Nontrivial solutions for resonance quasilinear elliptic systems

Natalino Borgia; Silvia Cingolani; Giuseppina Vannella

doi:10.1515/anona-2024-0005

Article Open Access

Nontrivial solutions for resonance quasilinear elliptic systems

Natalino Borgia , Silvia Cingolani and Giuseppina Vannella

Published/Copyright: June 3, 2024

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From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

We establish an Amann-Zehnder-type result for resonance systems of quasilinear elliptic equations with homogeneous Dirichlet boundary conditions, involving nonlinearities growing asymptotically ( p , q ) -linear at infinity. The proof relies on a cohomological linking in a product Banach space where the properties of cones of the sublevels are missing, differently from the single quasilinear equation. We also perform critical group computations of the energy functional at the origin, in spite of the lack of its C 2 regularity, to exclude that the found mini-max solution is trivial. Finally, we furnish a local condition which guarantees that the found solution is not semi-trivial.

Keywords: quasilinear elliptic systems; Fadell Rabinowitz index; asymptotically (p, q) linear; Morse index; resonance; critical groups

MSC 2010: 35J70; 35J60; 35B06; 35B65; 46E35

1 Introduction

The interaction between a nonlinearity g and the spectrum of − Δ at 0 and infinity has been used in the celebrated article [2], under the assumption of nonresonance at infinity, namely, λ ¯ is not an eigenvalue of − Δ . Precisely, Amann and Zehnder [2] proved the existence of a nontrivial solution for the nonresonance asymptotically linear elliptic equation:

(1.1) − Δ u = g ( u ) , in Ω , u = 0 , on ∂ Ω ,

where Ω is a bounded domain with smooth boundary, g ∈ C 1 ( R , R ) such that g ( 0 ) = 0 , and there exists λ ¯ ∈ R such that lim ∣ s ∣ → ∞ g ′ ( s ) = λ ¯ . Using Morse theory for manifolds with boundary, the same result has been found by Chang [7]. Successively, Lazer and Solimini [35] recognized that such nontrivial solution can be detected, combining mini-max characterization of the critical point and Morse index estimates (see also [41]).

Recently, the quasilinear counterpart of the Amann-Zehnder existence result has been obtained in [12] (see also [10,26]) for a class of quasilinear elliptic equations, involving as principal part either the p -Laplace operator or the operator related to the p -area functional, and a nonlinearity with p -linear growth at infinity, exploiting new techniques of Morse theory in Banach spaces (see [13–15,46]), and introducing a Saddle theorem where linear subspaces are substituted by symmetric cones (see Theorem 7.1 in [12]).

In this article, we are interested to derive Amann-Zehnder-type results for a class of quasilinear elliptic systems, having as principal parts ( p , q ) -Laplace operators or ( p , q ) -area-type operators, and nonlinearities growing ( p , q ) -linearly at infinity.

Precisely, we will seek for nontrivial solutions for the following autonomous quasilinear system:

(1.2) − div ( α + ∣ ∇ u ∣ 2 ) p − 2 2 ∇ u = G s ( u , v ) , x ∈ Ω , − div ( α + ∣ ∇ v ∣ 2 ) q − 2 2 ∇ v = G t ( u , v ) , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω ,

where 2 ≤ p < N , 2 ≤ q < N , α ≥ 0 , Ω is a smooth bounded domain of R N , and G ∈ C 1 ( R 2 , R ) satisfies the following conditions:

∇ G ( 0 , 0 ) = ( 0 , 0 ) and there exists λ ¯ ∈ R such that
lim ∣ ( s , t ) ∣ → ∞ G s ( s , t ) − λ ¯ F s ( s , t ) ∣ s ∣ p − 1 + ∣ t ∣ q p − 1 p = lim ∣ ( s , t ) ∣ → ∞ G t ( s , t ) − λ ¯ F t ( s , t ) ∣ s ∣ p q − 1 q + ∣ t ∣ q − 1 = 0 ,
where F ∈ C 1 ( R 2 , R ) is defined by
F ( s , t ) = 1 p ∣ s ∣ p + 1 q ∣ t ∣ q + 1 ( β + 1 ) ( γ + 1 ) ∣ s ∣ β ∣ t ∣ γ s t ,
with β , γ > 0 such that ( β + 1 ) ∕ p + ( γ + 1 ) ∕ q = 1 ;
there is a suitable neighborhood U of ( 0 , 0 ) such that G ∈ C 2 ( U , R ) .

Let X be the product space W 0 1 , p ( Ω ) × W 0 1 , q ( Ω ) endowed with the norm:

‖ z ‖ = ‖ u ‖ 1 , p + ‖ v ‖ 1 , q , z = ( u , v ) ∈ X ,

where ‖ ⋅ ‖ s and ‖ ⋅ ‖ 1 , s denote the usual norms in L s ( Ω ) and W 0 1 , s ( Ω ) , respectively.

Weak solutions of Problem (1.2) correspond to critical points of the Euler functional J α on X defined by setting

(1.3) J α ( z ) = J α ( u , v ) = 1 p ∫ Ω ( α + ∣ ∇ u ( x ) ∣ 2 ) p 2 d x + 1 q ∫ Ω ( α + ∣ ∇ v ( x ) ∣ 2 ) q 2 d x − ∫ Ω G ( u ( x ) , v ( x ) ) d x , z = ( u , v ) ∈ X .

By ( a 1 ) , the functional J α is C 1 on X and for any z 0 = ( u 0 , v 0 ) ∈ X , and z = ( u , v ) ∈ X , it results

⟨ J α ′ ( z 0 ) , z ⟩ = ∫ Ω ( α + ∣ ∇ u 0 ∣ 2 ) p − 2 2 ∇ u 0 ∇ u d x + ∫ Ω ( α + ∣ ∇ v 0 ∣ 2 ) q − 2 2 ∇ v 0 ∇ v d x − ∫ Ω ( G s ( u 0 , v 0 ) u + G t ( u 0 , v 0 ) v ) d x .

These systems model some phenomena in non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology (see [24,31,38,40,43]). Existence, nonexistence, and regularity results for such quasilinear elliptic systems are obtained by various authors (see, for instance, [3–5,18,20,25,37,39,48]).

In this article, we address the study of the interaction between the spectrum of the operators involved in System (1.2) and nonlinearities that grow ( p , q ) -linearly at infinity, in both the asymptotic resonant and nonresonant case.

To this aim, we consider the following nonlinear eigenvalue problem:

(1.4) − Δ p u = λ ∣ u ∣ p − 2 u + λ γ + 1 ∣ u ∣ β ∣ v ∣ γ v , x ∈ Ω , − Δ q v = λ ∣ v ∣ q − 2 v + λ β + 1 ∣ u ∣ β ∣ v ∣ γ u , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω ,

where Ω is a smooth bounded domain of R N , 2 ≤ p < N , 2 ≤ q < N , β > 0 , and γ > 0 are the real numbers satisfying ( β + 1 ) ∕ p + ( γ + 1 ) ∕ q = 1 .

The real number λ is called an eigenvalue of (1.4) if there exists a nontrivial solution ( u , v ) of (1.4).

In [42], the existence of an unbounded sequence of mini-max eigenvalues was proved using Z 2 -cohomological index of Fadell and Rabinowitz [28,29].

Precisely, let Φ : X → R , Ψ : X → R be the following functionals:

Φ ( u , v ) = 1 p ∫ Ω ∣ ∇ u ( x ) ∣ p d x + 1 q ∫ Ω ∣ ∇ v ( x ) ∣ q d x ,

Ψ ( u , v ) = ∫ Ω F ( u , v ) d x .

By Young’s inequality, we infer that

(1.5) 1 p γ γ + 1 ∣ s ∣ p + 1 q β β + 1 ∣ t ∣ q ≤ F ( s , t ) ≤ 1 p γ + 2 γ + 1 ∣ s ∣ p + 1 q β + 2 β + 1 ∣ t ∣ q .

Note that Φ and Ψ are ( p , q ) -homogeneous, i.e.,

(1.6) Φ t 1 p u , t 1 q v = t Φ ( u , v ) and Ψ t 1 p u , t 1 q v = t Ψ ( u , v ) ,

for all t > 0 and ( u , v ) ∈ X .

Moreover, λ is an eigenvalue iff there is ( u , v ) ∈ X \ { ( 0 , 0 ) } such that

Φ ′ ( u , v ) = λ Ψ ′ ( u , v ) .

Ou and Tang [42] proved that (1.4) has a nondecreasing and unbounded sequence of eigenvalues with the variational characterization:

(1.7) λ k = inf A ∈ Σ k sup ( u , v ) ∈ A Φ ( u , v ) ,

where Σ is the C 1 manifold

Σ ≔ { ( u , v ) ∈ X ∣ Ψ ( u , v ) = 1 }

and

Σ k ≔ { A ⊂ Σ ∣ A is symmetric, compact and i ( A ) ≥ k } ,

where i ( A ) denotes the Z 2 -cohomological index of A , introduced by Fadell and Rabinowitz [28,29]. For a matter of convenience, we also put λ 0 = − ∞ .

In [45], it is shown that the first eigenvalue λ 1 is simple and isolated with a first strictly positive eigenfunction ψ = ( ψ 1 , ψ 2 ) , i.e., ψ 1 > 0 , ψ 2 > 0 in Ω (see also [3,19,20,27]). However, it is not clear if the set of the eigenvalues described by (1.7) contains all the eigenvalues of Problem (1.4). Really, also for the single scalar eigenvalue problem − Δ p u = λ ∣ u ∣ p − 2 u , the spectral properties of − Δ p are not yet well understood. Denoted σ ( − Δ p ) the set of such eigenvalues λ , one can define in at least three different ways a diverging sequence ( λ m ) of eigenvalues of − Δ p [10,26], but it is not known if they agree for m ≥ 3 and if the whole set σ ( − Δ p ) is covered.

