Multiple solutions for asymptotically $q$-linear $(p,q)$-Laplacian problems

We investigate the existence and the multiplicity of solutions of the problem $$ \begin{cases} -\Delta_p u-\Delta_q u = g(x, u)\quad&\mbox{in } \Omega,\\ \displaystyle{u=0}&\mbox{on } \partial\Omega, \end{cases} $$ where $\Omega$ is a smooth, bounded domain of $\mathbb R^N$, $1<p<q<\infty$, and the nonlinearity $g$ behaves as $u^{q-1}$ at infinity. We use variational methods and find multiple solutions as minimax critical points of the associated energy functional. Under suitable assumptions on the nonlinearity, we cover also the resonant case.


Introduction
We consider the following Dirichlet (p, q)-Laplacian problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain of R N (N ≥ 2) with Lipschitz boundary ∂Ω, 1 < p < q < ∞, and the nonlinearity g(x, •) is asymptotically q-linear, meaning that g(x, t) ∼ ℓ ∞ |t| q−2 t as |t| → ∞ for some constant ℓ ∞ .Some of the difficulties arising in the study of this problem come from the nonhomogeneity of the operator −∆ p − ∆ q .The interest in (p, q)-Laplacians and, in general, in non-homogeneous operators has considerably increased since the seminal papers [19,20] by Marcellini, in late eighties, on the regularity of minimizers of the so-called functionals with non-standard growth.In this setting, both (p, q) and p(x)-growth conditions have been widely cosidered, see [14] and the references therein for a comprehesive monograph on the Lebesgue and Sobolev spaces involved in the p(x) variable exponent case.More recently, in [1,13], many others progresses were achieved in the study of energy functionals related to the operators −div(|∇u| p−2 ∇u + a(x)|∇u| q−2 ∇u).In these papers, the weight function a(x) ≥ 0 switches two different elliptic behaviors, justifying the name of double phase functionals.These functionals were first introduced by Zhikov in [22] to provide models for strongly anisotropic materials.In that setting, the exponents p and q cannot be too far from each other, the bound q/p < (1 + α/N ) for some α ∈ (0, 1] is needed both to develop a regularity theory and to prove some classical inequalities like a Poincaré-type one, in suitable Orlicz-Sobolev spaces, see [12,Remark 2.19].On the other hand, (2, q)-Laplacian operators naturally arise in the Born-Infeld theory of nonlinear electromagnetism, where the leading operator, i.e., the Minkowskicurvature operator −div ∇u √ 1−|∇u| 2 , can be approximated by a truncated series of 2h-Laplacians, h ∈ N, cf.[7,8,16].In this case, the highest exponent in the truncated series should morally go to infinity, and so no bounds on q/2 are admissible.In the same spirit, in the problem under consideration, the exponents p and q can be arbitrarily far.We observe that all the arguments in this paper can be adapted to operators of the form −div(|∇u| p−2 ∇u + a(x)|∇u| q−2 ∇u), with the weight function a(x) satisfying a ∈ C 1 ( Ω), a > 0 (it is enough to endow the functional space W 1,q 0 (Ω) with the equivalent norm a(x)∇u q ).On the contrary, things change a lot for the more general case in which a(x) can vanish somewhere, this case should be treated in suitable Orlicz-Sobolev spaces, the above mentioned bound on q/p should be required, and, in particular, being p(1 + 1/N ) < p * , the exponent q should be taken p-subcritical.While in our setting, since a ≡ 1 never vanishes, we are allowed not to require any relation between q and p (note that q > p is not an assumption, since the roles of p and q are interchangeable).
Let us now introduce in details the hypotheses required on the nonlinearity of the problem.We assume that there exist ℓ ∞ ∈ R and f : so that the problem can be written as In what follows, we denote by σ(−∆ q ) the spectrum of the Dirichlet q-Laplacian operator, namely the set of λ ′ s in R for which the problem in Ω, u = 0 on ∂Ω has a nontrivial weak solution.The nonlinearity f satisfies the following assumptions: Problem (1.3) has a variational structure, so the solutions are found as critical points of the associated energy functional I : W 1,q 0 (Ω) → R defined as cf. Subsection 2.2.We observe that, beyond the non-homogeneity of the operator, another feature of the problem that makes the analysis more interesting is the behavior of the nonlinearity at infinity.Indeed, the asymptotic q-linearity of g(x, •) in particular implies that the Ambrosetti-Rabinowitz condition -which is responsible for Palais-Smale sequences to be bounded-is not satisfied.This prevents the use of the classical Mountain Pass or Symmetric Mountain Pass Theorem for the existence of solutions to problem (1.1), see the Introduction of [17] for interesting comments on the topic.For existence results in the case of q-superlinear and subcritical nonlinearities in the (p, q)-setting, we refer to [21] in bounded domains, and to [4] in the whole space R N .Due to the behavior of g(x, •) at infinity, one can expect some interaction with the spectrum of the q-Laplacian, cf.[3,17].In particular, when ℓ ∞ is an eigenvalue of −∆ q , a stronger assumption on f is needed to get compactness, see (f r ).Furthermore, due to the symmetry condition (f sym ), the energy functional I is even, so that if u is a critical point of I at some critical value c, also −u has the same property.This is the reason why in the statement of the main theorem below, we always refer to pairs of solutions.We believe that even without the symmetry condition (f sym ), it is possible to obtain existence results by applying a Linking Theorem as in [3].
