Existence results for a Neumann problem involving the p(x) -Laplacian with discontinuous nonlinearities

Linear Algebra its Applications 466312–325 (2015) 102–116 Nonlinear Analysis: Real Worldand Applications 27 (2016)

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Linear Algebra and its Applications Nonlinear Analysis: Real World Applications www.elsevier.com/locate/laa www.elsevier.com/locate/nonrwa

Inverse Jacobi matrix Existence results for aeigenvalue Neumann problem problem of involving the data p(x)-Laplacianwith withmixed discontinuous nonlinearities a Wei 1 Ying Giuseppina Barletta , Antonia Chinn`ıb,∗ , Donal O’Regan c of di Mathematics, Nanjing University of Aeronautics and Astronautics, Universit` a degli StudiDepartment Mediterranea Reggio Calabria, DICEAM—Dipartimento di Ingegneria Civile e Nanjing 210016, China Ambientale, Via Graziella, Localit` a Feo PR di Vito, 89100 Reggio Calabria, Italy b Department of Civil, Information Technology, Construction, Environmental Engineering and Applied Mathematics, University of Messina, 98166 Messina, Italy c School of Mathematics, National a Statistics r t i c land e Applied i n fMathematics, o a b s University t r a c tof Ireland, Galway, Ireland a

article

in

Article history: Received 16 January 2014 20 September a b2014 stra fAccepted o Available online 22 October 2014 Submitted by Y. Wei

Article history: Received 13 May 2015 MSC: Accepted 1 August 201515A18 Available online 28 August 2015 15A57

Keywords: Keywords: p(x)-Laplacian Jacobi matrix Variable exponent Sobolev spaces Eigenvalue Critical points of non-smooth Inverse problem functions Submatrix

In this paper, the inverse eigenvalue problem of reconstructing a Jacobi matrix from its eigenvalues, its leading principal submatrix and part of the eigenvalues of its submatrix ct is considered. The necessary and sufficient conditions for the existence and uniqueness solution areNeumann derived. In this paper the existence of a nontrivial solutionoftothe a parametric a numerical algorithmthe and some numerical problem for a class of Furthermore, nonlinear elliptic equations involving p(x)-Laplacian and examples given. a discontinuous nonlinear term are is established. Under a suitable condition on the © 2014 ]0, Published bythat, Elsevier behavior of the potential at 0+ , we obtain an interval λ∗ ], such for Inc. any λ ∈ ]0, λ∗ ] our problem admits at least one nontrivial weak solution. The solution is obtained as a critical point of a locally Lipschitz functional. In addition to providing a new conclusion on the existence of a solution even for λ = λ∗ , our theorem also includes other results in the literature for regular problems. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction In this paper we are interested in the existence of non-trivial weak solutions of the Neumann problem  −∆p(x) u + a(x)|u|p(x)−2 u = λf (x, u) in Ω (Nλ )  ∂u = 0 on ∂Ω ∂ν N ¯ ), ∆p(x) u := div where Ω ⊂ R is an open bounded domain with smooth boundary ∂Ω , p ∈ C(Ω (|∇u|p(x)−2 ∇u) denotes the address: p(x)-Laplace operator, a is a function in L∞ (Ω ), λ is a positive parameter E-mail [email protected]. 1 Tel.: +86 13914485239. and ν is the outward unit normal to ∂Ω . The p(x)-Laplacian operator arises in generalized Lebesgue and Sobolev spaces (see for example [1–5]). http://dx.doi.org/10.1016/j.laa.2014.09.031 0024-3795/© 2014 problems Published by Elsevier Inc. Contributions to nonlinear elliptic associated with the p(x)-Laplacian from various view points can ∗ Corresponding author. Tel.: +39 090 3977324. E-mail addresses: [email protected] (G. Barletta), [email protected] (A. Chinn`ı), [email protected] (D. O’Regan).

http://dx.doi.org/10.1016/j.nonrwa.2015.08.002 1468-1218/© 2015 Elsevier Ltd. All rights reserved.