Furthermore, we remark that in the context of quasilinear systems, with p ≠ q , the level set E λ k = { ( u , v ) ∈ X ∣ Φ ( u , v ) = λ k Ψ ( u , v ) } is not a linear space or a cone, but, by (1.6), we have t 1 p u , t 1 q v ∈ E λ k for any t ≥ 0 and ( u , v ) ∈ E λ k . This makes delicate to recognize a saddle-type geometry for the energy functional associated with (1.2).

In this article, we derive a quite general abstract linking theorem (see Theorem 3.3), that extends Theorem 7.1 in [12], which deals with symmetric cones. This abstract result can be successfully applied in the framework of systems and allows us to detect the existence of a mini-max critical point for J α with a suitable nontrivial critical group. Finally, in order to prove that the solution we found is not trivial, we need to compute the critical groups at the origin in terms of differential notions. However, the applicability of Morse arguments in Banach spaces presents severe difficulties since classical Morse lemma and generalized Morse lemma [32] are so far known, also due to the lack of Fredholm properties of the second derivative of the functionals. Moreover, the energy functional J α is not C 2 , so that we cannot define the classical notion of Morse index. In order to overcome this lack of regularity, the first idea is to consider the quadratic form Q : X → R defined as

(1.8) Q ( z ) = Q ( u , v ) = α p − 2 2 ∫ Ω ∣ ∇ u ∣ 2 d x + α q − 2 2 ∫ Ω ∣ ∇ v ∣ 2 d x − ∫ Ω ( G s s ( 0 , 0 ) u 2 + G t t ( 0 , 0 ) v 2 + 2 G s t ( 0 , 0 ) u v ) d x = α p − 2 2 ‖ u ‖ 1 , 2 2 + α q − 2 2 ‖ v ‖ 1 , 2 2 − ∫ Ω H G ( 0 , 0 ) [ z ] 2 d x ,

where H G ( 0 , 0 ) is the Hessian matrix of G at ( 0 , 0 ) and consider m 0 and m 0 * defined as the supremum of the dimensions of the subspaces of X , where Q is negative definite and negative semidefinite, respectively. These objects will play the role of Morse index and large Morse index, respectively.

The second problem becomes to relate such differential notions m 0 and m 0 * to the behavior of the functional J α near the origin. This delicate issue will be obtained, by means of a penalized functional of J α , which is C 2 just in a ball centered at the origin. We stress that even if p ≥ 2 and q ≥ 2 , the functional F is just C 1 on X , differently from the case of a single equation. Roughly speaking, we say that the eigenvalue Problem (1.4) is intrinsically less regular than the eigenvalue problem for the single p -Laplace equation, due to the presence of the right-hand side coupled terms.

Our main results are the following:

Theorem 1.1

Suppose that 2 ≤ p < N , 2 ≤ q < N , α ≥ 0 , and ( a 1 ) − ( a 2 ) hold. Assume that λ ¯ is not an eigenvalue for System (1.4) and denote by m ∞ the integer such that

λ m ∞ < λ ¯ < λ m ∞ + 1 .

m ∞ ∉ [ m 0 , m 0 * ] ,

then there exists a nontrivial weak solution z = ( u , v ) of (1.2).

We remark that in Theorem 1.1, the parameter λ ¯ is not an eigenvalue of (1.4) and λ m < λ ¯ < λ m + 1 for some m ∈ N . This condition seems to be possible, for instance, if m = 1 taking into account that λ 1 is isolated and it is the only eigenvalue of (1.4) to which corresponds a componentwise positive eigenfunction [45]. However, such nonresonant restriction would be quite severe, and it becomes relevant to face with systems at resonance. In the following theorems, the value λ ¯ is allowed to be an eigenvalue of (1.4), under an additional condition of G at infinity.

Theorem 1.2

Suppose that 2 ≤ p < N , 2 ≤ q < N , α ≥ 0 , and ( a 1 ) – ( a 2 ) hold. Denote by m ∞ the integer such that

λ m ∞ < λ ¯ ≤ λ m ∞ + 1 .

m ∞ ∉ [ m 0 , m 0 * ]

and

( b − ) lim ∣ ( s , t ) ∣ → ∞ G ( s , t ) − 1 p G s ( s , t ) s − 1 q G t ( s , t ) t = − ∞ ,

then there exists a nontrivial weak solution z = ( u , v ) of (1.2).

Theorem 1.3

Suppose that 2 ≤ p < N , 2 ≤ q < N , α = 0 , and ( a 1 ) − ( a 2 ) hold. Denote by m ∞ the integer such that

λ m ∞ ≤ λ ¯ < λ m ∞ + 1 .

m ∞ ∉ [ m 0 , m 0 * ]

and

( b + ) lim ∣ ( s , t ) ∣ → ∞ G ( s , t ) − 1 p G s ( s , t ) s − 1 q G t ( s , t ) t = + ∞ ,

then there exists a nontrivial weak solution z = ( u , v ) of (1.2).

We remark that the nontrivial solutions of (1.2), including those found through Theorems 1.1–1.3, cannot be semitrivial if we assume that G t ( s , 0 ) ≠ 0 for any s ∈ ( − ε , ε ) \ { 0 } and G s ( 0 , t ) ≠ 0 for any t ∈ ( − ε , ε ) \ { 0 } , for an arbitrary ε > 0 (see Proposition 5.2 and also [9, Lemma 3.1]).

Finally, we note that if γ = β = 0 , then p = q = 2 . In this case, mini-max arguments and Morse theory are applied in [30] for deriving the existence of nontrivial solutions of a semilinear elliptic system with a C 2 nonlinear function G .

2 Palais-Smale (PS) and Cerami-Palais-Smale (CPS) conditions

We begin to recall a classical definition in a reflexive Banach space.

Definition 2.1

Let X be a reflexive Banach space and D ⊂ X . A map H : D → X ′ is said to be of class ( S ) + , if, for every sequence u k in D weakly convergent to u in X with

limsup k → ∞ ⟨ H ( u k ) , u k − u ⟩ ≤ 0 ,

we have ‖ u k − u ‖ → 0 .

Lemma 2.2

For any α ≥ 0 and r ≥ 2 , the map H α , r : W 0 1 , r ( Ω ) → W − 1 , r ′ ( Ω ) defined by

H α , r ( u ) = − div ( α + ∣ ∇ u ∣ 2 ) r − 2 2 ∇ u

is of class ( S ) + .

Proof

Let us denote by a α , r : R n → R n the map

a α , r ( ξ ) = ( α + ∣ ξ ∣ 2 ) r − 2 2 ξ .

We can see that for any α ≥ 0 , there is C > 0 such that

(2.1) ∣ a α , r ( ξ ) ∣ ≤ C ( 1 + ∣ ξ ∣ r − 1 ) ,

(2.2) a α , r ( ξ ) ξ ≥ ∣ ξ ∣ r ,

(2.3) ( a α , r ( ξ ) − a α , r ( η ) ) ( ξ − η ) > 0 ,

for any ξ ∈ R n and η ≠ ξ ∈ R n .

Let us consider η ≠ ξ ∈ R N . If ∣ ξ ∣ = ∣ η ∣ , then

( a α , r ( ξ ) − a α , r ( η ) ) ⋅ ( ξ − η ) = ( α + ∣ η ∣ 2 ) r − 2 2 ∣ ξ − η ∣ 2 > 0 .

Otherwise, if ∣ ξ ∣ ≠ ∣ η ∣ , by the monotonicity of the real function t ∈ R ↦ t ( α + t 2 ) r − 2 2 , we obtain

∣ ξ ∣ ∣ η ∣ ( α + ∣ ξ ∣ 2 ) r − 2 2 + ( α + ∣ η ∣ 2 ) r − 2 2 < ∣ ξ ∣ 2 ( α + ∣ ξ ∣ 2 ) r − 2 2 + ∣ η ∣ 2 ( α + ∣ η ∣ 2 ) r − 2 2 ,

which gives (2.3). As (2.1) and (2.2) are trivial and H α , r ( u ) = − div ( a α , r ( ∇ u ) ) , applying [1, Theorem 3.5], we complete the proof.□

Proposition 2.3

If z n is a bounded sequence in X such that ‖ J α ′ ( z n ) ‖ → 0 , then z n has a convergent subsequence in X.

Proof

Up to a subsequence, z n = ( u n , v n ) converges to some ( u ¯ , v ¯ ) weakly in X and strongly in L p ( Ω ) × L q ( Ω ) ; hence, ∫ Ω G s ( z n ( x ) ) ( u n ( x ) − u ¯ ( x ) ) d x → 0 .

Moreover, ⟨ J α ′ ( z n ) , ( u n − u ¯ , 0 ) ⟩ → 0 and, by Lemma 2.2, H α , p is of class ( S ) + , so that u n → u ¯ strongly in W 0 1 , p ( Ω ) .

In the same way, we obtain that also, v n → v ¯ strongly in W 0 1 , q ( Ω ) .□

Proposition 2.4

If λ ¯ is not an eigenvalue of (1.4), then J α satisfies the ( P S ) condition at any level c ∈ R , namely, if z n = ( u n , v n ) is a sequence in X such that J α ( z n ) → c and ‖ J α ′ ( z n ) ‖ → 0 , then z n has a convergent subsequence in X.

Proof

Let z n = ( u n , v n ) be a sequence in X such that J α ( z n ) is bounded and ‖ J α ′ ( z n ) ‖ → 0 . Due to the previous proposition, it is sufficient to show that

(2.4) ‖ z n ‖ is bounded .

By contradiction, suppose ‖ ( u n , v n ) ‖ → ∞ and, denoting by r n = ‖ u n ‖ 1 , p p + ‖ v n ‖ 1 , q q , set u n ′ = u n ∕ r n 1 ∕ p and v n ′ = v n ∕ r n 1 ∕ q .

It is immediate that ‖ u n ′ ‖ 1 , p , ‖ v n ′ ‖ 1 , q ≤ 1 ; hence, up to a subsequence, ( u n ′ , v n ′ ) converges to some z ′ = ( u ′ , v ′ ) weakly in X and strongly in L p ( Ω ) × L q ( Ω ) .