In Section 2, we will introduce the definitions of two suitable sequences (η h ) of quasi-eigenvalues related to the (p, q)-Laplacian operator, with α = 0 or 1.For the exact definitions of (η h ), we refer to (2.5) and (2.11), respectively.These sequences play an important role in the proof of the geometry conditions for the functional I.
We are now ready to state our main result.
Theorem 1.1.Assume that (f ), (f ∞ ), (f cpt ), and (f sym ) hold.Suppose further that one of the following assumptions is fulfilled: k .Then, problem (1.3) has at least k − h + 1 distinct pairs of non-trivial solutions.
All the solutions found are minimax critical points of the associated energy functional.The proof of our main theorem is based on an abstract result proved in [2,3] by using the pseudo-index theory related to the Krasnosel'skiȋ genus, see Theorem 2.2.In order to apply this result, we prove that the energy functional I satisfies a compactness assumption, the Cerami's weaker variant of the Palais-Smale condition, both in the non-resonant case and in the resonant one under (f r ), cf.Lemma 3.2.In the proof of this lemma, neither the symmetry nor the behavior near zero of f enter at all.Moreover, as usual, in order to find multiple minimax critical points, it is also needed to show that the energy functional I has the right geometry.This part of the proof involves the behavior of the nonlinearity both at infinity and at zero, and is responsible for the assumptions on ℓ ∞ , ℓ 0 , and ℓ ′ 0 in (H − ) and (H + ).Some remarks on the statement are now in order.In both cases (H − ) and (H + ), the larger |ℓ 0 |, ℓ ′ 0 , the higher the possibility of finding solutions.In fact, in the limit cases |ℓ 0 | = ℓ ′ 0 = ∞, we get the highest number of solutions.
is satisfied for every k ∈ N.Moreover, being (η h ) divergent (see Proposition 2.5), there always exists h ∈ N for which ℓ ∞ < η (0) h .Therefore, in this case (1.3) admits infinitely many pairs of distinct solutions.
On the other hand, it is not clear whether the case ℓ 0 = 0 can be covered, see Remarks 3.4 and 3.8.The cases ℓ 0 = −∞ and ℓ ′ 0 < ∞ include the p-behavior at zero.We refer to [5,6,18], for problems with f ∼ |t| p−2 t in a neighborhood of 0, and the so-called two-parameter eigenvalue problem for the (p, q)-Laplacian.
The paper is organized as follows.In Section 2, we present the abstract result that we will apply, and introduce the variational setting and the two sequences of quasi-eigenvalues (η (α) h ) and (ν (α) h ).In Section 3, we prove the multiplicity result through the intermediate steps of showing that the energy functional satisfies the compactness condition and has the right geometry.

Abstract results.
Definition 2.1.Let X be a Banach space.A C 1 -functional I : X → R satisfies the Cerami-Palais-Smale condition ((CPS)-condition for short) if every sequence (u n ) ⊂ X such that admits a convergent subsequence.
We will apply the following multiplicity result, see [2, Theorem 2.9] for a proof in Hilbert spaces and [3, Theorems 2.6 and 2.7] for Banach spaces.In particular, for the version that appears as (resp....) in the statement, we refer to [3, Remark 2.8].
Theorem 2.2.Let X be a Banach space and for ρ > 0 denote S ρ := {u ∈ X : u X = ρ}.Suppose that the functional I ∈ C 1 (X, R) satisfies the following properties (i) I is even; (ii) I satisfies (CP S) in (0, ∞), and I(0) ≥ 0; (iii) there exist two closed subspaces V, W ⊂ X such that dimV < ∞ and codimW < ∞, and two constants c ∞ > c 0 > I(0) for which the following assumptions hold (a) If furthermore dimV > codimW , then I possesses at least m = dimV − codimW distinct pairs of critical points, whose corresponding critical values belong to [c 0 , c ∞ ].
