Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth

It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by $\Phi$-Laplacian operator. One of these solutions is built as a ground state solution. In order to prove our main results we apply the Nehari method combined with the concentration compactness theorem in an Orlicz-Sobolev framework. One of the difficulties in dealing with this kind of operator is the lost of homogeneity properties.

(3) Other examples, for instance involving anisotropic elliptic problems, can be seen in [8] and references therein.
The main difficulty in dealing with this kind of operator is because it is inhomogeneous, which requires some aditional effort to overcome the estimates. As is mentioned in [24] the problem has many physical applications, for instance, in nonlinear elasticity, plasticity, generalized Newtonian fluids, etc. We refer the reader to the following related papers [2,15,16,17,21,24] and in references therein, where there have handled handled different types of nonlinearities involving this kind of operator. Problems like above was started in a beautiful work due to Brézis and Nirenberg [3], when ∆ Φ = ∆, where they treated a nonhomogeneous problem with critical growth obtaining existence result, assuming that f ≥ 0 =0 , together with some aditional conditions. Then Tarantello [26] treated the same problem getting existence and multiplicity results under a stronger hypothesis that made in [3]. These works were extended in [20],which was obtained four weak solutions, at least one of them is sign changing solution. On the other hand, in [12] is proved some multiplicity results for symmetric domain by using the category theory. There are only few works involving p− Laplacian, that is, when ∆ Φ = ∆ p , extending results in [26]. We would like to mention [7,11] and references therein.
Due to the nature of the operator ∆ Φ we shall work in the framework of Orlicz-Sobolev spaces W 1,Φ 0 (Ω). Throughout this paper we define which is extended as even function, Φ(t) = Φ(−t), for all t < 0. Recall that hypotheses (φ 1 ) − (φ 2 ) allow us to use the Orlicz and Orlicz-Sobolev spaces, while the hypothesis (φ 3 ) ensures that the Orlicz-Sobolev spaces are Banach reflexive spaces. There are several publications on Orlicz-Sobolev spaces, we would like to recommend the reader to [1,13,16,19,23,24]. However, for the sake of completeness, we recall some definitions and properties in the Appendix.
In order to perfom our precise hypotheses for our results, we will consider the functions g α : [0, ∞) → R, α ∈ {ℓ, m} defined by It is easy to see that there exists t α > 0 such that Inspired by [27], given u ∈ W 1,φ 0 (Ω), with ||u|| ℓ * = 1, we assume the following assumptions on f.
We have a second solution to the problem (1.1) considering a more restrictive condition given by: Our first main result can be read as follows teorem1 Theorem 1.1. In addition to (φ 1 )−(φ 3 ) and (H), suppose f ≥ 0
Now we shall consider the following result teorema2 , and f ∈ L ℓ * ′ (Ω), and either (f 1 ) or (f 2 ) ′ holds. Then there exists Λ 2 > 0 in such way that problem (1.1) admits at least one positive solution u − satisfying J(u − ) > 0 for any f verifying Putting together the all results established just above and using a regularity result for quasilinear elliptic problems we can state the following multiplicity result. , and f ∈ L ℓ * ′ (Ω).
Remark 1.2. We point out that concerning just existence of solution, f can change sign, see Lemma 2.6. However in such case the solution could change sign, as well.

Preliminary results
In this section we give some basic results involving the Nehari manifold method, including the fibering maps associated with the functional J, which will give information on the critical points of Euler-Lagrange functional J. We suggest the reader to the book due to Willem [28], for an overview on the Nehari method. The proofs of our results follow closely the arguments used in [9,10].
The Nehari manifold associated with the functional J is given by It will be proved later on that N is a C 1 -submanifold of W 1,Φ 0 (Ω). Initially, note that if u ∈ N , by (2.4), we have that or equivalently First of all we shall prove some geometric properties of functional J, which allows us to find a critical point for J. coercive Proposition 2.1. The functional J is coercive and bounded from below on N .

