On a functional satisfying a weak Palais-Smale condition

In this paper we study a quasilinear elliptic problem whose functional satisfies a weak version of the well known Palais-Smale condition. An existence result is proved under general assumptions on the nonlinearities.


Introduction
The aim of this paper is to generalize a recent result obtained in [2] concerning the following quasilinear elliptic problem −∇ · [φ ′ (|∇u| 2 )∇u] + |u| α−2 u = f (u), x ∈ R N , u(x) → 0, as |x| → ∞, (1) where N 2, φ(t) behaves like t q/2 for small t and t p/2 for large t, 1 < p < q < min{p * , N}, 1 < α p * q ′ /p ′ , being p * = pN N −p and p ′ and q ′ the conjugate exponents, respectively, of p and q. In [2] the authors have proved that if f (t) = |t| s−2 t grows as t goes to +∞ more than max{t a−1 , t q−1 } and less than t p * −1 and φ ∈ C 1 (R + , R + ) satisfies    then the problem possesses infinitely many solutions. As remarked in that paper, the same result can be obtained if we assume more general hypotheses on the nonlinearity f . In particular, apart from the local assumptions related with the behaviour at 0 and at infinity, it is required the following global condition where θ > α and F (t) = t 0 f (z) dz. This assumption, known as the Ambrosetti-Rabinowitz condition, (AR) in short, is quite classical in the field of critical points theory and typically occurs when we try to prove the boundedness of the Palais-Smale sequences related with the functional of the action. However, some papers have shown that there are many situations in which (AR) can be successfully bypassed. In [6], for instance, the equation is solved without (AR) in two steps. First the authors reduce the problem to that of minimizing a constrained (bounded below) functional, obtaining a solution to the equation, up to a Lagrange multiplier. Then they exploit the behaviour of the equation with respect to the rescaling to make the Lagrange multiplier disappear. More recently, in [13] and [11] it has been shown a method, named monotonicity trick, which exploits the differentiability a.e. of the monotone functions to get bounded Palais Smale sequences for functionals related with approximating equations. If there is no problem of compactness, this method allows us to get a Palais Smale sequence constituted by solutions of approximating equations. Afterwards, getting some more information on the elements of the sequence using the fact that they solve an equation, we could prove the boundedness of the Palais-Smale sequence. Usually, the additional information we look for is the well known Pohozaev identity, an equality satisfied by sufficiently regular solutions of elliptic equations in the divergence form.
In our situation, a different approach is required. Consider the problem where we will assume on g hypotheses similar to those in [6].
Observe that, since φ ′ is not homogeneous, we can not proceed as in [6]. On the other hand, also the monotonicity trick does not seem to be of use. Indeed, since no regularity result on the solutions of (3) is available, we can not obtain a Pohozaev identity in a standard way. To overcome these difficulties, we use a result contained in [10], where an alternative way of approaching (2) is showed. The method presented consists in adding a dimension to the space where the problem is set, and constructing a Palais-Smale sequence for a suitable auxiliary functional defined in this new space. Such a technique, which we call the adding dimension technique, permits to get additional information on a Palais-Smale sequence of the original functional and, possibly, to prove it is bounded. We remark the fact that this method does not require any regularity assumption on the solutions of the equations.
