiction that V(y) > V 0 . Then, we have Using that vn → v in W 1,p (R N ) ∩ W 1,q (R N ) we obtain, from Fatou's Lemma, and therefore Proof. Up to a subsequence, From Lemma 5.3 we have that I ϵn (uϵ n ) → c 0 , and there exists a positive constant C such that uϵ n ≤ C, ∀ n ∈ N , for some C > 0.
Since v ≠ 0, by Proposition 5.1 we have y n = 0, for n ∈ N. Moreover, recalling that V is continuous, we have We claim that V(x) = V 0 . Indeed, Suppose by contradiction that V(x) > V 0 , then Thus, by (5.4) and Fatou's Lemma, we have  Hence, and then there exists a sequence (xn) ⊂ ∂Ωϵ n , such that uϵ n (xn) ≥ δ 2 .
Repeating the above arguments, we will get an absurd. Thus, the proof is finished.

Exponential decay of the solution u ϵ
Finally, we are going to prove the exponential decay. First technical results Proof. Note that this prove the first item.
In order to show the unicity of t u, consider f (t) = t τ and note that f (t) t q is increasing.