Now by combining
with the Hölder inequality and the continuous embedding where S is given by (1.2). Thus, J is coercive and bounded from below on N . The proposition is proved. Now, define the fibering map γ u : (0, +∞) → R given by From (φ 1 ) − (φ 2 ) it follows that γ u is of C 1 , and its Gateaux derivative is given by The main feature of the fibering map is the knowledge of the geometry of γ u , which will give information about the existence and multiplicity of solutions. This method was introduced in [14], then it was also employed, for instance, in [4,5,6,26,27,29,30] and references therein.
Remark 2.1. Notice that tu ∈ N if, and only if, γ ′ u (t) = 0. Therefore, u ∈ N if, and only if, γ ′ u (1) = 0. Thus, the stationary points of fibering map are the critical points of J on N .
The next result is the crucial step in our argument to prove the main result.
Proof of item (2). Suppose without loss of generality that, u ∈ N + . Define G(u) := J ′ (u), u . We can see that Furthermore, using (2.4), we also have that J ′ (u), u = 0. Hence, 0 ∈ R is a regular value for G and N + = G −1 (0). That is, N + is a C 1 -manifold. Similarly , we may show that N − is a C 1 -manifold. Hence, since we are supposing (f 1 ) and (f 2 ), the proof of item (2) follows in virtue of N 0 = ∅.
Next we are going to prove that any critical point for J on N λ is a free critical point, i.e, is a critical point in the whole space W 1,Φ 0 (Ω). Actually, the proof of the Lemma below is fairly standard and we include it for the sake of completeness.
Proof. Suppose without any loss of generality that u 0 is a local minimum of J. Define the function Then u 0 is a solution for the minimization problem min {J(u), θ(u) = 0} .
(2.14) lagrange Proceeding as in Carvalho et al. [10], we have holds true for all u, v ∈ W 1,Φ 0 (Ω). Making u = v = u 0 , since u 0 ∈ N + , by (2.4) and (2.10), we get From Lemma 2.1, the problem (2.14) has a solution verifying where µ ∈ R which is given by Lagrange multipliers Theorem. Notice that θ ′ (u 0 ), u 0 = 0, then µ = 0, i.e, u 0 is a critical point for J on W 1,Φ 0 (Ω). The proof of lemma is complete. Now we give a complete description on the geometry for the fibering map associated with problem (1.1), where we will foccus on the sign of Ω f u.

Consider the auxiliary function
where the points tu ∈ N will compared with the function m u .
Proof. Fix t > 0 in such may that tu ∈ N . Then From the definition of m u , the proof of the result follows.
The next lemma will give a precise information on the function m u and the fibering map.

m_u-comp
Lemma 2.4. There exists an unique critical point for m u , i.e, there is an unique pointt > 0 in such way that m ′ u (t) = 0. Furthermore, we know thatt > 0 is a global maximum point for m u and m u (∞) = −∞.
Taking into account (φ 3 ) it is easy to verify that Firstly, we prove that m u is increasing for t > 0 small enough and lim Since m < ℓ * we mention that m ′ u (t) > 0 for any t > 0 small enough. Arguing as above we obtain Therefore, since m < ℓ * , we infer that lim t→∞ m u (t) = −∞.
Next, we will show that m u has an unique critical pointt > 0. Observe that m ′ u (t) = 0 if, and only if, Define the auxiliary function η u : R → R by Using the inequality below it is easy to see that lim On the other hand, from Proposition 5.2, for any t > 1, we have Hence ( Using hypothesis (φ 3 ) we have The proof of this lemma is now complete.
Next we will estimate max t>0 m u (t). To do this, consider g α , α = ℓ, m, defined in (1.3). As in the proof of the previous Lemma, there exists t α > 0, given by Proof. If We will consider three possibilities, namely: On the other hand, using Proposition 5.1 and inequality This finishes the proof of lemma.
fib Lemma 2.6. Let u ∈ W 1,Φ 0 (Ω)/{0} be a fixed function. Then we shall consider the following assertions: (1) there exists an unique Proof. First of all, notice that arguing as in [5], it is easy to see that if tu ∈ N , The case We emphasize that m ′ u (t 1 ) < 0, because m u is a decreasing function in (t, ∞). Therefore, using Lemma 2.3, we have t 1 u ∈ N , proving that γ ′ u (t 1 ) = 0. Additionally, by the identity (2.25) The case Ω f u > 0. We can consider Lemma 2.5 and we get which m u is increasing in (0,t) and decreasing in (t, ∞). It is not hard to verify that there exist exactly two points 0 < t 1 = t 1 (u) <t < t 2 = t 2 (u) such that As in the previous step we infer that t 1 u ∈ N + and t 2 u ∈ N − . This completes the proof. nehari- Proof. Since u ∈ N − , we have that ψ ′ (u), u < 0. Arguing as in the proof of Lemma 2.1, we obtain Moreover, in view of (2.7) and the Sobolev imbedding, we have that By the above inequality, we get given by (f 1 ). On the other hand, if (f 2 ) holds, we have Hence, in either case (f 1 ) ou (f 2 ) ′ , we conclude that J(u) ≥ δ 1 , for all u ∈ N − . Thus, Consequently, On the other hand, if u ∈ N , using the above inequality and (φ 3 ), we get Since N = N − ∪ N + and α − > 0, we have that α + = α, and the Lemma is proved.