It is worthy of note that, differently from the functional related with (2), the functional of the action associated with (3) will be defined on a particular Orlicz-Sobolev space. Treating with this space carries some more complications when we try to solve (3) with a nonlinearity modeled on that in [6]. To explain better what we mean, we list our assumptions on g and state the main result.
Comparing the main result in [2] with ours, we note that the prize we have to pay to generalize the nonlinearity is a more restrictive assumption on α, which we require is not too close to 1. This fact arises from a significant difference between the classical embedding results known for Sobolev spaces, and the embedding results available for the Orlicz-Sobolev space where we set our problem. To clarify this point, we recall a well known fact. Consider D(R N ), the set of all C ∞ function in R N with a compact support and set 1 < p < N. Define the norm · D 1,p such that for all u ∈ D, It is well known that D 1,p (R N ) ֒→ L p * (R N ), so that, for any u ∈ D 1,p (R N ), If 1 < α and · α is the usual Lebesgue norm, we define Of course, since V is continuously embedded in D 1,p (R N ), inequality (5) holds true for any u ∈ V. Observe that, if φ(t) = t p 2 , the space V would be a nice set to study problem (1). If we want to proceed analogously in our situation, we have to construct a space W taking into account (Φ2) and (Φ3). We have to substitute the norm · D 1,p , with a Luxemburg norm to be computed on ∇u. Assumptions (Φ2) and (Φ3) suggest to use the norm of the space L p (R N ) + L q (R N ), which we call · p,q and to replace Sobolev space D 1,p (R N ) with the Orlicz-Sobolev there is no continuous embedding of D 1,p,q (R N ) in any Lebesgue space (see Remark 1.8). However, in [2] it has been proved that, if we define the analogous of V in the following way then the following inequality holds true From this, we deduce that W ֒→ L r (R N ) for any α r p * , even if, differently from V, it is not possible to control the L p * (R N ) norm just with the L p (R N ) + L q (R N ) norm of the gradient. This fact translates to a technical difficulty in proving the Palais Smale condition. Precisely, we will show that the functional of the action satisfies a compactness condition weaker than the Palais-Smale if α is not too close to 1.
Finally, we point out that, since we do not require assumption (Φ5), our existence result holds for function φ more general than those treated in [2].
The paper is so organized: in section 1 we will introduce the functional setting, and the related properties we will use in our variational approach to the problem. For the most part, the results contained in this section are proved in [2] and [4] so we only recall them. In section 2 we will define a new weakened version of the Palais-Smale condition, and we will introduce the definition of a particular type of Palais-Smale sequences. Finally, in section 3, we will prove our main result by means of the adding dimension technique introduced in [10].
• If r > 0, we denote by B r the ball of center 0 and radius r.
• Everytime we consider a subset of R N , we assume it is measurable and we denote by | · | its measure.
• We denote by D the space of all functions in C ∞ (R N ) with compact support.
• C and c will denote generic constants which would change from line to line.