In particular,

(2.5) 1 r n p − 1 p ⟨ J α ′ ( z n ) , ( u n ′ − u ′ , 0 ) ⟩ → 0

and

(2.6) 1 r n q − 1 q ⟨ J α ′ ( z n ) , ( 0 , v n ′ − v ′ ) ⟩ → 0 .

Considering that β p − 1 + ( γ + 1 ) p q ( p − 1 ) = 1 , by assumption ( a 1 ) and Young’s inequality, there is c > 0 such that

∣ G s ( s , t ) ∣ ≤ c 1 + ∣ s ∣ p − 1 + ∣ t ∣ q ( p − 1 ) p

so that

1 r n p − 1 p ∫ Ω G s ( z n ) ( u n ′ − u ′ ) d x ≤ c ∫ Ω ∣ u n ′ − u ′ ∣ d x + ∫ Ω ∣ u n ′ ∣ p − 1 ∣ u n ′ − u ′ ∣ d x + ∫ Ω ∣ v n ′ ∣ q ( p − 1 ) p ∣ u n ′ − u ′ ∣ d x .

Hence, by Hölder inequality, we obtain

lim n → ∞ 1 r n p − 1 p ∫ Ω G s ( z n ) ( u n ′ − u ′ ) d x = 0 ,

which, combined with (2.5), gives

∫ Ω α r n 2 ∕ p + ∣ ∇ u n ′ ∣ 2 p − 2 2 ∇ u n ′ ∇ ( u n ′ − u ′ ) d x → 0 .

Therefore, using the convexity of the function ξ ∈ R n ↦ α r n 2 ∕ p + ∣ ξ ∣ 2 p 2 , we see that

‖ u ′ ‖ 1 , p p ≤ liminf n → ∞ ‖ u n ′ ‖ 1 , p p ≤ limsup n → ∞ ‖ u n ′ ‖ 1 , p p ≤ limsup n → ∞ ∫ Ω α r n 2 ∕ p + ∣ ∇ u n ′ ∣ 2 p 2 ≤ limsup n → ∞ ∫ Ω α r n 2 ∕ p + ∣ ∇ u ′ ∣ 2 p 2 + p ∫ Ω α r n 2 ∕ p + ∣ ∇ u n ′ ∣ 2 p − 2 2 ∇ u n ′ ∇ ( u n ′ − u ′ ) ≤ ‖ u ′ ‖ 1 , p p .

Hence u n ′ → u ′ also strongly in W 0 1 , p ( Ω ) . Analogously, by (2.6), we obtain that v n ′ → v ′ strongly in W 0 1 , q ( Ω ) .

In particular,

‖ u ′ ‖ 1 , p p + ‖ v ′ ‖ 1 , q q = lim n → ∞ ‖ u n ′ ‖ 1 , p p + ‖ v n ′ ‖ 1 , q q = lim n → ∞ ‖ u n ‖ 1 , p p + ‖ v n ‖ 1 , q q r n = 1

so that ( u ′ , v ′ ) ≠ ( 0 , 0 ) .

By assumption ( a 1 ) ,

G s ( s , t ) = λ ¯ ∣ s ∣ p − 2 s + 1 γ + 1 ∣ s ∣ β ∣ t ∣ γ t + r 1 ( s , t ) ,

where

(2.7) lim ∣ ( s , t ) ∣ → ∞ r 1 ( s , t ) ∣ s ∣ p − 1 + ∣ t ∣ q p − 1 p = 0 .

Hence, there is c > 0 such that

(2.8) ∣ r 1 ( s , t ) ∣ ≤ c ∣ s ∣ p − 1 + ∣ t ∣ q p − 1 p + 1 ,

for any ( s , t ) ∈ R 2 . Let us fix u ˜ ∈ W 0 1 , p ( Ω ) , we want to prove that

(2.9) lim n → ∞ 1 r n ( p − 1 ) ∕ p ∫ Ω ∣ r 1 ( z n ( x ) ) ∣ ∣ u ˜ ( x ) ∣ d x = 0 .

Recalling that ‖ u n ′ ‖ 1 , p , ‖ v n ′ ‖ 1 , q ≤ 1 , let c ¯ > 0 be such that

(2.10) ‖ u n ′ ‖ p p − 1 + ‖ v n ′ ‖ q q p − 1 p ≤ c ¯ , for any n ∈ N .

Let us fix ε > 0 . By (2.7), there is δ ε > 0 such that

(2.11) ∣ ( s , t ) ∣ > δ ε ⇒ ∣ r 1 ( s , t ) ∣ < ε 2 c ¯ ∣ s ∣ p − 1 + ∣ t ∣ q p − 1 p .

Denoting by Ω n , ε − = { x ∈ Ω : ∣ z n ( x ) ∣ ≤ δ ε } and Ω n , ε + = Ω \ Ω n , ε − ,

x ∈ Ω n , ε − ⇒ ∣ u n ( x ) ∣ , ∣ v n ( x ) ∣ ≤ δ ε

so, by (2.8),

1 r n ( p − 1 ) ∕ p ∫ Ω n , ε − ∣ r 1 ( z n ( x ) ) ∣ ∣ u ˜ ( x ) ∣ d x ≤ c r n ( p − 1 ) ∕ p ∣ δ ε ∣ p − 1 + ∣ δ ε ∣ q p − 1 p + 1 ‖ u ˜ ‖ 1

and, choosing n is big enough,

(2.12) 1 r n ( p − 1 ) ∕ p ∫ Ω n , ε − ∣ r 1 ( z n ( x ) ) ∣ ∣ u ˜ ( x ) ∣ d x < ε 2 ‖ u ˜ ‖ p .

On the other hand, by (2.11), Hölder inequality, and (2.10),

1 r n ( p − 1 ) ∕ p ∫ Ω n , ε + ∣ r 1 ( z n ( x ) ) ∣ ∣ u ˜ ( x ) ∣ d x ≤ 1 r n ( p − 1 ) ∕ p ε 2 c ¯ ∫ Ω n , ε + ∣ u n ( x ) ∣ p − 1 + ∣ v n ( x ) ∣ q p − 1 p ∣ u ˜ ( x ) ∣ d x ≤ ε 2 c ¯ ∫ Ω ∣ u n ′ ( x ) ∣ p − 1 + ∣ v n ′ ( x ) ∣ q p − 1 p ∣ u ˜ ( x ) ∣ d x ≤ ε 2 ‖ u ˜ ‖ p ,

which, together with (2.12), proves (2.9). As u n ′ → u ′ in W 0 1 , p ( Ω ) , we infer

lim n → ∞ 1 r n ( p − 1 ) ∕ p ∫ Ω ( α + ∣ ∇ u n ( x ) ∣ 2 ) p − 2 2 ∇ u n ( x ) ∇ u ˜ ( x ) d x = lim n → ∞ ∫ Ω α r n 2 ∕ p + ∣ ∇ u n ′ ( x ) ∣ 2 p − 2 2 ∇ u n ′ ( x ) ∇ u ˜ ( x ) d x = ∫ Ω ∣ ∇ u ′ ( x ) ∣ p − 2 ∇ u ′ ( x ) ∇ u ˜ ( x ) d x

and

lim n → ∞ 1 r n ( p − 1 ) ∕ p ∫ Ω ∣ u n ( x ) ∣ p − 2 u n ( x ) + 1 γ + 1 ∣ u n ( x ) ∣ β ∣ v n ( x ) ∣ γ v n ( x ) u ˜ ( x ) d x = ∫ Ω ∣ u ′ ( x ) ∣ p − 2 u ′ ( x ) + 1 γ + 1 ∣ u ′ ( x ) ∣ β ∣ v ′ ( x ) ∣ γ v ′ ( x ) u ˜ ( x ) d x ,

which, together with (2.9), gives

∫ Ω ∣ ∇ u ′ ∣ p − 2 ∇ u ′ ∇ u ˜ d x − λ ¯ ∫ Ω ∣ u ′ ∣ p − 2 u ′ + 1 γ + 1 ∣ u ′ ∣ β ∣ v ′ ∣ γ v ′ u ˜ d x = lim n → ∞ 1 r n ( p − 1 ) ∕ p ⟨ J α ′ ( z n ) , ( u ˜ , 0 ) ⟩ = 0 .

In the same manner, we can see that, for any v ˜ ∈ W 0 1 , q ( Ω ) ,

∫ Ω ∣ ∇ v ′ ∣ q − 2 ∇ v ′ ∇ v ˜ d x − λ ¯ ∫ Ω ∣ v ′ ∣ q − 2 v ′ + 1 β + 1 ∣ u ′ ∣ β ∣ v ′ ∣ γ u ′ v ˜ d x = lim n → ∞ 1 r n ( q − 1 ) ∕ q ⟨ J α ′ ( z n ) , ( 0 , v ˜ ) ⟩ = 0 .

In other words, ( u ′ , v ′ ) ≠ ( 0 , 0 ) solves (1.4) with λ = λ ¯ , so the contradiction proves (2.4).□

Proposition 2.5

If λ ¯ is an eigenvalue of (1.4) and ( b − ) holds, so that

lim ∣ ( s , t ) ∣ → ∞ G ( s , t ) − 1 p G s ( s , t ) s − 1 q G t ( s , t ) t = − ∞ ,

then J α satisfies the CPS condition at any level c ∈ R , namely, any sequence z n = ( u n , v n ) in X such that J α ( z n ) → c and ‖ J α ′ ( z n ) ‖ ( 1 + ‖ z n ‖ ) → 0 has a convergent subsequence in X.

Moreover, if λ ¯ is an eigenvalue of (1.4) and ( b + ) holds, so that

lim ∣ ( s , t ) ∣ → ∞ G ( s , t ) − 1 p G s ( s , t ) s − 1 q G t ( s , t ) t = + ∞ ,

then J 0 satisfies the CPS condition at any level c ∈ R .