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be found in a large number of papers in the literature. The main result of this paper, Theorem 3.1, establishes the existence of one non-trivial weak solution for problem (Nλ ) when the nonlinear term f = f (x, t) is almost everywhere continuous with respect the variable t. We note that equations involving discontinuous nonlinear terms arise frequently in free boundary problems and obstacle problems. Chang in [6] extended variational methods to a class of non-differentiable functionals and later S.A. Marano and D. Motreanu in [7,8] obtained a non-smooth version of B. Ricceri’s three critical-points theorem (see [9,10]). To study problem (Nλ ) via a variational approach, we use an abstract nonsmooth critical point result established in [11] in which a recent critical point result of Bonanno (see [12]) has been extended to the nonsmooth framework. ¯ ) verifies Hereafter we suppose that p ∈ C(Ω 1 < p− := inf p(x) ≤ p(x) ≤ p+ := sup p(x) < +∞, x∈Ω

(1)

x∈Ω

while for f : Ω × R → R we require that (f1 ) f is measurable with respect to each variable separately; ¯ ) with 1 < q(x) < p∗ (x) for each x ∈ Ω ¯ , such that (f2 ) there exist a1 , a2 ∈ [0, +∞[ and q ∈ C(Ω |f (x, t)| ≤ a1 + a2 |t|q(x)−1

for each (x, t) ∈ Ω × R, where    N p(x) ∗ p (x) := N − p(x)  ∞

if p(x) < N

(2)

if p(x) ≥ N.

Our result, Theorem 3.1, extends Theorem 3.1 of [13] where the existence of one non trivial weak solution for problem (Nλ ) was established under continuity assumptions on the nonlinear term f . We note as in [13], that we remove the assumption p− > N which appears in most papers in the literature involving the p(x)-Laplacian (see for example [14–23,23–27]). In the growth condition (f2 ) there is no relation between ¯ . Finally, we observe that Theorem 3.1 provides functions q and p except that q(x) < p∗ (x) for each x ∈ Ω a precise estimate of the interval of parameters λ where a trivial weak solution exists. We now present a simple consequence of the main result. Theorem 1.1. Let f : R → R be a locally essentially bounded and almost everywhere continuous function satisfying (f˜2 ) |f (t)| ≤ a1 + a2 |t|q−1 − for each t ∈ R, for some a1 , a2 ∈ [0, +∞[, 1 < q < (p∗ ) and max{a1 , a2 } < a1 k1 (p+ )

1 p−

q p−

+ aq2 (kq ) (p+ ) ; t f (ξ) dξ ˜ (f4 ) lim supt→0+ 0 tp− = +∞. q

Moreover, put Df := {z ∈ R : f (·) is discontinuous at z}, and we suppose that (f˜3 ) for each z ∈ Df f − (z) := lim+ ess inf δ→0

|ξ−z|<δ f (ξ)

> [|z|]

p−2

:= max{|z|p

−

−2

, |z|p

+

−2

}.

1 ¯ k

where k¯ :=

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Then the problem  −∆p(x) u + |u|p(x)−2 u = f (u)  ∂u = 0 on ∂Ω ∂ν admits at least one non trivial weak solution.

in Ω ˜λ ) (N

The paper is arranged as follows. In Section 2 we present some auxiliary results that we use in Section 3 to prove our main theorem. In Section 4 we give some examples of functions satisfying the assumptions in our main result. 2. Preliminaries ¯ ) satisfies condition (1). The variable exponent Lebesgue Here and in the sequel, we suppose that p ∈ C(Ω p(x) space L (Ω ) is defined as    Lp(x) (Ω ) = u : Ω → R : u is measurable and ρp (u) := |u(x)|p(x) dx < +∞ . Ω p(x)

On L

(Ω ) we consider the norm      u(x) p(x)   dx ≤ 1 . λ>0:  λ  Ω

 ∥u∥Lp(x) (Ω) := inf

The generalized Lebesgue–Sobolev space W 1,p(x) (Ω ) is defined as   W 1,p(x) (Ω ) := u ∈ Lp(x) (Ω ) : |∇u| ∈ Lp(x) (Ω ) with the norm ∥u∥W 1,p(x) (Ω) := ∥u∥Lp(x) (Ω) + ∥|∇u|∥Lp(x) (Ω) .

