Multiple Solutions to (p,q)-Laplacian Problems with Resonant Concave Nonlinearity

1 Introduction

Let Ω be a bounded domain in ℝN with a smooth boundary ∂⁡Ω, let 1<q≤p<+∞, and let μ≥0. Consider the Dirichlet problem

where Δr, r>1, denotes the r-Laplacian, namely,

Δ r ⁢ u := div ⁡ ( ∥ ∇ ⁡ u ∥ r - 2 ⁢ ∇ ⁡ u )   for all ⁢ u ∈ W 0 1 , r ⁢ ( Ω ) ,

the reaction term f:Ω×ℝ→ℝ satisfies Carathéodory’s conditions, while, as usual, p=q if and only if μ=0. Elliptic equations involving differential operators of the form

A ⁢ u := - Δ p ⁢ u - Δ q ⁢ u ,

often called (p,q)-Laplacian, occur in many important concrete situations. For instance, this happens when one seeks stationary solutions to the reaction-diffusion system

u t + A ⁢ u = c ⁢ ( x , u ) ,

which exhibits a wide range of applications in physics and related sciences such as biophysics, quantum and plasma physics, and chemical reaction design; see [3, 6]. Consequently, they have been the subject of numerous investigations, both in bounded domains and on the whole space, mainly concerning the multiplicity of solutions or bifurcation-type results.

This paper falls within the first framework. We show that if, roughly speaking, f has a subcritical growth and, moreover,

1. lim|t|→+∞⁡p|t|p⁢∫0tf⁢(x,ξ)⁢𝑑ξ=λ1,p  uniformly in x∈Ω, where λ1,p denotes the first eigenvalue of (-Δp,W01,p⁢(Ω)),

2. lim|t|→+∞⁡[f⁢(x,t)⁢t-p⁢∫0tf⁢(x,ξ)⁢𝑑ξ]=+∞  uniformly in x∈Ω,

3. c⁢|t|θ≤f⁢(x,t)⁢t≤θ⁢∫0tf⁢(x,ξ)⁢𝑑ξ  for all (x,t)∈Ω×[-δ,δ], where c>0, θ∈(1,q), while δ>0,

then (1.1) possesses at least three nontrivial solutions in C01⁢(Ω¯), one greatest negative v-, another smallest positive u+, and a third nodal u0, such that v-≤u0≤u+; see Theorem 3.9 below.

Assumptions (i)–(ii) directly give

see the proof of Lemma 3.1. Hence, resonance with respect to λ1,p from the left occurs and, a fortiori, the energy functional φ associated with (1.1) is coercive.

Now, the question of investigating what happens if there is resonance from the right of λ1,p, i.e., the limit in (1.2) equals -∞, naturally arises. Accordingly, φ turns out to be indefinite and direct methods no longer work. However, via linking arguments and, in place of (ii), via the hypothesis that

1. either μ>0 and

   lim inf | t | → + ∞ ⁡ 1 | t | η ⁢ [ p ⁢ ∫ 0 t f ⁢ ( x , ξ ) ⁢ 𝑑 ξ - f ⁢ ( x , t ) ⁢ t ] ≥ C > 0   uniformly in ⁢ x ∈ Ω ,

   where η∈(q,p], or μ=0 and

   lim | t | → + ∞ ⁡ [ p ⁢ ∫ 0 t f ⁢ ( x , ξ ) ⁢ 𝑑 ξ - f ⁢ ( x , t ) ⁢ t ] = + ∞   uniformly in ⁢ x ∈ Ω ,

we still obtain a nontrivial solution u0∈C01⁢(Ω¯) of (1.1); cf. Theorem 4.5 below.

It should be also noted that, in both settings, due to (ii), the nonlinearity f⁢(x,⋅) exhibits a concave behavior at the origin. Such a type of growth rate has been widely studied, also combined with further conditions, provided p=2 and μ=0, i.e., the equation is semilinear. As an example, besides the seminal paper [2], let us mention [21, 22, 16, 8]. A similar comment holds true also when p≠2 but μ=0, in which case the literature looks to be daily increasing; see for instance the very recent papers [12, 19, 14, 18] and, concerning the nonsmooth framework, [17, 13].