2.2.Variational Setting.Throughout the paper, for 1 ≤ r ≤ ∞, we denote with • r , the usual norm in the Lebesgue space L r (Ω).We look for solutions of (1.3) in the Sobolev space W 1,q 0 (Ω) endowed with the equivalent norm u := ∇u q .
Definition 2.3.A function u ∈ W 1,q 0 (Ω) is a weak solution of (1.3) if for every ϕ ∈ W 1,q 0 (Ω) the following distributional identity holds We observe that, due to the boundedness of Ω, all the integrals above are finite.Indeed, for every r < q, by Hölder's inequality we have The problem (1.3) has a variational structure, its associated energy functional I : W 1,q 0 (Ω) → R is defined as follows for every u ∈ W 1,q 0 (Ω) It is straightforward to verify that I is of class C 1 and that u ∈ W 1,q 0 (Ω) is a weak solution of (1.3) if and only if it is a critical point of I.
Moreover, for future use, we introduce the Sobolev critical exponent for the embedding W 1,q (Ω) ֒→ L r (Ω) to be the conjugate exponent q ′ of q, and the dual space of (W 1,q 0 (Ω)) ′ =: W −1,q ′ (Ω), with its operatorial norm denoted by • −1,q ′ .2.3.Two sequences of quasi-eigenvalues.Inspired by [9] and [17], we define below two sequences, denoted by (η h ), of quasi-eigenvalues for a (p, q)-Laplacian-type operator.Compared with the arguments in [9,17] for the classical p-Laplacian, here the arguments are slightly more delicate due to the lack of homogeneity of the operator.
Since in this subsection α ≥ 0 is fixed, for simplicity in notation, throughout this subsection we will drop the superscript (α) and denote the sequences simply by (η h ) and (ν h ).
• Proof of the claim.Let (u n ) ⊂ S be such that Φ(u n ) → η 1 , then for every n ∈ N , where o(1) → 0 as n → ∞.Then, (u n ) is bounded in the reflexive Banach space W 1,q 0 (Ω) and so, up to a subsequence, u n ⇀ u in W 1,q 0 (Ω).Now, the function Φ is convex and continuous w.r.t. the strong topology in W 1,q 0 (Ω), then it is weakly lower semicontinuous in W 1,q 0 (Ω).Hence, On the other hand, by the compact embedding W 1,q 0 (Ω) ֒→֒→ L q (Ω), u n → u in L q (Ω), and so u ∈ S. Therefore, u is an admissible competitor for the infimum defining η 1 , so that the only possibility for (2.2) to hold is that Φ(u) = η 1 .
• Proof of the claim.The proof is the same as for the previous claim, with the only difference that one has to prove also that L 1 (u) = 0, u being the weak limit of the minimizing sequence By iterating this procedure, we introduce a sequence of positive numbers (η h ), a sequence of functions (ϕ h ) ⊂ S and, in correspondence, a sequence of linear operators (L h ) ⊂ L q ′ (Ω) ∩ W −1,q ′ (Ω) defined by More precisely, denoted S 0 := S, we define the following weakly closed subspaces of W 1,q 0 (Ω) and, for every h ∈ N, the corresponding constrained infimum each one achieved on the corresponding function ϕ h ∈ S h−1 .From the definition, it easily follows that 0 (2.6) In the spirit of [9, Lemma 5.2], we prove the following proposition.
Proof.Suppose by contradiction that there exists a number η ∈ (0, ∞) such that As a consequence, for every h ∈ N is bounded in the reflexive Banach space W 1,q 0 (Ω).Thus, there exist a subsequence, still denoted by (ϕ h ), and a function φ ∈ W 1,q 0 (Ω) such that ϕ h ⇀ φ ∈ W 1,q 0 (Ω) and ϕ h → φ in L q (Ω).In particular, (ϕ h ) is a Cauchy sequence in L q (Ω), thus for any positive ε < 1, there is h 0 ∈ N such that Therefore, (2.6) and the Hölder inequality imply Let us recall that if V ⊆ X is a closed subspace of a Banach space X, a subspace W ⊂ X is a (topological) complement of V if W is closed and every x ∈ X can be uniquely written as v + w, with v ∈ V and w ∈ W ; furthermore the projection operators onto V and W are (linear and) continuous.When this happens and V has finite dimension, we say that W has finite codimension, with codimW = dimV .Lemma 2.6.Let α ≥ 0 be fixed.For every h ∈ N, let us set ) (2.8) Reasoning as in [9, Lemma 5.3], we point out that for any u ∈ V h , by the linear independence of ϕ 1 , . . ., ϕ h , we can write uniquely with (c 1 , . . ., c h ) ∈ R h and by (2.6) it results that

Therefore, given a function
Which means that V h ∩ W h = {0}.Now, fixed any u ∈ W 1,q 0 (Ω), we put Taken w := u − v, we have that and for every i ∈ {2, . . ., h} Hence, w ∈ W h and we conclude the proof.