The functional setting
This section is devoted to the construction of the functional setting. As a first step, we have to recall some known facts on the sum of Lebesgue spaces.
In the next proposition we give a list of properties that will be useful in the rest of the paper. The reader can find the proofs in [2] and [4] We have: We can now define the Orlicz-Sobolev space where we will study our problem. Definition 1.3. We assume the following definition Moreover, if α > 1, we denote with W the following space where · = · α + ∇ · p,q .
We again refer to [2] for the proofs of the following propositions and theorems on the space W where C is a positive constant which does not depend on u and t > 1 satisfies the Theorem 1.6. If 1 < p < min{q, N} and 1 < p * q ′ p ′ then, for every α ∈ 1, p * q ′ p ′ , the space (W, · ) is continuously embedded into L p * (R N ).

Remark 1.7. By interpolation we have that
Remark 1.8. The normed space (W, ∇ · p,q ) is not continuously embedded in any Lebesgue space. Indeed, consider ϕ ∈ D(R N ) and for any t > 0 set Of course for any t > 0 the function ϕ t ∈ W and we have that Since ∇ϕ t p,q ∇ϕ t p , we deduce that the family (ϕ t ) t>0 is bounded in (W, ∇ · p,q ). On the other hand, if r = p * , we can make the Lebesgue norm as large as we want just taking large t, if 1 < r < p * and small t, if p * < r. So (W, ∇ · p,q ) does not embed in any L r (R N ), for 1 < r = p * . We see that (W, ∇ · p,q ) does not embed even in L p * (R N ) just observing that, if for any s > 0 we set ϕ s = s q−N q ϕ( · s ), then sup s>0 ∇ϕ s p,q sup s>0 ∇ϕ s q < +∞, sup s>0 ϕ s p * = +∞.
Now we define the functional of the action related with our problem. For any u ∈ W we set (from now on, we omit the symbol dx in the integration) where G : R → R is defined as in assumption (BL). To make the functional well defined and C 1 , we modify g according to the following two possibilities: 1st case: lim inf s→+∞ g(s) Sinceg satisfies (g3) lim s→∞ |g(s)| |s| p * −1 = 0, by [4] and Theorem 1.6 we can prove J is well defined and C 1 in W if we replace g withg. On the other hand, we point out that, if u ∈ W solved equation (3) withg in the place of g, then 0 u and, if the second case occurred, then we also would have u s 0 . As a consequence, we deduce that no loss of generality would arise supposing that g is defined asg and (g3) holds.
Some simple computations show that for functions g 1 and g 2 the following properties hold (i) g 1 and g 2 are nonnegative in R + , (iv) there exists a positive constant a such that at α−1 g 2 (t), for any t ∈ R + , (v) for any ε > 0 there exists C ε > 0 such that g 1 (t) εg 2 (t) + C ε t p * −1 , for any t ∈ R + .
Once we have set G i (z) := z 0 g i (s) ds > 0 for i = 1, 2, we have that the functional can be written In order to have compactness, we introduce a symmetry requirement on our space. Indeed, take u ∈ W r , and consider the set Λ ∇u . Since ∇u p,q < +∞, certainly ∇u ∈ L p (Λ ∇u ). On the other hand, since u ∈ L α (R N ), we have that By symmetry of u, the set Λ ∇u has a smooth boundary so, by standard arguments (see for example [1]), there exists a continuous extension operator T : E(Ω) → E(R N ). Then embedding theorems hold in the domain Λ ∇u so we deduce that u ∈ L s (Λ ∇u ) for any s ∈ [α, p * ]. Analogously u ∈ L s (Λ c ∇u ), for any s ∈ [α, q * ]. Since p * < q * , we conclude. At the present stage of our knowledge, we do not know if, for p * q ′ p ′ < α < p * , these embeddings are also continuous.
The following compactness result holds. Theorem 1.11. If 1 < p < q < N and 1 < α p * q ′ p ′ , then the functionals are weakly continuous.
Proof We prove the weak continuity of the first functional. By Lemma 2.13 in [2], for any u ∈ W r , Now, consider a sequence (u n ) n in W r weakly convergent to u 0 . By Theorem 2.11 in [2], (u n ) n possesses a subsequence strongly convergent to u 0 in L τ (R N ) for any τ ∈]α, p * [. So, up to subsequences, we can assume that (u n ) n converges almost everywhere to u 0 .
. By property (iii) of the function g 1 , remark 1.7 and (10), we can apply the Strauss compactness Lemma in the version as it appears in [6] and conclude. In a similar way we prove the rest of the statement.

A weak Palais-Smale condition
As it is well known, the Palais-Smale condition is a compactness property related to a functional defined on a Banach space. It states as follows: let I : E → R be a C 1 functional on the Banach space E and c ∈ R. If for any given (x n ) n in E such that I(x n ) → c and I ′ (x n ) → 0 there exists a converging subsequence of (x n ) n , we say that I satisfies the Palais Smale condition at the level c. Usually, in the calculus of variation, testing the Palais Smale condition consists in two steps: first we check if every Palais Smale sequence (namely a sequence verifying the previous assumptions) is bounded, second we handle with compact embedding theorems to prove strong convergence (up to a subsequence) in the Banach space. Many times it happens that the main problem in verifying Palais-Smale condition is related with the first step. In such cases, one tries to prove that the functional satisfies at least a weakened version of the Palais-Smale condition, and look for the existence of at least one sequence to which that condition can be applied.
In this direction, we recall, for example, the well-known Cerami version of the Palais-Smale condition (see [8]), and the problem in [5] where this condition is applied.
Here we introduce a new version of a weakened Palais-Smale condition.
there exists a converging subsequence (in the topology of E).