Proof

Let us show that z n = ( u n , v n ) is bounded in X when ( b − ) holds and α ≥ 0 .

By contradiction, assume that ‖ z n ‖ → ∞ and set r n = ‖ u n ‖ 1 , p p + ‖ v n ‖ 1 , q q , u n ′ = u n ∕ r n 1 ∕ p , v n ′ = v n ∕ r n 1 ∕ q . Reasoning as in the proof of Proposition 2.4, we infer that, up to a subsequence, ( u n ′ , v n ′ ) converges to some z ′ = ( u ′ , v ′ ) ≠ ( 0 , 0 ) strongly in X .

Denoting by K ( s , t ) = G ( s , t ) − 1 p G s ( s , t ) s − 1 q G t ( s , t ) t , CPS condition gives that, for a suitable C ∈ R ,

C ≥ J α ( z n ) − 1 p ⟨ J α ′ ( z n ) , ( u n , 0 ) ⟩ − 1 q ⟨ J α ′ ( z n ) , ( 0 , v n ) ⟩ = ∫ Ω α p ( α + ∣ ∇ u n ( x ) ∣ 2 ) ( p − 2 ) ∕ 2 + α q ( α + ∣ ∇ v n ( x ) ∣ 2 ) ( q − 2 ) ∕ 2 d x − ∫ Ω K ( z n ( x ) ) d x ≥ − ∫ Ω K ( z n ( x ) ) d x .

From ( b − ) , there is c 1 ∈ R such that

K ( s , t ) ≤ c 1 ∀ ( s , t ) ∈ R 2 .

Hence, by Fatou’s lemma,

(2.13) − C ≤ ∫ Ω limsup n → ∞ K ( z n ( x ) ) d x .

As we are assuming r n → + ∞ , by ( b − ) , we obtain that, for almost every x ∈ Ω ,

z ′ ( x ) ≠ ( 0 , 0 ) ⇒ ∣ z n ( x ) ∣ = ∣ ( r n 1 ∕ p u n ′ ( x ) , r n 1 ∕ q v n ′ ( x ) ) ∣ → ∞ ⇒ K ( z n ( x ) ) → − ∞ .

So, from (2.13), we infer that z ′ ( x ) = ( 0 , 0 ) almost everywhere in Ω , which contradicts z ′ ≠ ( 0 , 0 ) . Consequently, z n = ( u n , v n ) is bounded and, by Proposition 2.3, has a convergent subsequence in X .

If instead ( b + ) holds, J 0 ( z n ) → c ∈ R and ‖ J ′ ( z n ) ‖ ( 1 + ‖ z n ‖ ) → 0 , then

lim n → ∞ ∫ Ω K ( z n ( x ) ) d x = − c .

Then, reasoning as earlier, we infer again that z n has a convergent subsequence in X .□

3 Cohomological linking

Throughout this section, Y denotes a Banach space and f : Y → R a C 1 function. We also denote by H * the Alexander-Spanier cohomology [44] with coefficients in Z 2 = { − 1 , 1 } .

Let us recall next definitions (see [6,8,21,22]).

Definition 3.1

Let D , S , and A be three subsets of Y and m a nonnegative integer.

We say that ( D , S ) links A cohomologically in dimension m (over Z 2 ), if S ⊆ D , S ∩ A = ∅ and the restriction homomorphism H m ( Y , Y \ A ) → H m ( D , S ) is not identically zero.

If ( D , S ) links A cohomologically in some dimension, then ( D , S ) links A (c.f. [10, Definition 5.1]). Moreover, if ( D , S ) links A , then D ∩ A ≠ ∅ .

Definition 3.2

Let G be an abelian group. Let the m th critical group of f at z ∈ Y with coefficient in G be defined by

C m ( f ; z ) = H m ( f c , f c \ { z } ) ,

where c = f ( z ) , f c = { u ∈ Y ∣ f ( u ) ≤ c } .

In general, it can happen that C m ( f , z ) is not finitely generated for some m and that C m ( f , z ) ≠ 0 for infinitely many m ’s.

Now, we assume that f is Gauteax differentiable and we denote U an open subset of X . If z is an isolated critical point of f and f ′ : U → X ′ is a demicontinuous function (namely, it is continuous from the strong topology to the weak topology) and of class ( S ) + in a neighborhood of u , then C * ( f , z ) is of finite type (see [11, Theorem 1.1] and [1, Theorem 3.4].

Let us introduce a general result concerning a C 1 functional in a Banach space that extends Theorem 3.2 in [10] to the case in which f satisfies only the CPS condition.

Theorem 3.3

Let Y be a Banach space, f ∈ C 1 ( Y , R ) , D , S , and A be three subsets of Y , m ∈ N .

Assume that

( D , S ) links A cohomologically in dimension m over Z 2 ,

sup S f < inf A f = a and b = sup D f < + ∞ ,

f verifies the (CPS) condition at any level c ∈ [ a , b ] and f − 1 ( [ a , b ] ) contains a finite number of critical points.

Then, there exists a critical point z of f with a ≤ f ( z ) ≤ b and C m ( f ; z ) ≠ { 0 } .

Proof

Since D ∩ A ≠ ∅ , a ≤ b . We aim to apply [21, Theorem 5.2]. If ( E , d ) is a metric space, f : E → R is a continuous function and z ∈ E , let us consider the notion of weak slope ∣ d f ∣ ( z ) as defined in [23, Definition 2.1] (see also [33, Definition 5.1]). Consequently, we say that

z is a critical point if ∣ d f ∣ ( z ) = 0
f satisfies the PS condition at a level c ∈ R if any sequence z n in E such that f ( z n ) → c and ∣ d f ∣ ( z n ) → 0 has a convergent subsequence in E .

In particular, if E is also a Banach space and f is C 1 , we have

∣ d f ∣ ( z ) = ‖ f ′ ( z ) ‖ , ∀ z ∈ E .

Due to [16, Theorem 4.1, Remark 4.4], there exists a metric d ˜ on Y , topologically equivalent to the metric induced by ‖ ‖ Y , such that ( Y , d ˜ ) is complete and, denoting by ∣ d ˜ f ∣ the weak slope of f with respect to the metric d ˜ ,

∣ d ˜ f ∣ ( z ) = ‖ f ′ ( z ) ‖ Y ′ ( 1 + ‖ z ‖ Y ) , ∀ z ∈ Y .

Therefore, as f satisfies the C P S condition at any level c ∈ [ a , b ] , then f satisfies the P S condition at any level c ∈ [ a , b ] , with respect to d ˜ .

By [21, Theorem 7.5], the assumptions of [21, Theorem 5.2] are satisfied. Then, the assertion follows, taking into account [21, Proposition 7.3 and Remark 5.3].□

Denoting by A the class of symmetric subsets of Y , Fadell and Rabinowitz (see [28,29]) constructed an index theory i : A → N ∪ { ∞ } with the following properties:

Definiteness: i ( A ) ≥ 0 , i ( A ) = 0 ⇔ A = ∅ ;
Monotonicity: If there is an odd continuous map A → A ′ , then
i ( A ) ≤ i ( A ′ ) .
In particular, equality holds if A and A ′ are homeomorphic;
Subadditivity:
i ( A ∪ A ′ ) ≤ i ( A ) + i ( A ′ ) ;
Continuity: If A is closed, there is a neighborhood U ∈ A of A such that
i ( U ) = i ( A ) ;
Neighborhood of zero: If U is a bounded symmetric neighborhood of 0 in Y ,
i ( ∂ U ) = dim Y ;
Stability: If A is closed and A * Z 2 is the join of A with Z 2 , realized in Y ⊕ R ,
i ( A * Z 2 ) = i ( A ) + 1 ,
where A * Z 2 is the union of all line segments in Y ⊕ R , joining { 1 } and { − 1 } to points of A ;
Piercing property: Assume that A , A 0 , and A 1 are closed and
φ : A × [ 0 , 1 ] → A 0 ∪ A 1
is an odd continuous map such that φ ( A × [ 0 , 1 ] ) is closed, and
φ ( A × { 0 } ) ⊂ A 0 , φ ( A × { 1 } ) ⊂ A 1 .
Then,
i ( φ ( A × [ 0 , 1 ] ) ∩ A 0 ∩ A 1 ) ≥ i ( A ) .

We can state the next theorem whose proof follows from [10, Theorem 3.6] and [22, Theorem 2.7]. For reader’s convenience, we give a sketch of the proof.

Theorem 3.4

Let S and A be two symmetric subsets of Y with S ∩ A = ∅ . Assume that 0 ∈ A and i ( S ) = i ( Y \ A ) = m < ∞ .

Then, ( Y , S ) links A cohomologically in dimension m over Z 2 .

Proof

We want to prove that H m ( Y , Y \ A ) → H m ( Y , S ) is not identically zero. So, considering the exact sequence

H m ( Y , Y \ A ) → H m ( Y , S ) → H m ( Y \ A , S ) ,

it is sufficient to show that

(3.1) H m ( Y , S ) → H m ( Y \ A , S ) is not injective.

If m = 0 , then S = ∅ and A = Y , so H 0 ( Y , S ) ≠ { 0 } , while H 0 ( Y \ A , S ) = { 0 } , so (3.1) is proved.

Let us denote by j : H m − 1 ( Y \ A ) → H m − 1 ( S ) and i : H m − 1 ( Y ) → H m − 1 ( S ) the homomorphisms induced by inclusions. We recall that H q ( Y ) = { 0 } if q ≠ 0 , while the dimension of H 0 ( Y ) is 1.

By Lemma 3.4 in [10], we infer that

if m = 1 , the dimension of i m ( j ) is at least 2, while the dimension of i m ( i ) is at most 1.
if m ≥ 2 , the dimension of i m ( j ) is at least 1, while i m ( i ) = { 0 } .

Hence, i m ( i ) is a proper subset of i m ( j ) , for any m ≥ 1 .