(3)

With such norms, Lp(x) (Ω ) and W 1,p(x) (Ω ) are separable, reflexive and uniformly convex Banach spaces. If we assume that (H1 ) a ∈ L∞ (Ω ), with a− := ess inf Ω a(x) > 0, then on W 1,p(x) (Ω ) it is possible to consider the following norm:          ∇u(x) p(x)  u(x) p(x)     ∥u∥a = inf σ > 0 : + a(x) dx ≤ 1 ,  σ   σ  Ω which is equivalent to that introduced in (3). In particular (see [13]) [a− ] 1

1

p

1 + [a− ] 1

∥u∥W 1,p(x) (Ω) ≤ ∥u∥a ≤ (1 + ∥a∥∞ ) p− ∥u∥W 1,p(x) (Ω)

p

¯ ) with 1 < h− , we put for each u ∈ W 1,p(x) (Ω ), where, for α > 0 and h ∈ C(Ω h

−

+

[α] := max{αh , αh } and −

+

[α]h := min{αh , αh }. The following result generalizes the well-known Sobolev embedding theorem.

(4)

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315

¯ ) with p(x) > 1 for each x ∈ Ω ¯ . If r ∈ C(Ω ¯) Theorem 2.1 ([20, Proposition 2.5]). Assume that p ∈ C(Ω ∗ 1,p(x) and 1 < r(x) < p (x) for all x ∈ Ω , then there exists a continuous and compact embedding W (Ω ) ↩→ r(x) ∗ L (Ω ) where p is the critical exponent related to p defined in (2). In the sequel, we will denote by kr the best constant for which one has ∥u∥Lr(x) (Ω) ≤ kr ∥u∥a

for all u ∈ W 1,p(x) (Ω ).

We define the functional Φ : W 1,p(x) (Ω ) → R as    1 p(x) p(x) Φ(u) = |∇u| + a(x)|u| dx Ω p(x)

for all u ∈ W 1,p(x) (Ω ).

(5)

(6)

Proposition 2.1 ([16, Proposition 2.2]). Let u ∈ W 1,p(x) (Ω ). (j) If ∥u∥a < 1 then (jj) If ∥u∥a > 1 then

1 p+ p+ ∥u∥a 1 p− p+ ∥u∥a

≤ Φ(u) ≤ ≤ Φ(u) ≤

1 p− p− ∥u∥a . 1 p+ p− ∥u∥a .

Standard arguments (see for example [24,19]) imply that Φ is sequentially weakly lower semi-continuous and that it is a C 1 functional on W 1,p(x) (Ω ), with derivative    ′ ⟨Φ (u), v⟩ = |∇u|p(x)−2 ∇u∇v + a(x)|u|p(x)−2 uv dx, (7) Ω 1,p(x)

for any u, v ∈ W (Ω ). Moreover (see [16, Lemma 3.1]), Φ ′ is an homeomorphism. Given any Banach space X, by X ∗ we denote the topological dual of X, and by ⟨·, ·⟩ the duality brackets for the pair (X ∗ , X). Let (X, ∥ · ∥) be a real Banach space. A functional h : X → R is said to be locally Lipschitz when, to each x ∈ X, there corresponds a neighborhood U of x and a constant L ≥ 0, depending on x, such that |h(y) − h(z)| ≤ L∥y − z∥ for each y, z ∈ U . For a locally Lipschitz function h : X → R, the generalized directional derivative of h at point x ∈ X, in the direction v ∈ X, is defined as h0 (x; v) := lim sup

u→x;t→0+

h(u + tv) − h(u) , t

while the generalized gradient of h in x is the set ∂h(x) := {w∗ ∈ X ∗ : ⟨w∗ , v⟩ ≤ h0 (x; v) ∀v ∈ X}. Basic properties of generalized directional derivative and generalized gradient can be found in [28] and in [6]. If h : X → R is a locally Lipschitz functional and x ∈ X, then we say that x is a critical point of h if it satisfies the inequality h0 (x; y) ≥ 0 for all y ∈ X or, equivalently, 0 ∈ ∂h(x). Definition 2.1. If h : X → R is a locally Lipschitz functional that can be written as h = Φ − Υ , with Φ and Υ locally Lipschitz, and r ∈ R, we say that h verifies the Palais–Smale condition cut off upper at r ((P S)[r] -condition for short) if any sequence {un }n∈N in X such that