Another meaningful feature of (1.1) is the following. If μ>0, then the differential operator u↦-Δp⁢u-μ⁢Δq⁢u turns out to be nonhomogeneous. Hence, standard results for the p-Laplacian not always extend in a simple way to it.

Our approach is variational, based on critical point theory, together with appropriate truncation-comparison arguments and results from Morse theory.

2 Mathematical Background

Let (X,∥⋅∥) be a real Banach space. Given a set V⊆X, write V¯ for the closure of V and ∂⁡V for the boundary of V. If x∈X, δ>0, then Bδ⁢(x):={z∈X:∥z-x∥<δ}, while Bδ:=Bδ⁢(0). The symbol X* denotes the dual space of X, 〈⋅,⋅〉 indicates the duality brackets for the pair (X*,X), and xn→x (respectively, xn⇀x) in X means that ‘the sequence {xn} converges strongly (respectively, weakly) in X’. An operator A:X→X* is called of type (S)+ provided

x n ⇀ x   in X ,   lim sup n → + ∞ ⁡ 〈 A ⁢ ( x n ) , x n - x 〉 ≤ 0   imply   x n → x   in X .

Let φ∈C1⁢(X) and let c∈ℝ. Put

φ c := { x ∈ X : φ ⁢ ( x ) ≤ c } , φ c := { x ∈ X : φ ⁢ ( x ) ≥ c } , K φ := { x ∈ X : φ ′ ⁢ ( x ) = 0 } , K φ c := { x ∈ K φ : φ ⁢ ( x ) = c } .

We say that φ satisfies the Cerami condition when

1. every sequence {xn}⊆X such that {φ⁢(xn)} is bounded and

   lim n → + ∞ ⁡ ( 1 + ∥ x n ∥ ) ⁢ φ ′ ⁢ ( x n ) = 0   in ⁢ X *

   admits a strongly convergent subsequence.

This compactness-type assumption turns out to be weaker than the usual Palais–Smale condition. Nevertheless, it suffices to prove a deformation theorem, from which the minimax theory for the critical values of φ follows. In such a framework, the topological notion of linking sets plays a key role.

The following general minimax principle is well known; see, e.g., [10, Theorem 5.2.5].

Appropriate choices of linking sets in Theorem 2.2 produce meaningful critical point results. For later use, we state here the famous Ambrosetti–Rabinowitz mountain pass theorem.

Let (Y1,Y2) be a topological pair such that Y2⊆Y1⊆X and let k be any nonnegative integer. We denote by Hk⁢(Y1,Y2) the k-th relative singular homology group for the pair (Y1,Y2) with integer coefficients. Given an isolated critical point x0∈Kφc,

C k ⁢ ( φ , x 0 ) := H k ⁢ ( φ c ∩ U , φ c ∩ U ∖ { x 0 } ) , k ∈ ℕ 0 ,

is the k-th critical group of φ at x0. Here, U indicates any neighborhood of x0 fulfilling Kφ∩φc∩U={x0}. The excision property of singular homology ensures that this definition does not depend on the choice of U. The monographs [5, 11] are general references on this subject.

Hereafter, ∥⋅∥ stands for the ℝN-norm, while |A| denotes the N-dimensional Lebesgue measure of A⊆ℝN. If p∈[1,+∞), then p′ indicates the conjugate exponent of p and ∥⋅∥p is the usual norm of the Sobolev space W01,p⁢(Ω), namely, thanks to the Poincaré inequality,

∥ u ∥ p := ∥ ∇ ⁡ u ∥ L p ⁢ ( Ω )   for all ⁢ u ∈ W 0 1 , p ⁢ ( Ω ) .