The (PS) condition
Here we follow same ideas discussed in Tarantello [26], in order to prove some auxiliary results to get the Palais-Smale conditon for the functional J constrained to the Nehari manifold.
Here we observe that F u (1, 0) = ψ(u). As a consequence, for each u ∈ N , we have By using the Inverse Function Theorem, there exist ǫ > 0 and a differentiable function ξ : Here ∂ 1 F u and ∂ 2 F u denote the partial derivatives on the first and second variable, respectively.
On the other hand, after some manipulations, putting w = 0 and ξ = ξ(0) = 1, we have Here was used the fact that ∂ 1 F u (1, 0) = ψ ′ (u), u holds for any u ∈ N . The proof is complete.
Similarly, we have the following Furthermore, we obtain Next, we shall prove that any minimizing sequences on the Nehari manifold in N + or N + provides us a Palais-Smale sequence.
Proof: Remember that (u n ) ⊂ N , mΦ(t) ≤ φ(t)t 2 and arquing as in the proof of Lemma 2.7, we infer that holds for any n ∈ N large enough. By using the above inequality and the continuous embedding W 1,Φ 0 (Ω) ֒→ L ℓ * (Ω), we deduce that and (3.28) holds. Furthermore, using (3.30) and arguing as in (2.12), we obtain that (3.31) Hence the last assertions give us where α ∈ {ℓ, m}. Now we will prove two technical results, which will be used to prove that any minimizing sequence for J constrained to the Nehari manifold is a Palais-Smale sequence.
Using Lemma 3.1 we infer that Notice also that we have the following convergences as ρ → 0, for any n ∈ N. Applying Mean Value Theorem, there exists t ∈ (0, 1) in such way that Remind that ||u n − µ ρ || → 0 as ρ → 0. Since µ ρ ∈ N + and using (3.33) and (3.34), we obtain where o ρ (.) denotes a quantity that goes to zero as ρ goes to zero. Using that J ′ (µ ρ ), µ ρ = 0, we have From the above estimates and (3.34) we obtain Noticing that from this inequality we have ||µ ρ −u n || ≤ ρ|ξ n (w ρ )|+|ξ n (w ρ )−1| ||u n || and lim Therefore, using the fact that (u n ) is bounded and (3.35), we infer that On the other hand, since ξ n (w ρ ) − 1 ρ and ξ n (w ρ ) are bounded for ρ > 0 small enough, we obtain Since (u n ) is bounded there exists a constant C > 0 in such that Putting all these estimates together we prove (3.32) holds. Proof: Firstly notice that the numerator in (3.26) is bounded from below away from zero by b||v|| where b > 0 is a constant. Define the auxiliary function χ n : W 1,Φ 0 (Ω) → R given by
To sum up, using the estimate (3.36), we can be shown that Arguing as in the proof of Lemma 2.1, by the above inequality and (3.37) we have a contradiction since either (f 1 ) or (f 2 ) hold. This completes the proof. there exist C > 0 independent on n ∈ N such that ξ n (0) ≤ C. This estimate together with Proposition 3.3 This implies that J ′ (u n ) → 0 as n → ∞. This finishes the proof.
4. The proof of our main theorems 4.1. The proof of Theorem 1.1. We are going to apply the following result, whose proof is made by using the concentration compactness principle due to Lions for Orlicz -Sobolev framework, see [28] or else in [9,16].
From Lemma 2.8 we infer that We will find a function u ∈ N + in such that J(u) = min u∈N + J(u) =: α + and J ′ (u) ≡ 0. First of all, using Proposition 3.1, there exists a minimizing sequence denoted by Since the functional J is coercive in N + , this implies that (u n ) is bounded in N + . Therefore, there exists a function u ∈ W 1,Φ 0 (Ω) such that u n ⇀ u in W 1,Φ 0 (Ω), u n → u a.e. in Ω, u n → u in L Φ (Ω). (4.39) convergencia We shall prove that u is a weak solution for the problem elliptic problem (1.1). Notice that, by (4.38), we mention that holds for any v ∈ W 1,Φ 0 (Ω). In view of (4.39) and Lemma 4.1 we get for any v ∈ W 1,Φ (Ω) proving that u is a weak solution to the elliptic problem (1.1).