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(α) {h(un )}n∈N is bounded; (β) there exists a sequence {ϵn }n∈N in ]0, +∞[, ϵn → 0+ such that h0 (un ; v) ≥ −ϵn ∥v∥

for all v ∈ X

(γ) Φ(un ) < r for all n ∈ N; has a convergent subsequence. The energy functional associated to (Nλ ) can be written as Iλ = Φ − λΨ where Φ is that in (6) and Ψ : W

1,p(x)

(Ω ) → R is defined as

 Ψ (u) =

F (x, u(x))dx

for all u ∈ W 1,p(x) (Ω ),

(8)

Ω

ξ with F (x, ξ) := 0 f (x, t) dt for each (x, ξ) ∈ Ω × R. If f satisfies the growth condition (f2 ) then Proposition 2.2 of [29] ensures that the functional Ψ is locally Lipschitz on W 1,p(x) (Ω ) and so Iλ is locally Lipschitz rather than smooth. We recall that, for fixed λ > 0, a point u ∈ W 1,p(x) (Ω ) is a weak solution to (Nλ ) if   p(x)−2 p(x)−2 (|∇u| ∇u∇v + a(x)|u| uv)dx = λ f (x, u(x))v(x)dx (9) Ω

Ω

1,p(x)

holds for each v ∈ W (Ω ). In Section 3, Lemma 3.1, we will show that any critical point of Iλ satisfies (9) and is hence a solution to (Nλ ). From this lemma, it thus suffices to show that Iλ has a non trivial critical point and to do this we shall use the next result. Theorem 2.2 ([11, Theorem 2.5]). Let X be a real Banach space, Φ, Ψ : X → R be two locally Lipschitz continuous functionals such that inf x∈X Φ(x) = Φ(0) = Ψ (0) = 0. Assume that there exist r > 0 and x ¯ ∈ X, with 0 < Φ(¯ x) < r, such that: (a1 ) (a2 )

supΦ(x)≤r Ψ (x) < Ψ (¯x) , r  Φ(¯x)  Φ(¯ x) r for each λ ∈ Ψ , (¯ x) supΦ(x)≤r Ψ (x)

Then, for each λ ∈ Λr :=



the functional Iλ := Φ − λΨ satisfies (P S)[r] condition.

Φ(¯ x) r Ψ (¯ x) , supΦ(x)≤r Ψ (x) −1

that Iλ (x0,λ ) ≤ Iλ (x) for all x ∈ Φ

 , there exists a critical point of Iλ , x0,λ ∈ Φ −1 (]0, r[) such

(]0, r[).

We conclude this section by specifying the type of discontinuity allowed on the nonlinear term f in (Nλ ). We denote by G the family of highly discontinuous functions, that is the family of all locally bounded functions g : Ω × R → R satisfying the following conditions: (m1 ) g(·, z) is measurable for every z ∈ R; (m2 ) there exists a set Ω0 ⊆ Ω with m(Ω0 ) = 0 such that the set Dg :=

 x∈Ω\Ω0

has measure zero;

{z ∈ R : g(x, ·) is discontinuous at z}

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317

(m3 ) the functions g − (x, z) := lim+ ess inf δ→0

|ξ−z|<δ g(x, ξ),

g + (x, z) := lim+ ess sup δ→0

|ξ−z|<δ g(x, ξ)

are superpositionally measurable i.e. g − (·, u(·)) and g + (·, u(·)) are measurable provided u : Ω → R is measurable too. Clearly, if f ∈ G then f satisfies (f1 ). 3. Main result A first result obtained from Theorem 2.2 is the existence of one nontrivial solution for problem (Nλ ) when the function f is discontinuous with respect to the second variable. Theorem 3.1. Let f ∈ G, satisfy (f2 ), (f3 ) for each λ > 0, for a.e. x ∈ Ω and each z ∈ Df the condition λf − (x, z) ≤ a(x)|z|p(x)−2 z ≤ λf + (x, z) implies λf (x, z) = a(x)|z|p(x)−2 z, (f4 )  lim sup t→0+