We remark that, by definition, for each h ∈ N, W h is a closed subspace of W 1,q 0 (Ω) of codimension h.Moreover, reasoning as in Remark 2.4, the following inequalities hold in W h−1 : with B = {u ∈ W 1,q 0 (Ω) : u q ≤ 1}, and in particular, if α = 0 or p = q, η h u q q ≤ (α + 1) (2.10) The sequence of quasi-eigenvalues (η h ) satisfies (2.9).In order to prove multiplicity results, it is useful to have also a reversed inequality on finite dimensional subspaces of W 1,q 0 (Ω).Hence, we introduce (ν h ), another sequence of quasi-eigenvalues satisfying exactly this property.
For all h ∈ N we set with ϕ 1 defined above.Since W h+1 ⊂ W h , ν h ≤ ν h+1 for every h.

Main results
We first observe that, by (f ) and (f ∞ ), for every ε > 0 there exists R ε > 0 such that |f (x, t)| ≤ ε|t| q−1 + A ε for a.e.x ∈ Ω and every t ∈ R, where Throughout this section, we will denote by the same symbol C various positive constants whose values are not important for the proof itself and may change from line to line.

Compactness condition. We introduce here the following operator
Lemma 3.1.The operator −∆ p − ∆ q satisfies the (S + )-property, i.e., if u n ⇀ u in W 1,q 0 (Ω) and Au n , u n − u → 0, then u n → u in W 1,q 0 (Ω).Proof.This result is contained in [21, Proposition 2.2], but for the sake of clarity, we prefer to report here its proof.Since ∇u n p−1 p − ∇u p−1 p and ∇u n p − ∇u p have the same sign, and the same holds true for the q-norms, On the other hand, for every u, v ∈ W 1,q 0 (Ω), by Hölder's inequality Au, v ≤ ∇u p−1 p ∇v p + ∇u q−1 q ∇v q .
Proof.Let (u n ) ⊂ W 1,q 0 (Ω) be a sequence satisfying We claim that it is enough to show that (u n ) is convergent, provided it is bounded in W 1,q 0 (Ω).This is a quite standard consequence of Lemma 3.1, cf. for instance [10, Lemma 3.1] and [15,Lemma 2].However, we prefer to write here all the details of the proof of this claim, because most of the arguments therein will be useful also in the rest of the proof of the present lemma.Suppose that ( ∇u n q ) is bounded.Since W 1,q 0 (Ω) is a reflexive Banach space, there exist a subsequence, still denoted by (u n ), and a function u ∈ W 1,q 0 (Ω) for which u n ⇀ u in W 1,q 0 (Ω).Moreover, for every n ∈ N we have and, by (3.3) and by the boundedness of (u n ) in W 1,q 0 (Ω), where o(1) → 0 as n → ∞.On the other hand, by the compact embedding W 1,q 0 (Ω) ֒→֒→ L r (Ω), u n → u in L r (Ω) for every r ∈ [1, q * ).Therefore, by Hölder's inequality and, using (3.1) with ε = 1 and (3.5), Hence, inserting the last three estimates in (3.4), we get Therefore, by Lemma 3.1, u n → u in W 1,q 0 (Ω).It remains to prove that (u n ) is bounded in W 1,q 0 (Ω).We will follow the guidelines of [3, Proposition 3.1-(i)] and of [17,].
• We first consider the non-resonant case in which (f nr ) holds.We argue by contradiction and suppose that there exists a subsequence, still denoted by (u n ), such that u n → ∞.Thus, without loss of generality, we can assume that u n > 0 for every n and define w n := u n / u n .Clearly, (w n ) is bounded in W 1,q 0 (Ω), hence up to a subsequence, w n ⇀ w in W 1,q 0 (Ω) and w n → w in L r (Ω) for every r ∈ [1, q * ) and some w ∈ W 1,q 0 (Ω).We claim that w = 0. Indeed, as a consequence of the second convergence in (3.3), the following relation holds Dividing (3.8) by u n q = ∇u n q q , we get ∇u n p p Moreover, by Hölder's inequality, Assume by contradiction that w = 0.Then, since w n → w in L q (Ω), by (3.9) we get On the other hand, by (3.1), (2.1) with r = 1, and Poincaré's inequality, we have This contradicts (3.11) and proves that w = 0.