Let u,v:Ω→ℝ and let t∈ℝ. The symbol u≤v means u⁢(x)≤v⁢(x) for almost every x∈Ω, t±:=max⁡{±t,0}, as well as u±⁢(⋅):=u⁢(⋅)±. It is known that u±∈W01,p⁢(Ω) provided u∈W01,p⁢(Ω). Next, define

C 0 1 ⁢ ( Ω ¯ ) := { u ∈ C 1 ⁢ ( Ω ¯ ) : u | ∂ ⁡ Ω = 0 } .

With the standard norm of C1⁢(Ω¯), this set is an ordered Banach space whose positive cone

C + := { u ∈ C 0 1 ⁢ ( Ω ¯ ) : u ⁢ ( x ) ≥ 0 ⁢ in ⁢ Ω ¯ }

has nonempty interior given by

int ⁡ ( C + ) = { u ∈ C + : u ⁢ ( x ) > 0 ⁢ for all ⁢ x ∈ Ω , ∂ ⁡ u ∂ ⁡ n ⁢ ( x ) < 0 ⁢ for all ⁢ x ∈ ∂ ⁡ Ω } ,

where n⁢(⋅) denotes the outward unit normal on ∂⁡Ω; see [10, Remark 6.2.10]. If

p ≤ r < p * := { N ⁢ p N - p for p < N , + ∞ otherwise,

then, due to the continuous embedding W01,p⁢(Ω)⊆Lr⁢(Ω) and the Poincaré inequality, one has

Let W-1,p′⁢(Ω) be the dual space of W01,p⁢(Ω) and let Ap:W01,p⁢(Ω)→W-1,p′⁢(Ω) be the nonlinear operator stemming from the negative p-Laplacian, i.e.,

〈 A p ⁢ ( u ) , v 〉 := ∫ Ω ∥ ∇ ⁡ u ⁢ ( x ) ∥ p - 2 ⁢ ∇ ⁡ u ⁢ ( x ) ⋅ ∇ ⁡ v ⁢ ( x ) ⁢ 𝑑 x   for all ⁢ u , v ∈ W 0 1 , p ⁢ ( Ω ) .

Denote by λ1,p (respectively, λ2,p) the first (respectively, second) eigenvalue of the operator -Δp in W01,p⁢(Ω). The following properties of λ1,p, λ2,p, and Ap can be found in [7, 15]; see also [10, Section 6.2].

1. 0<λ1,p<λ2,p.

2. ∥u∥Lp⁢(Ω)p≤1λ1,p⁢∥u∥pp for all u∈W01,p⁢(Ω).

3. There is a unique eigenfunction u1,p corresponding to λ1,p such that

   u 1 , p ∈ int ⁡ ( C 0 1 ⁢ ( Ω ¯ ) + ) , ∥ u 1 , p ∥ L p ⁢ ( Ω ) = 1 .

   Any other eigenfunction is a scalar multiple of u1,p.

4. If U:={u∈W01,p⁢(Ω):∥u∥Lp⁢(Ω)=1} and

   Γ 0 := { γ ∈ C 0 ⁢ ( [ - 1 , 1 ] , U ) : γ ⁢ ( - 1 ) = - u 1 , r = - γ ⁢ ( 1 ) } ,

   then

   λ 2 , p = inf γ ∈ Γ 0 ⁡ max u ∈ γ ⁢ ( [ - 1 , 1 ] ) ⁢ ∥ u ∥ p p .

5. The operator Ap is bounded, continuous, strictly monotone, and of type (S)+.

Now, with p,q,μ, and f as in Section 1, suppose that

for appropriate c>0, put

and consider the C1-functional φ:W01,p⁢(Ω)→ℝ given by

φ ⁢ ( u ) := 1 p ⁢ ∥ u ∥ p p + μ q ⁢ ∥ u ∥ q q - ∫ Ω F ⁢ ( x , u ⁢ ( x ) ) ⁢ 𝑑 x   for all ⁢ u ∈ W 0 1 , p ⁢ ( Ω ) .