Then, with λ∗ =

1

1

a1 k1 (p+ ) p− +

a2 q−

q+

Ω

F (x, t) dx = +∞. t p−

, where k1 and kq are given by (5), for each λ ∈]0, λ∗ ] prob-

[kq ]q (p+ ) p−

lem (Nλ ) admits at least one non trivial weak solution. The proof of Theorem 3.1 needs two preliminary lemmas. Lemma 3.1. If f ∈ G, satisfies (f2 ) and (f3 ) then for each λ > 0, the critical points of the functional Iλ are weak solutions for problem (Nλ ). Proof. Put X := W 1,p(x) (Ω ). Fixed λ > 0, if u0 ∈ X is a critical point of Φ − λΨ , then one has (Φ − λΨ )0 (u0 ; v) ≥ 0

for all v ∈ X.

(10)

Since Φ ∈ C 1 (X), we have, in particular 0 ≤ (Φ − λΨ )0 (u0 ; v) ≤ Φ 0 (u0 ; v) + (−λΨ )0 (u0 ; v) = ⟨Φ ′ (u0 ), v⟩ + (−λΨ )0 (u0 ; v), whence −⟨Φ ′ (u0 ), v⟩ ≤ (−λΨ )0 (u0 ; v)

for all v ∈ X.

This means −Φ ′ (u0 ) ∈ ∂(−λΨ )(u0 ), that is (see [28]) Φ ′ (u0 ) ∈ ∂(λΨ )(u0 ).

(11)

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Now, from Theorem 1.8 of [30], X is embedded and dense in Lq(x) (Ω ) and so, from Theorem 2.2 of [6] one has   ∂(−λΨ )(u0 ) ⊆ ∂ −λΨ|Lq(x) (Ω) (u0 ). (12) From (f2 ) and because −λf ∈ G too, we deduce that f + , f − , −λf + and −λf − satisfy all the assumptions in Theorem 2.1 of [6]. Thus ∂(λΨ )Lq(x) (Ω) (u0 ) ⊆ λ[f − (x, u0 (x)), f + (x, u0 (x))]q′ where (see [6]) q ′ is the conjugate function of q, i.e.

1 q(x)

+

1 q ′ (x)

= 1 and

[f − (·, u0 (·)), f + (·, u0 (·))]q′ = {W ∈ Lq (x) (Ω ) : W (x) ∈ [f − (x, u0 (x)), f + (x, u0 (x))] a.e. in Ω }. ′

Using (11), Φ ′ (u0 ) ∈ λ[f − (·, u0 (·)), f + (·, u0 (·))]q′ and then by (7) we have −∆p(x) u ∈ [−a(·)|u0 (·)|p(x)−2 u0 (·) + λf − (·, u0 (·)), −a(x)|u0 (·)|p(x)−2 u0 (·) + λf + (·, u0 (·))]q′ . Thus, there exists a unique W0 ∈ [−a(·)|u0 (·)|p(x)−2 u0 (·) + λf − (·, u0 (·)), −a(·)|u0 (·)|p(x)−2 u0 (·) + λf + (·, u0 (·))]q′ such that 

|∇u0 |p(x)−2 ∇u0 ∇vdx =

Ω

 W0 vdx

(13)

Ω

for each v ∈ Lq(x) (Ω ). Now, to conclude, we show that W0 (x) = −a(x)|u0 (x)|p(x)−2 u0 (x) + λf (x, u0 (x))

a.e. in Ω .