Remark 2.2. Consider the functional of the action related with the equation
where g is as in [6]. After having produced a suitable modification of the function g, we can see that finite energy solutions of the equation are critical points of being G a primitive of g. The properties on g 1 and g 2 listed in (i) . . . (v) hold, except that we have to replace α with 2 and p * with 2 * . We show that I verifies a weak Palais-Smale condition with respect to H 1 r (R N ) and D 1,2 r (R N ), each one provided with its natural norm. Indeed suppose (u n ) n is a Palais Smale sequence for which ( ∇u 2 ) n is bounded. Since I(u n ) is bounded, for a certain M > 0 and any ε ∈]0, 1[ there exists a suitable C ε > 0 for which we have S is the Sobolev constant for the embedding D 1,2 (R N ) ֒→ L 2 * (R N ). Then and, since ( ∇u n 2 ) n is bounded, we conclude that ( u n 2 ) n is also bounded since we have At this point the arguments are quite standard: we extract a weakly convergent (in H 1 -norm) subsequence and we use radial symmetry of functions in our space and a Strauss compactness lemma to find a strong convergent sequence.
We have the following result Proof Suppose (u n ) n is a sequence of functions in W r such that 1, 2 and 3 of definition 2.1 hold. We first prove that the sequence is bounded. By computations analogous to those in remark 2.2, there exist M > 0 and C 1 > 0, such that Now, if (u n ) n is bounded in the L p * −norm, we have concluded. By (9) and 3 of definition 2.1, for some C 2 > 0. Suppose that ( u n p * ) n diverges (up to a subsequence). Then, by (13), certainly there exists a constant C such that, definitely, Comparing (12) and (14), taking into account that t t−1 = p ′ N ′ and a u n α α R N G 2 (u n ), we have, for some positive constant C, and then, since α > N ′ p * p ′ , the sequence ( u n α ) n is bounded. Therefore, by Proposition 1.4 and Theorem 1.11, there exists u 0 ∈ W r such that, up to subsequences, and, by [2,Theorem 2.11], From this point till the end, the proof follows the scheme of Proposition 3.3 in [2] and of Proposition 2 in [7], step 9.1c (see also [3,Lemma 3.5]). We point out only the key passages. By (15) and arguing as in [12, page 208], we have u n ⇀ u 0 , weakly in L α (R N ).
As in [3] we prove that for any z ∈ C ∞ 0 (R N ), we have Set

By (20), and since
By convexity, we have and then, passing to the limit, Since, by weak lower semicontinuity of A 1 we also have we conclude that lim By (18) and (21), we can deduce (see [9]) Moreover, since we are able also to prove that u n → u 0 in L α (R N ) and we conclude.

Proof of the main Theorem
In view of Lemma 2.3, we have just to find a level for which we can find a Palais-Smale sequence satisfying the boundedness assumption 3 in the Definition 2.1.

Lemma 3.1. The set
is nonempty.
Proof Starting from the function z ∈ D(R N ) for which R N G(z) > 0 (the existence of such a function is proved in [6]), the proof is standard. Indeed consider z l (·) = z(·/l) for a value of l > 0 to be established and compute We deduce that J(z l ) < 0 if l is sufficiently large. At this point any continuous path connecting 0 with z l is in Γ. Proof Of course it is enough to verify the following geometrical mountain pass assumptions: there exist δ, ρ > 0 such that • J(u) δ, for all u ∈ W r such that u = ρ • J(u) 0, for all u ∈ W r such that u ρ.
By (Φ2), (vi) of Proposition 1.2 and since W ֒→ L p * (R N ), we have that, if u is sufficiently small (note that p < q) Taking respectively u = ρ or u ρ with ρ > 0 sufficiently small we conclude.
We introduce the following auxiliary functional on the space R × W r In analogy with Γ and c mp , we define  Proof We estimate the functional J. Since φ is increasing in R + , by similar computations as those in Lemma 3.2, for small u we have: So we deduce that there exists δ > 0 such that I(θ, u) is nonnegative if θ 2 + u 2 δ, and it is positive for θ 2 + u 2 = δ. As in Lemma 3.1 we can prove the existence of (θ,ū) ∈ R × W r for which J(θ,ū) < 0.
Taking into account that θ n → 0, from (29) we deduce that J ′ (ũ n ) → 0 in W ′ r . Then (ũ n ) n satisfies 1, 2, and 3 of Definition 2.1. We conclude with the proof of our main Theorem Proof [Proof of Theorem 0.1] Let (u n ) n be a sequence as in Proposition 3.4. By Theorem 2.3, we can extract a subsequence, relabeled (u n ) n , strongly convergent to some u 0 ∈ W r . Finally, it is enough to observe that, by Lemma 3.2 and Proposition 3.3, u 0 = 0. Moreover u 0 0 by definition of g 1 .