Consider now the commutative diagram

where the horizontal rows are exact sequences and the vertical rows are induced by inclusions.

Let ω ∈ i m ( j ) \ i m ( i ) , then δ ω ≠ 0 in H m ( Y , S ) , while δ ¯ ω = ( δ ω ) ∣ ( Y \ A , S ) = 0 .

Hence, (3.1) is proved also for any m ≥ 1 , as H m ( Y , S ) → H m ( Y \ A , S ) is not injective.□

Now, we consider the Banach space X = W 0 1 , p ( Ω ) × W 0 1 , q ( Ω ) . We recall that the sequence ( λ m ) is defined by (1.7), i.e.,

λ m = inf A ∈ Σ m sup ( u , v ) ∈ A Φ ( u , v ) ,

where Σ m = { A ⊂ Σ ∣ A is symmetric, compact, and i ( A ) ≥ m } . Let us recall the following lemma, proved in [22].

Lemma 3.5

For every symmetric and open subset A of Σ

i ( A ) = sup { i ( K ) ∣ K is c o m p a c t a n d s y m m e t r i c w i t h K ⊆ A } .

Remark 3.6

Denoting by ℬ m = { A ⊂ Σ ∣ A is symmetric and i ( A ) ≥ m } , the previous lemma assures that

λ m = inf A ∈ ℬ m sup ( u , v ) ∈ A Φ ( u , v ) .

Proposition 3.7

Let α ∈ R and r > 0 and set

X α = { z ∈ X ∣ Φ ( z ) ≤ α Ψ ( z ) } , Σ α = Σ ∩ X α ,

S r α = { z ∈ X α ∣ Φ ( z ) = r } ,

X α ∘ = { z ∈ X ∣ Φ ( z ) < α Ψ ( z ) } , Σ α ∘ = Σ ∩ X α ∘ .

Then,

i ( Σ α ) ≥ m , for any α ≥ λ m ;
λ m < λ m + 1 ⇒ i ( Σ α ) = m , for any α ∈ [ λ m , λ m + 1 ) ;
λ m < λ m + 1 ⇒ i ( Σ λ m + 1 ∘ ) = m ;
Σ α is an odd strong deformation retract of X α \ { 0 } ;
Σ α ∘ is an odd strong deformation retract of X α ∘ \ { 0 } ;
S r α is an odd strong deformation retract of X α \ { 0 } .

Proof

Let us consider α > λ 1 ; otherwise, the proof is even simpler.

By contradiction, assume that i ( Σ λ m ) ≤ m − 1 . Continuity assures that there is a closed neighborhood N of Σ λ m such that i ( N ) = i ( Σ λ m ) . As N is also a neighborhood of the critical set { z ∣ Φ ∣ Σ ( z ) = λ m } , by the equivariant deformation theorem, there exist ε > 0 and an odd continuous map η : Σ λ m + ε → Σ λ m − ε ∪ N = N . So, due to monotonicity, i ( Σ λ m + ε ) ≤ i ( N ) = i ( Σ λ m ) ≤ m − 1 . By definition of λ m , there is A ¯ ∈ Σ m such that sup Φ ( A ¯ ) < λ m + ε . Hence, A ¯ ⊂ Σ λ m + ε , and we have the contradiction m ≤ i ( A ¯ ) ≤ i ( Σ λ m + ε ) ≤ m − 1 , which proves ( a ) .

Now, if λ m < λ m + 1 and α ∈ [ λ m , λ m + 1 ) , by ( a ) we know that i ( Σ α ) ≥ m . By contradiction, assume that i ( Σ α ) ≥ m + 1 . Recalling Remark 3.6, Σ α ∈ ℬ m + 1 and

λ m + 1 = inf A ∈ ℬ m + 1 sup A Φ ≤ sup Σ α Φ = α < λ m + 1 ,

so we conclude that i ( Σ α ) = m .

Moreover, still due to ( a ) , i ( Σ λ m + 1 ∘ ) ≥ m . Let K be a symmetric and compact subset of Σ λ m + 1 ∘ , so that max Φ ( K ) ≤ α for a suitable α ∈ [ λ m , λ m + 1 ) . ( b ) gives that i ( K ) ≤ i ( Σ α ) = m ; hence, ( c ) comes from Lemma 3.5.

For any s ∈ [ 0 , 1 ] and z = ( u , v ) ∈ X \ { 0 } , we define γ s , z = 1 − s + s Ψ ( z ) and η ( s , ( u , v ) ) = ( γ s , z 1 ∕ p u , γ s , z 1 ∕ q v ) .

η is clearly continuous and odd. In addition, due to the ( p , q ) -homogeneity of Φ and Ψ , shown in (1.6),

Φ ( η ( s , z ) ) = γ s , z Φ ( z ) and Ψ ( η ( s , z ) ) = γ s , z Ψ ( z ) .

Hence, ( d ) and ( e ) are proved as

z ∈ X α \ { 0 } ⇒ η ( 0 , z ) = z , η ( 1 , z ) ∈ Σ α and η ( s , z ) ∈ X α \ { 0 } ∀ s ∈ [ 0 , 1 ]

z ∈ Σ ⇒ η ( s , z ) = z , for any s ∈ [ 0 , 1 ] .

Analogously, we obtain ( f ) setting

δ s , z = 1 − s + s r Φ ( z ) and η ¯ ( s , ( u , v ) ) = ( δ s , z 1 ∕ p u , δ s , z 1 ∕ q v ) .□

Let m ∈ N be such that λ m < λ m + 1 .

If m ≥ 1 , we set

X − m ≔ { z = ( u , v ) ∈ X ∣ Φ ( u , v ) ≤ λ m Ψ ( u , v ) } ,

X + m ≔ { z = ( u , v ) ∈ X ∣ Φ ( u , v ) ≥ λ m + 1 Ψ ( u , v ) } ,

and if m = 0

X − 0 ≔ { 0 } X + 0 ≔ X .

We underline that the sets X − m and X + m are symmetric and ( p , q ) -homogeneous, i.e.,

( u , v ) ∈ X ± m ⇒ δ 1 p u , δ 1 q v ∈ X ± m , for any δ > 0 .

In particular, if p ≠ q and m ≥ 1 , the sets X + m and X − m are not cones.

Theorem 3.8

Setting

D r = { z ∈ X − m ∣ Φ ( z ) ≤ r } and S r = { z ∈ X − m ∣ Φ ( z ) = r } ,

( D r , S r ) links X + m cohomologically in dimension m over Z 2 , for any r > 0 .

Proof

From Proposition 3.7, we obtain that

i ( S r ) = i ( X \ X + m ) = m .

As S r and X + m are symmetric, 0 ∈ X + m and S r ∩ X + m = ∅ , Theorem 3.4 gives that

(3.2) ( X , S r ) links X + m cohomologically in dimension m over Z 2 .

Consider the commutative diagram

H m − 1 ( X ) → H m − 1 ( X − m \ { 0 } ) → H m ( X , X − m \ { 0 } ) → H m ( X ) → H m ( X − m \ { 0 } ) ↓ γ 1 ↓ γ 2 ↓ γ 3 ↓ γ 4 ↓ γ 5 H m − 1 ( D r ) ⟶ H m − 1 ( S r ) ⟶ H m ( D r , S r ) ⟶ H m ( D r ) ⟶ H m ( S r ) ,

where the horizontal rows are exact sequences and the vertical rows are induced by inclusions.

In particular, γ 1 , γ 2 , γ 4 , and γ 5 are isomorphisms, as D r is homotopic to X and S r is homotopic to X − m \ { 0 } . Hence, due to the five lemma [44], also γ 3 is an isomorphism, which, combined with (3.2), completes the proof.□

4 Critical group estimates at zero

From now on in this section, we assume that p , q ∈ [ 2 , N ) , α ≥ 0 , H ∈ C 1 ( R 2 , R ) and there exist C > 0 , p ′ ∈ ( p , p * ) , and q ′ ∈ ( q , q * ) such that

(4.1) ∣ H s ( s , t ) ∣ ≤ C ∣ s ∣ p ′ − 1 + ∣ t ∣ q ′ p ′ − 1 p ′ + 1 , ∣ H t ( s , t ) ∣ ≤ C ∣ s ∣ p ′ q ′ − 1 q ′ + ∣ t ∣ q ′ − 1 + 1 .

We denote by I α , H the C 1 -functional defined by

(4.2) I α , H ( z ) = 1 p ∫ Ω ( α + ∣ ∇ u ( x ) ∣ 2 ) p 2 d x + 1 q ∫ Ω ( α + ∣ ∇ v ( x ) ∣ 2 ) q 2 d x − ∫ Ω H ( u ( x ) , v ( x ) ) d x ,

for any z = ( u , v ) ∈ X .

Theorem 4.1

Let p , q ∈ [ 2 , N ) , α ≥ 0 . Assume that H ∈ C 1 ( R 2 , R ) satisfies (4.1) and ∇ H ( 0 , 0 ) = ( 0 , 0 ) .

If 0 = ( 0 , 0 ) is an isolated critical point for I α , H , there are H ¯ ∈ C 1 ( R 2 , R ) and η ¯ > 0 such that:

H ¯ ( s , t ) = H ( s , t ) when ∣ s ∣ , ∣ t ∣ ≤ η ¯ .
0 is an isolated critical point for I α , H ¯ and C m ( I α , H , 0 ) = C m ( I α , H ¯ , 0 ) , for any m ∈ N .
H ¯ ( R 2 ) = H ¯ ( [ − 2 η ¯ , 2 η ¯ ] 2 ) ⊂ H ( [ − 2 η ¯ , 2 η ¯ ] 2 ) .

In particular, if ( s , t ) ∉ [ − 2 η ¯ , 2 η ¯ ] 2 , then

H ¯ ( s , t ) = H ( 0 , 0 ) , i f ∣ s ∣ ≥ 2 η ¯ , ∣ t ∣ ≥ 2 η ¯ , H ¯ ( 0 , t ) , i f ∣ s ∣ ≥ 2 η ¯ , ∣ t ∣ ≤ 2 η ¯ , H ¯ ( s , 0 ) , i f ∣ s ∣ ≤ 2 η ¯ , ∣ t ∣ ≥ 2 η ¯ .