In addition, the weak solution u is not zero. In fact, using the fact that u n ∈ N + , we obtain From (4.38) and (4.39) we obtain Hence u ≡ 0.
We shall prove that J(u) = α + and u n → u in W 1,Φ 0 (Ω). Since u ∈ N we also see that Notice that t → Φ(t) − 1 ℓ * φ(t)t 2 is a convex function. In fact, by hypothesis (φ 3 ) and m < ℓ * , we infer that In addition, the last assertion says that is weakly lower semicontinuous function. Therefore we obtain This implies that J(u) = α + . Additionally, using (4.39), we also have From the last identity In view of Brezis-Lieb Lemma, choosing v n = u n − u, we infer that The previous assertion implies that Therefore, we obtain that lim Ω Φ(|∇v n |) = 0 and u n → u in W 1,Φ (Ω). Hence we conclude that u n → u in W 1,Φ 0 (Ω).
We shall prove that u ∈ N + . Arguing by contradiction we have that u / ∈ N + . Using Lemma 2.6 there are unique t + 0 , t − 0 > 0 in such way that t + 0 u ∈ N + and t − 0 u ∈ N − . In particular, we know that t + 0 < t − 0 = 1. Since d dt J(t + 0 u) = 0 and using (4.40) together the Lemma 2.6 we have that which is a contradiction to the fact that u is a minimizer in N + . So that u is in N + . To conclude the proof of theorem it remains to show that u ≥ 0 when f ≥ 0. For this we will argue as in [26]. Since u ∈ N + , by Lemma 2.6 there exists a t 0 ≥ 1 such that t 0 |u| ∈ N + and t 0 |u| ≥ |u|. Therefore if f ≥ 0, we get . So we can assume without loss of generality that u ≥ 0. Since J is coercive in N and so on N − , using Lemma 2.1, we have that (v n ) is bounded sequence in W 1,Φ 0 (Ω). Up to a subsequence we assume that v n ⇀ v in W 1,Φ 0 (Ω) holds for some v ∈ W 1,Φ 0 (Ω). Additionally, using the fact that ℓ * > 1, we get t << Φ * (t) and W 1,Φ 0 (Ω) ֒→ L 1 (Ω) is also a compact embedding. This fact implies that v n → v in L 1 (Ω). In this way, we can obtain Now we claim that v ∈ W 1,Φ 0 (Ω) given just above is a weak solution to the elliptic problem (1.1). In fact, using (4.42), we infer that holds for any w ∈ W 1,Φ 0 (Ω). Now using Lemma 4.1 we get So that v is a critical point for the functional J. Without any loss of generality, changing the sequence (v n ) by (|v n |), we can assume that v ≥ 0 in Ω.
Next we claim that v = 0. The proof for this claim follows arguing by contradiction assuming that v ≡ 0. Recall that J(tv n ) ≤ J(v n ) for any t ≥ 0 and n ∈ N. These facts together with Lemma 5.1 imply that Using the above estimate, Lemma 5.1 and the fact that (v n ) is bounded, we obtain holds for some C > 0. These inequalities give us It is no hard to verify that v n ≥ c > 0 for any n ∈ N. Using Proposition 5.1 we get min(t ℓ , t m ) ≤ o n (1)t + C holds for any t ≥ 0 where C = C(ℓ, m, ℓ * , Ω, a, b) > 0 where o n (1) denotes a quantity that goes to zero as n → ∞. Here was used the fact v n → 0 in L 1 (Ω). This estimate does not make sense for any t > 0 big enough. Hence v = 0 as claimed. Hence v is in N = N + ∪ N + . Next, we shall prove that v n → v in W 1,Φ 0 (Ω). The proof follows arguing by contradiction. Assume that lim inf n→∞ Ω Φ(|∇v n − ∇v|) ≥ δ holds for some δ > 0.
Recall that Ψ : R → R given by Since v ∈ N there exists unique t 0 in (0, ∞) such that t 0 v ∈ N − . It is easy to verify that This implies that This is a contradiction proving that v n → v in W 1,Φ 0 (Ω). Therefore v is in N − . This follows from the strong convergence and the fact that t = 1 is the unique maximum point for the fibering map γ v for any v ∈ N − . Hence using the same ideas discussed in the proof of Theorem 1.1 we infer that In particular, α − = J(v) and lim Ω Φ(|∇v n |) − 1 ℓ * φ(|∇v n |)|∇v n | 2 = Ω Φ(|∇v|) − 1 ℓ * φ(|∇v|)|∇v| 2 .