The next result establishes a relation between local C01⁢(Ω¯)-minimizers and local W01,p⁢(Ω)-minimizers of φ. Its proof is the same as that of [1, Proposition 2], with the (p,q)-Laplacian instead of the differential operator considered therein. This idea goes back to the pioneering works of Brézis and Nirenberg [4] for p=2 and García Azorero, Manfredi, and Peral Alonso [9] when p≠2.

Finally, we shall write Nf⁢(u)⁢(⋅):=f⁢(⋅,u⁢(⋅)) for every u∈Lp⁢(Ω). The function

N f : L p ⁢ ( Ω ) → L p ′ ⁢ ( Ω )

is often called the Nemytskii operator associated with f. Moreover, given u:Ω→ℝ and c∈ℝ,

Ω ( u ≤ c ) := { x ∈ Ω : u ( x ) ≤ c } .

The meaning of Ω(u>c) etc. is analogous.

3 Resonance from the Left

To avoid unnecessary technicalities, ‘for every x∈Ω’ will take the place of ‘for almost every x∈Ω’ and the variable x will be omitted when no confusion may arise. Moreover, p=q if and only if μ=0 and f⁢(x,0)≡0. We will posit the following assumptions, where F is given by (2.3).

1. For appropriate c>0, one has

   | f ⁢ ( x , t ) | ≤ c ⁢ ( 1 + | t | p - 1 )   for all ⁢ ( x , t ) ∈ Ω × ℝ .

2. lim|t|→+∞⁡p⁢F⁢(x,t)|t|p=λ1,p uniformly in x∈Ω.

3. lim|t|→+∞⁡[f⁢(x,t)⁢t-p⁢F⁢(x,t)]=+∞ uniformly in x∈Ω.

4. There exist θ∈(1,q) and δ0, c0>0 such that

   c 0 ⁢ | t | θ ≤ f ⁢ ( x , t ) ⁢ t ≤ θ ⁢ F ⁢ ( x , t ) , ( x , t ) ∈ Ω × [ - δ 0 , δ 0 ] .

The energy functional φ:W01,p⁢(Ω)→ℝ stemming from (1.1) is defined by

φ ⁢ ( u ) := 1 p ⁢ ∥ u ∥ p p + μ q ⁢ ∥ u ∥ q q - ∫ Ω F ⁢ ( x , u ⁢ ( x ) ) ⁢ 𝑑 x   for all ⁢ u ∈ W 0 1 , p ⁢ ( Ω ) .

Clearly, φ∈C1⁢(W01,p⁢(Ω)). Moreover, once

f + ⁢ ( x , t ) := f ⁢ ( x , t + ) , f - ⁢ ( x , t ) := f ⁢ ( x , - t - ) , F ± ⁢ ( x , t ) := ∫ 0 t f ± ⁢ ( x , ξ ) ⁢ 𝑑 ξ ,

one has F+⁢(x,t)=F⁢(x,t+), F-⁢(x,t)=F⁢(x,-t-), while the associated truncated functionals

φ ± ⁢ ( u ) := 1 p ⁢ ∥ u ∥ p p + μ q ⁢ ∥ u ∥ q q - ∫ Ω F ± ⁢ ( x , u ⁢ ( x ) ) ⁢ 𝑑 x , u ∈ W 0 1 , p ⁢ ( Ω ) ,

turn out to be C1 as well.

To establish the existence of a third nodal solution, we will first show that there exist two extremal constant-sign solutions, i.e., a smallest positive one and a biggest negative one. In fact, through (h1) and (h4) one has

where c1>0. Thus, it is quite natural to compare solutions of (1.1) with those of the auxiliary problem

which, by [20, Lemma 2.2], possesses a unique positive solution u¯∈int⁡(C+) and a unique negative solution v¯=-u¯. Reasoning as in the proof of [20, Lemma 2.2] yields the next result.

From now on, Σ will denote the set of all solutions to (1.1), while

Σ + := Σ ∩ int ⁡ ( C + ) , Σ - := Σ ∩ ( - int ⁡ ( C + ) ) .

Proceeding exactly as in the proof of [20, Lemma 4.2], one obtains the next result.