In addition, if H is C 2 at least in an open set containing [ − 2 η ¯ , 2 η ¯ ] 2 , then H ¯ ∈ C 2 ( R 2 , R ) .

Proof

Consider a C ∞ -function θ : R → [ 0 , 1 ] such that θ ( τ ) = 1 for ∣ τ ∣ ≤ 1 and θ ( τ ) = 0 for ∣ τ ∣ ≥ 2 .

For every δ ∈ [ 0 , 1 ] , we define H δ ( s , t ) = H ( θ ( δ s ) s , θ ( δ t ) t ) , so that

(4.3) H s δ ( s , t ) = H s ( θ ( δ s ) s , θ ( δ t ) t ) ( θ ′ ( δ s ) δ s + θ ( δ s ) ) , H t δ ( s , t ) = H s ( θ ( δ s ) s , θ ( δ t ) t ) ( θ ′ ( δ t ) δ t + θ ( δ t ) ) .

As the function τ ↦ ∣ θ ′ ( τ ) τ + θ ( τ ) ∣ is bounded in R , by (4.1) and (4.3), there is a constant C 0 > 0 , independent from δ , such that

(4.4) ∣ H s δ ( s , t ) ∣ ≤ C 0 ∣ s ∣ p ′ − 1 + ∣ t ∣ q ′ p ′ − 1 p ′ + 1 , ∣ H t δ ( s , t ) ∣ ≤ C 0 ∣ s ∣ p ′ q ′ − 1 q ′ + ∣ t ∣ q ′ − 1 + 1 .

Hence, the functional I α , H δ : X → R is C 1 for any α ≥ 0 and δ ∈ [ 0 , 1 ] and for any z 0 = ( u 0 , v 0 ) ∈ X , z = ( u , v ) ∈ X , it results in

⟨ I α , H δ ′ ( z 0 ) , z ⟩ = ∫ Ω ( α + ∣ ∇ u 0 ∣ 2 ) p − 2 2 ∇ u 0 ∇ u d x + ∫ Ω ( α + ∣ ∇ v 0 ∣ 2 ) q − 2 2 ∇ v 0 ∇ v d x − ∫ Ω ( H s δ ( u 0 , v 0 ) u + H t δ ( u 0 , v 0 ) v ) d x .

Let r > 0 be such that 0 is the unique critical point of I α , H 0 = I α , H in

D r = { z = ( u , v ) ∈ X : ‖ ∇ u ‖ p + ‖ ∇ v ‖ q ≤ r } .

The map δ ↦ I α , H δ is continuous from [ 0 , 1 ] to C b 1 ( D r , R ) .

Moreover, reasoning as in the proof of Proposition 2.3, we see that I α , H δ satisfies the Palais-Smale condition in D r , for any α ≥ 0 and δ ∈ [ 0 , 1 ] .

We claim that there exists δ ¯ ∈ ( 0 , 1 ] such that 0 is the unique critical point of I α , H δ in D r , for any δ ∈ [ 0 , δ ¯ ] . Assume, by contradiction, that δ j → 0 and z j = ( u j , v j ) ∈ D r \ { 0 } is a critical point of I α , H δ j .

Taking account of (4.4), by Theorem 1.1 in [47], we infer that every u j and v j are in L ∞ ( Ω ) and there is C 1 > 0 , depending on Ω , p , q , p ′ , q ′ , and r but independent from j , such that

‖ u j ‖ ∞ , ‖ v j ‖ ∞ ≤ C 1 .

Thus, ∣ δ j u j ( x ) ∣ < 1 and ∣ δ j v j ( x ) ∣ < 1 in Ω , when j is big enough.

Therefore, by (4.3), z j = ( u j , v j ) is a critical point of I α , H and a contradiction follows.

Setting H ¯ = H δ ¯ , from [17, Theorem 5.2], we deduce that C m ( I α , H , 0 ) = C m ( I α , H ¯ , 0 ) (for related results, see also [8, Theorem I.5.6]) and the assertion follows.□

Assume that H ∈ C 2 ( R 2 , R ) , and there exist C > 0 , p ′ ∈ ( p , p * ) , q ′ ∈ ( q , q * ) such that

(4.5) c ∣ H s s ( s , t ) ∣ ≤ C ∣ s ∣ p ′ − 2 + ∣ t ∣ q ′ p ′ − 2 p ′ + 1 , ∣ H s t ( s , t ) ∣ , ∣ H t s ( s , t ) ∣ ≤ C ∣ s ∣ p ′ − 1 − p ′ q ′ + ∣ t ∣ q ′ − 1 − q ′ p ′ + 1 , ∣ H t t ( s , t ) ∣ ≤ C ∣ s ∣ p ′ q ′ − 2 q ′ + ∣ t ∣ q ′ − 2 + 1 .

Then, I α , H , defined through (4.2), is a C 2 -functional, and for any z 0 = ( u 0 , v 0 ) , z 1 = ( u 1 , v 1 ) , z 2 = ( u 2 , v 2 ) ∈ X , we have

⟨ I α , H ″ ( z 0 ) z 1 , z 2 ⟩ = ∫ Ω ( ( α + ∣ ∇ u 0 ∣ 2 ) ( p − 2 ) ∕ 2 ( ∇ u 1 ∣ ∇ u 2 ) + ( p − 2 ) ( α + ∣ ∇ u 0 ∣ 2 ) ( p − 4 ) ∕ 2 ( ∇ u 0 ∣ ∇ u 1 ) ( ∇ u 0 ∣ ∇ u 2 ) ) d x + ∫ Ω ( ( α + ∣ ∇ v 0 ∣ 2 ) ( q − 2 ) ∕ 2 ( ∇ v 1 ∣ ∇ v 2 ) + ( q − 2 ) ( α + ∣ ∇ v 0 ∣ 2 ) ( q − 4 ) ∕ 2 ( ∇ v 0 ∣ ∇ v 1 ) ( ∇ v 0 ∣ ∇ v 2 ) ) d x − ∫ Ω ( H s s ( u 0 , v 0 ) u 1 u 2 + H t t ( u 0 , v 0 ) v 1 v 2 + H s t ( u 0 , v 0 ) u 1 v 2 + H t s ( u 0 , v 0 ) u 2 v 1 ) d x .

We recall the following notion.

Definition 4.2

If Y is a Banach space, I ∈ C 2 ( Y , R ) and z 0 a critical point of I , and the Morse index m ( I , z 0 ) of I at z 0 is the supremum of the dimensions of the subspaces of Y , where I ″ ( z 0 ) is negative definite. The large Morse index m * ( I , z 0 ) is the sum of m ( I , z 0 ) and the dimension of the kernel of I ″ ( z 0 ) .

Arguing as in [6, Theorem 1.4], we can establish the following critical group estimates for the C 2 functionals I α associated with systems of ( p , q ) -area equations. Precisely, we have

Theorem 4.3

Let α > 0 and H ∈ C 2 ( R 2 , R ) satisfying assumptions (4.5). If z 0 is a critical point of the functional I α , H , then m ( I α , H , z 0 ) and m * ( I α , H , z 0 ) are finite and

C m ( I α , H , z 0 ) = { 0 } ,

whenever m < m ( I α , H , z 0 ) or m > m * ( I α , H , z 0 ) .

In the following theorem, we refer to m 0 and m 0 * , which have been defined through (1.8).

Theorem 4.4

Let α ≥ 0 and p , q ∈ [ 2 , N ) . Let G ∈ C 1 ( R 2 , R ) a function that satisfies the conditions ( a 1 ) and ( a 2 ) . If 0 is an isolated critical point of the functional J α , defined as in (1.3), then

C m ( J α , 0 ) = { 0 } , whenever m < m 0 or m > m 0 * .

Proof

From (4.2), J α = I α , G . Due to assumption ( a 1 ) and Theorem 4.1, there is G ¯ ∈ C 1 ( R 2 , R ) and η ¯ > 0 such that

(4.6) ∣ s ∣ , ∣ t ∣ ≤ η ¯ ⇒ G ¯ ( s , t ) = G ( s , t ) ,

(4.7) ∣ s ∣ , ∣ t ∣ ≥ 2 η ¯ ⇒ G ¯ ( s , t ) = G ( 0 , 0 ) ,

(4.8) C m ( J α , 0 ) = C m ( I α , G ¯ , 0 ) , for any m ∈ N .

Taking account of assumption ( a 2 ) , where U is an open set containing [ − 2 η ¯ , 2 η ¯ ] × [ − 2 η ¯ , 2 η ¯ ] , Theorem 4.1 assures that G ¯ ∈ C 2 ( R 2 , R ) . Furthermore, by (4.7), G ¯ satisfies (4.5), so that I α , G ¯ ∈ C 2 ( X , R ) .

Moreover, (4.6) and (1.8) give that ⟨ I α , G ¯ ″ ( 0 ) z , z ⟩ = Q ( z ) for any z in X ; thus, m ( I α , G ¯ , 0 ) = m 0 and m * ( I α , G ¯ , 0 ) = m 0 * .

If 0 , then Theorem 4.3 holds; thus, C m ( I α , G ¯ , 0 ) = 0 whenever m < m ( I α , G ¯ , 0 ) or m > m * ( I α , G ¯ , 0 ) , so by (4.8), the assertion follows.

If instead α = 0 , we have to make a further distinction.

If H G ( 0 , 0 ) is null or negative semidefinite, we have m 0 = 0 , m 0 * = + ∞ and the assertion is obvious.

If H G ( 0 , 0 ) is negative definite, we have m 0 = m 0 * = 0 and there is μ > 0 such that

(4.9) H G ( 0 , 0 ) [ ξ ] 2 ≤ − μ ∣ ξ ∣ 2 for any ξ ∈ R 2 .