• We now consider the resonant case and assume the validity of (f r ).We observe that in this case ℓ ∞ > 0. In this part of the proof, for brevity, we will write the nonlinear terms ℓ ∞ |t| q−2 t + f (x, t) as g(x, t).Let G(x, t) := t 0 g(x, s)ds.It is straightforward to check that g(x, t)t − qG(x, t) = f (x, t)t − qF (x, t) for a.e.x ∈ Ω and every t ∈ R. Thus, the limit in (f r ) can be rewritten equivalently in terms of g as follows As a consequence, there exists T 0 > 0 such that g(x, t)t − qG(x, t) ≥ 0 for a.e.x ∈ Ω and for every |t| ≥ T 0 .
By virtue of (f ), there exists where for simplicity in notation we have denoted Hence, by (3.22) and being q > p, qc+o (3.23) Now, let K > 0 be a constant to be specified later.By (3.20), there exists T K ≥ T 0 > 0 such that g(x, t)t − qG(x, t) ≥ K for a.e.x ∈ Ω and for every |t| ≥ T K . (3.24) Thus, continuing from (3.23), we have On the other hand, for every r > q > p, by (3.3), In view of the embedding W 1,q 0 (Ω) ֒→ L s (Ω), let C S > 0 be the best constant such that the following inequality holds for every u ∈ W 1,q 0 (Ω) u q s ≤ C S ∇u q q .Thus, continuing the previous estimate, we get This is possible thanks to (3.25), taking K = (qc + C 0 )[2C S (ℓ ∞ + 1)] s s−q .In conclusion, (3.26) gives for a suitable positive constant C ′ .This proves the boundedness of ( u n ) and concludes the proof.
• If ℓ 0 = −∞, clearly condition (3.27) is trivially verified also when h = 1.Again by the assumptions on f , for every M > 0 there exists a M > 0 such that F (x, t) ≤ − M q |t| q + a M |t| s+q for a.e.x ∈ Ω and every t ∈ R.
Hence, for every u ∈ W 1,q 0 (Ω), taking M > max{0, ℓ ∞ }, and the conclusion follows as in the previous case in S ρ ∩ W 0 = S ρ ∩ W 1,q 0 (Ω) = S ρ with ρ > 0 small enough.Remark 3.4.A careful inspection in the proof of the previous lemma shows that the result holds true also under the assumption h .
We are now ready to prove the first part of Theorem 1.1.
In particular, if ℓ ′ 0 = ∞, the conclusion holds for every k ∈ N. Proof.• We first consider the case ℓ ′ 0 < ∞.By (f ), (f ∞ ), and (f + 0 ), for every ε > 0 there exists a ′ ε > 0 such that k )/2 to get for every u ∈ W 1,q 0 (Ω) By (2.11), there exists a k-dimensional closed subspace V ε,(1) k ⊂ W 1,q 0 (Ω) such that for every u ∈ V where C, C ′ > 0 are two constants arising in the equivalence of the norms in the finite dimensional space V ε,(1) k . Since p < q and in view of the choice of ε, there exists ρ > 0 so small that for every u ∈ V is satisfied for every k ∈ N. Again, by the assumptions on f , for every M > 0, there exists a ′ M > 0 such that F (x, t) ≥ M p |t| p − a ′ M |t| q for a.e.x ∈ Ω and every t ∈ R.
We are now ready to prove the second part of Theorem 1.1.
• Proof of Theorem 1.1-(H + ).In view of Lemmas 3.2, 3.6 and 3.7, we can apply Theorem 2.2 to the functional −I, with X = W 1,q 0 (Ω), W = W h .Therefore, the two conditions required in (H − ), when ℓ 0 < 0, can be written as the following chain of inequalities ℓ ∞ + ℓ 0 < η k < ℓ ∞ from which it becomes apparent that for ℓ 0 = 0 they are never compatible.On the other hand, if we take into account Remark 3.4, under the weaker condition (3.29), in general we cannot exclude that the intersection of the two conditions in (H − ), when ℓ 0 = 0, is empty.

Remark 3 . 8 .
It has been proved in[3] that the sequences (η increasing and divergent and that, for every h ∈ N, η