Moreover, due to assumption ( a 1 ) , there are p ′ > p , q ′ > q , and C > 0 such that W 0 1 , p ( Ω ) ⊂ L p ′ ( Ω ) , W 0 1 , q ( Ω ) ⊂ L q ′ ( Ω ) , and

∣ G ( s , t ) ∣ ≤ C ( ∣ s ∣ p ′ + ∣ t ∣ q ′ + 1 ) .

Combining with (4.9) and redefining C > 0 , we obtain

− G ( s , t ) ≥ − G ( 0 , 0 ) − C ( ∣ s ∣ p ′ + ∣ t ∣ q ′ ) , for any ( s , t ) ∈ R 2 ,

Thus,

J 0 ( u , v ) − J 0 ( 0 ) ≥ 1 p ‖ u ‖ p p + 1 q ‖ v ‖ q q − C ( ∣ u ∣ p ′ p ′ + ∣ v ∣ q ′ q ′ )

≥ ‖ u ‖ p p 1 p − C ‖ u ‖ p p ′ − p + ‖ v ‖ q q 1 q − C ‖ v ‖ q q ′ − q ≥ 0 ,

if ‖ ( u , v ) ‖ is small enough.

Hence, 0 is a local minimum for J 0 , and, by the excision property, we have

C m ( J 0 , 0 ) ≈ H m ( { 0 } , ∅ ) ,

so the assertion follows.

If H G ( 0 , 0 ) is positive semidefinite or indefinite, we have m 0 = m 0 * = + ∞ .

As I α , G ¯ ∈ C 2 ( X , R ) , from [34, Theorem 3.1], we infer that C m ( I α , G ¯ , 0 ) = { 0 } for any m , so by (4.8), the assertion follows.□

5 Geometry of J α

Let us recall that F : R 2 → R is the C 1 -function defined by

F ( s , t ) = 1 p ∣ s ∣ p + 1 q ∣ t ∣ q + 1 ( β + 1 ) ( γ + 1 ) ∣ s ∣ β ∣ t ∣ γ s t .

Assuming that G ( 0 , 0 ) = 0 , from assumption ( a 1 ) , we see that

G ( s , t ) = λ ¯ F ( s , t ) + R ( s , t ) ,

where R ∈ C 1 ( R 2 , R ) , R ( 0 , 0 ) = 0 , and

(5.1) lim ∣ ( s , t ) ∣ → ∞ R s ( s , t ) ∣ s ∣ p − 1 + ∣ t ∣ q p − 1 p = 0 lim ∣ ( s , t ) ∣ → ∞ R t ( s , t ) ∣ s ∣ p q − 1 q + ∣ t ∣ q − 1 = 0 .

Theorem 5.1

Following the previous notations,

lim ∣ ( s , t ) ∣ → ∞ R ( s , t ) F ( s , t ) = 0 .

Proof

From Young’s inequality and (1.5), there exists c > 0 be such that

1 p ∣ s ∣ p + ∣ s ∣ ∣ t ∣ q p − 1 p + 1 q ∣ t ∣ q ≤ c F ( s , t ) , ∀ ( s , t ) ∈ R 2 .

By (5.1), for every ε > 0 , there are η ε , c ε , δ ε > 0 such that

∣ ( s , t ) ∣ > η ε ⇒ ∣ R s ( s , t ) ∣ ≤ ε 2 c ∣ s ∣ p − 1 + ∣ t ∣ q p − 1 p , ∣ R t ( s , t ) ∣ ≤ ε 2 c ∣ s ∣ p q − 1 q + ∣ t ∣ q − 1 ; ∣ ( s , t ) ∣ ≤ η ε ⇒ ∣ R s ( s , t ) ∣ ≤ c ε , ∣ R t ( s , t ) ∣ ≤ c ε ; ∣ ( s , t ) ∣ > δ ε ⇒ F ( s , t ) > 4 c ε η ε ε .

For any r ∈ R , let us denote by r + = max { 0 , r } and r − = min { 0 , r } . As

R ( s , t ) = ∫ 0 s R s ( σ , t ) d σ + ∫ 0 t R t ( 0 , τ ) d τ ,

we infer that

∣ R ( s , t ) ∣ ≤ ∫ s − s + ∣ R s ( σ , t ) ∣ d σ + ∫ t − t + ∣ R t ( 0 , τ ) ∣ d τ = ∫ { σ ∈ [ s − , s + ] : ∣ ( σ , t ) ∣ ≤ η ε } ∣ R s ( σ , t ) ∣ d σ + ∫ { σ ∈ [ s − , s + ] : ∣ ( σ , t ) ∣ > η ε } ∣ R s ( σ , t ) ∣ d σ + ∫ { τ ∈ [ t − , t + ] : ∣ τ ∣ ≤ η ε } ∣ R t ( 0 , τ ) ∣ d τ + ∫ { τ ∈ [ t − , t + ] : ∣ τ ∣ > η ε } ∣ R t ( 0 , τ ) ∣ d τ ≤ 2 c ε η ε + ε 2 F ( s , t ) .

Therefore,

□ ∣ ( s , t ) ∣ > δ ε ⇒ ∣ R ( s , t ) ∣ F ( s , t ) < ε .

Proposition 5.2

If ( a 1 ) holds and ( u , v ) is a weak solution to Problem (1.2), then

( u , v ) ∈ ( C 1 , η ( Ω ¯ ) ) 2 , for some η ∈ ( 0 , 1 ) ;
for every bounded set A ⊂ X , there exists η ∈ ( 0 , 1 ) and M > 0 such that ‖ u ‖ C 1 , η ( Ω ¯ ) ≤ M and ‖ v ‖ C 1 , η ( Ω ¯ ) ≤ M for every ( u , v ) ∈ A solving (1.2);
if there is ε > 0 such that
G s ( 0 , t ) ≠ 0 , ∀ t ∈ ( − ε , ε ) \ { 0 } , G t ( s , 0 ) ≠ 0 , ∀ s ∈ ( − ε , ε ) \ { 0 } , (*)
then every nontrivial solution ( u , v ) of (1.2) is also not semitrivial, i.e., u ≠ 0 and v ≠ 0 .

Proof

If ( a 1 ) holds and A is a bounded subset of X , by Theorem 1.1 in [47], all the weak solutions ( u , v ) ∈ A are in ( L ∞ ( Ω ) ) 2 and there is M 0 > 0 such that ‖ u ‖ ∞ ≤ M 0 and ‖ v ‖ ∞ ≤ M 0 . Hence, due to [36], we infer (i) and (ii).

Let us prove (iii) arguing by contradiction. Suppose that ( a 1 ) and ( * ) hold and that there is an ( u , 0 ) ≠ ( 0 , 0 ) solving (1.2). By (i), we obtain that u ∈ C ( Ω ¯ ) \ { 0 } , M = max Ω ¯ ∣ u ∣ > 0 , and there is x ¯ ∈ Ω such that 0 < ∣ u ( x ¯ ) ∣ < min { ε , M } . Hence, by ( * ) , G t ( u ( x ¯ ) , 0 ) ≠ 0 , in contradiction with (1.2).□

Lemma 5.3

If r ≥ 2 , α ≥ 0 , and ε > 0 , there is c ( r , α , ε ) > 0 such that

1 r ( α + t 2 ) r 2 ≤ 1 r ( 1 + ε ) ∣ t ∣ r + c ( r , α , ε ) ,

for any t ∈ R .

Proof

The assertion is trivial if r = 2 , so let us consider the case r > 2 .

Setting f α ( t ) = 1 r ( α + t 2 ) r 2 − 1 r ( 1 + ε ) ∣ t ∣ r , we see that lim ∣ t ∣ → + ∞ f α ( t ) = − ∞ .

Moreover, f α ′ ( t ) = t ∣ t ∣ r − 2 α t 2 + 1 r − 2 2 − 1 − ε and, denoting by t α = α ( 1 + ε ) 2 r − 2 − 1 1 ∕ 2 , we obtain

□ max t ∈ R f α ( t ) = f α ( ± t α ) = α r ∕ 2 r 1 + ε ( 1 + ε ) 2 r − 2 − 1 r − 2 2 .

Setting

L ( s , t ) = 1 p ∣ s ∣ p + 1 q ∣ t ∣ q and L α ( s , t ) = 1 p ( α + s 2 ) p ∕ 2 + 1 q ( α + t 2 ) q ∕ 2 ,

and recalling Theorem 5.1, we obtain the following result.

Corollary 5.4

For any ε > 0 , there is c ε > 0 such that

(5.2) L ( s , t ) ≤ L α ( s , t ) ≤ ( 1 + ε ) L ( s , t ) + c ε

(5.3) − ( λ ¯ + ε ) F ( s , t ) − c ε ≤ − G ( s , t ) ≤ − ( λ ¯ − ε ) F ( s , t ) + c ε ,

for any ( s , t ) ∈ R 2 .

Proof of Theorem 1.1

As λ ¯ is not an eigenvalue for System (1.4), let m ∞ ∈ N be such that

λ m ∞ < λ ¯ < λ m ∞ + 1 and assume m ∞ ∉ [ m 0 , m 0 * ] .

Setting

X − = { z ∈ X ∣ Φ ( z ) ≤ λ m ∞ Ψ ( z ) } , X + = { z ∈ X ∣ Φ ( z ) ≥ λ m ∞ + 1 Ψ ( z ) } , D r = { z ∈ X − ∣ Φ ( z ) ≤ r } S r = { z ∈ X − ∣ Φ ( z ) = r } .

Theorem 3.8 assures that ( D r , S r ) links X + cohomologically in dimension m ∞ over Z 2 , for any r > 0 .

We take into consideration only the case m ∞ ≥ 1 , as when m ∞ = 0 , the proof is similar and simpler.

Let α ′ , α ″ , β ′ be such that

λ m ∞ < α ′ < α ″ < λ ¯ < β ′ < λ m ∞ + 1 .

By Corollary 5.4, there is C > 0 such that

− β ′ F ( s , t ) − C ≤ − G ( s , t ) ≤ − α ″ F ( s , t ) + C L ( s , t ) ≤ L α ( s , t ) ≤ α ′ λ m ∞ L ( s , t ) + C .

Therefore, for any ( u , v ) ∈ X + ,

J α ( u , v ) = ∫ Ω L α ( ∣ ∇ u ( x ) ∣ , ∣ ∇ v ( x ) ∣ ) d x − ∫ Ω G ( u ( x ) , v ( x ) ) d x ≥ ∫ Ω L ( ∣ ∇ u ( x ) ∣ , ∣ ∇ v ( x ) ∣ ) d x − ∫ Ω ( β ′ F ( u ( x ) , v ( x ) ) + C ) d x ≥ ( λ m ∞ + 1 − β ′ ) ∫ Ω F ( u ( x ) , v ( x ) ) d x − C ∣ Ω ∣ ≥ − C ∣ Ω ∣ ,

so that

inf z ∈ X + J α ( z ) = a > − ∞ .

On the other hand, for any ( u , v ) ∈ X − ,

J α ( u , v ) = ∫ Ω L α ( ∣ ∇ u ( x ) ∣ , ∣ ∇ v ( x ) ∣ ) d x − ∫ Ω G ( u ( x ) , v ( x ) ) d x ≤ α ′ λ m ∞ ∫ Ω L ( ∣ ∇ u ( x ) ∣ , ∣ ∇ v ( x ) ∣ ) d x + C ∣ Ω ∣ − α ″ ∫ Ω F ( u ( x ) , v ( x ) ) d x + C ∣ Ω ∣ ≤ 2 C ∣ Ω ∣ + α ′ − α ″ λ m ∞ ∫ Ω L ( ∣ ∇ u ( x ) ∣ , ∣ ∇ v ( x ) ∣ ) d x ,

so that

lim ‖ z ‖ → ∞ z ∈ X − J α ( z ) = − ∞ ⇒ ∃ r > 0 : sup S r J α < a = inf X + J α .

Moreover, as D r is compact, b = max D r J α < + ∞ .

If J α − 1 ( [ a , b ] ) contains an infinite number of critical points, the theorem is obviously proved. Otherwise, applying Theorem 3.3 combined with Proposition 2.4, there exists a critical point z 0 of J α with a ≤ J α ( z 0 ) ≤ b and

C m ∞ ( J α , z 0 ) ≠ { 0 } .

As m ∞ ∉ [ m 0 , m 0 * ] , Theorem 4.4 gives that

C m ∞ ( J α , 0 ) = { 0 } .

Hence z 0 ≠ 0 .□

Lemma 5.5

Let Γ : R 2 → R be a C 1 function such that

(5.4) lim ∣ ( s , t ) ∣ → ∞ Γ ( s , t ) F ( s , t ) = 0

and

(5.5) lim ∣ ( s , t ) ∣ → ∞ Γ ( s , t ) − 1 p Γ s ( s , t ) s − 1 q Γ t ( s , t ) t = − ∞ ,

then

(5.6) lim ∣ ( s , t ) ∣ → ∞ Γ ( s , t ) = − ∞ .

In the same way, if Γ satisfies Condition (5.4) and

(5.7) lim ∣ ( s , t ) ∣ → ∞ Γ ( s , t ) − 1 p Γ s ( s , t ) s − 1 q Γ t ( s , t ) t = + ∞ ,

then

(5.8) lim ∣ ( s , t ) ∣ → ∞ Γ ( s , t ) = + ∞ .

Proof

Let us observe F ( x 1 ∕ p s , x 1 ∕ q t ) = x F ( s , t ) , for any x ≥ 0 and ( s , t ) ∈ R 2 .

Hence, if ( s , t ) ∈ R 2 \ { ( 0 , 0 ) } , then, by (5.4),

(5.9) lim x → + ∞ Γ ( x 1 ∕ p s , x 1 ∕ q t ) x = 0 .

Let us fix M > 0 . By (5.5), there exists δ > 0 such that

Γ ( s , t ) − 1 p Γ s ( s , t ) s − 1 q Γ t ( s , t ) t < − M when ∣ ( s , t ) ∣ > δ .

If ( s ′ , t ′ ) ∈ R 2 is such that ∣ ( s ′ , t ′ ) ∣ > δ , then

d d x Γ ( x 1 p s ′ , x 1 q t ′ ) + M x = 1 p Γ s ( x 1 p s ′ , x 1 q t ′ ) x 1 p s ′ + 1 q Γ t ( x 1 p s ′ , x 1 q t ′ ) x 1 q t ′ − Γ ( x 1 p s ′ , x 1 q t ′ ) − M x 2 > 0 , if x ≥ 1 ,

so that

Γ ( x 1 p s ′ , x 1 q t ′ ) + M x ≤ Γ ( x ¯ 1 p s ′ , x ¯ 1 q t ′ ) + M x ¯ , whenever 1 ≤ x ≤ x ¯ .

By (5.9), passing to the limit for x ¯ → + ∞ , we obtain

Γ ( x 1 p s ′ , x 1 q t ′ ) + M x ≤ 0 ∀ x ≥ 1 .

Hence, Γ ( s ′ , t ′ ) ≤ − M , which proves (5.6).

Finally, (5.8) can be proved applying (5.6) to the function Γ ¯ = − Γ .□

Proof of Theorem 1.2

lim ∣ ( s , t ) ∣ → ∞ G ( s , t ) − 1 p G s ( s , t ) s − 1 q G t ( s , t ) t = − ∞

and F ( s , t ) − 1 p F s ( s , t ) s − 1 q F t ( s , t ) t = 0 for any ( s , t ) ∈ R 2 , by Theorem 5.1 and Lemma 5.5, we obtain

lim ∣ ( s , t ) ∣ → ∞ G ( s , t ) − λ ¯ F ( s , t ) = − ∞ .

Hence, G ( s , t ) − λ ¯ F ( s , t ) is upperly bounded.

Therefore, recalling Corollary 5.4 and letting α ′ , α ″ be such that

λ m ∞ < α ′ < α ″ < λ ¯ ≤ λ m ∞ + 1 ,

there is C > 0 such that

− λ ¯ F ( s , t ) − C ≤ − G ( s , t ) ≤ − α ″ F ( s , t ) + C L ( s , t ) ≤ L α ( s , t ) ≤ α ′ λ m ∞ L ( s , t ) + C .

So we can conclude as in the proof of Theorem 1.1, defining X − and X + in the same way.□

Proof of Theorem 1.3

First, reasoning as in the previous proof, by assumption ( b + ) , we obtain

(5.10) lim ∣ ( s , t ) ∣ → ∞ G ( s , t ) − λ ¯ F ( s , t ) = + ∞ .

Let β ′ ∈ ( λ ¯ , λ m ∞ + 1 ) . By Corollary 5.4, there is C > 0 such that

− β ′ F ( s , t ) − C ≤ − G ( s , t ) .

We define X − and X + as in the proof of Theorem 1.1 and, reasoning in the same way, we see that

inf z ∈ X + J 0 ( z ) = a > − ∞ .

So it is sufficient to prove that, for any M > 0 , we can choose a suitably big r > 0 such that

sup z ∈ S r J 0 ( z ) < − M .

By contradiction, there is M > 0 and a sequence z n = ( u n , v n ) in X − such that Φ ( z n ) = n and

(5.11) J 0 ( z n ) ≥ − M .

As z n ∈ X − , we see that

(5.12) J 0 ( z n ) = Φ ( z n ) − ∫ Ω G ( z n ( x ) ) d x ≤ ∫ Ω λ ¯ F ( z n ( x ) ) − G ( z n ( x ) ) d x .

Let δ n = 1 ∕ Φ ( z n ) , u n ′ = δ n 1 ∕ p u n , and v n ′ = δ n 1 ∕ q v n , z n ′ = ( u n ′ , v n ′ ) .

As Φ ( z n ′ ) = 1 , z n ′ converges to some z ′ = ( u ′ , v ′ ) weakly in X and strongly in L p ( Ω ) × L q ( Ω ) , up to a subsequence.

It is easy to see that every z n ′ ∈ X − , as well as z ′ ∈ X − .

Moreover,

Ψ ( z ′ ) = lim n → ∞ Ψ ( z n ′ ) ≥ lim n → ∞ 1 λ m ∞ Φ ( z n ′ ) = 1 λ m ∞ > 0 .

Hence,

(5.13) z ′ ≠ 0 .

Since

∣ u n ( x ) ∣ p + ∣ v n ( x ) ∣ q = Φ ( z n ) ( ∣ δ n 1 ∕ p u n ( x ) ∣ p + ∣ δ n 1 ∕ q v n ( x ) ∣ q ) ≥ n ( ∣ u n ′ ( x ) ∣ p + ∣ v n ′ ( x ) ∣ q )

and

∣ u n ′ ( x ) ∣ p + ∣ v n ′ ( x ) ∣ q ⟶ ∣ u ′ ( x ) ∣ p + ∣ v ′ ( x ) ∣ q , a.e. in Ω ,

taking (5.10), (5.12), and (5.13) into account, we infer that, up to a subsequence,

lim n → ∞ J 0 ( z n ) = − ∞ ,

which contradicts (5.11).□

Acknowledgments

The authors thank Prof. Marco Degiovanni for some fruitful discussions.

Funding information: The authors were supported by PRIN PNRR P2022YFAJH “Linear and Nonlinear PDEs: New directions and applications,” and partially supported by INdAM-GNAMPA. The first and second authors thank PNRR MUR Project CN00000013 HUB - National Centre for HPC, Big Data and Quantum Computing (CUP H93C22000450007).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript.
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-08-04

Revised: 2024-01-12

Accepted: 2024-02-22

Published Online: 2024-06-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2024-0005

Keywords for this article

quasilinear elliptic systems; Fadell Rabinowitz index; asymptotically (p, q) linear; Morse index; resonance; critical groups

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