(14)

Let Ω1 ⊆ Ω be a set of measure zero such that W0 (x) ∈ [−a(x)|u0 (x)|p(x)−2 u0 (x) + λf − (x, u0 (x)), −a(x)|u0 (x)|p(x)−2 u0 (x) + λf + (x, u0 (x))]

(15)

for all x ∈ Ω \ Ω1 . Put Ωf := {x ∈ Ω : u0 (x) ∈ Df } and note that Ωf = u−1 0 (Df ). From (13), u0 is a weak solution of the problem  −∆p(x) u = W0 (x) in Ω (N0 )  ∂u = 0 on ∂Ω . ∂ν Now using Lemma 1 of [31] and arguing as in [29, Proposition 2.4], we obtain W0 (x) = 0 a.e. on Ωf . Let Ω3 ⊆ Ωf be a set of measure zero such that W0 (x) = 0

(16)

for all x ∈ Ωf \ Ω3 . From (f3 ), there exists a set Ω2 ⊆ Ω of measure zero such that for all x ∈ Ω \ Ω2 and each z ∈ Df the condition λf − (x, z) ≤ a(x)|z|p(x)−2 z ≤ λf + (x, z) implies λf (x, z) = a(x)|z|p(x)−2 z. Put Ω ∗ = ∪3i=0 Ωi , and we want to prove that W0 (x) = −a(x)|u0 (x)|p(x)−2 u0 (x) + λf (x, u0 (x))

(17)

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319

for each x ∈ Ω \ Ω ∗ . Clearly m(Ω ∗ ) = 0. Take x ∈ Ω \ Ω ∗ . If u0 (x) ̸∈ Df , taking into account that in particular x ∈ Ω \ Ω0 (see (m2 )), the definition of Df leads to continuity at x of the function f (·, u0 (·)). From (15) one has W0 (x) = −a(x)|u0 (x)|p(x)−2 u0 (x) + λf (x, u0 (x)). If u0 (x) ∈ Df , since x ∈ (Ω \ Ω1 ) ∩ (Ω \ Ω3 ) then 0 = W0 (x) ∈ [−a(x)|u0 (x)|p(x)−2 u0 (x) + λf − (x, u0 (x)), −a(x)|u0 (x)|p(x)−2 u0 (x) + λf + (x, u0 (x))] i.e. λf − (x, u0 (x)) ≤ a(x)|u0 (x)|p(x)−2 u0 (x) ≤ λf + (x, u0 (x)). Also, x ∈ Ω \ Ω2 and u0 (x) ∈ Df , and so from (17) one has λf (x, u0 (x)) = a(x)|u0 (x)|p(x)−2 u0 (x) and so W0 (x) = 0 = −a(x)|u0 (x)|p(x)−2 u0 (x) + λf (x, u0 (x)). We have proved (14) and the conclusion follows from (13) and (14).



Lemma 3.2. Assume that p ∈ C(Ω ) satisfies (1) and f : Ω × R → R is a function satisfying (f1 ) and (f2 ). Then, for each λ > 0, the functional Iλ satisfies the Palais–Smale condition (P S)[r] , for any r > 0. Proof. Fixed λ > 0, let {un }n∈N be a sequence in W 1,p(x) (Ω ), satisfying (α), (β) and (γ) in Definition 2.1. Then {un }n∈N is bounded and we can extract a subsequence (again call it {un }), converging to u ∈ W 1,p(x) (Ω ), weakly in W 1,p(x) (Ω ) and strongly in Lr(x) (Ω ), for any function r as in Theorem 2.1. From [28] we know that (−λΨ )0 is u.s.c. and  0 −λΨ|W 1,p(x) (Ω) (u; v) ≤ (−λΨ )0 (u; v) for all u, v ∈ W 1,p(x) (Ω ), so  0 lim sup −λΨ|W 1,p(x) (Ω) (un ; un − u) ≤ lim sup(−λΨ )0 (un ; un − u) = 0. n→+∞

(18)

n→+∞

If we choose v = un − u in the condition (β) of Definition 2.1 (Iλ0 (un ; v) ≥ −ϵn ∥v∥) and pass to the upper limit, then  0 lim sup⟨Φ ′ (un ), u − un ⟩ ≤ lim sup −λΨ|W 1,p(x) (Ω) (un ; un − u) ≤ 0. n→+∞

Thus (see [19]) un → u in W

n→+∞

1,p(x)

(Ω ).



Proof of Theorem 3.1. Put X := W 1,p(x) (Ω ) equipped by norm ∥ · ∥a , and we consider the functional Iλ (·) := Φ(·) − λΨ (·) introduced in the previous section. As before, Φ and Ψ satisfy the regularity assumptions required in Theorem 2.2. In particular Lemma 3.2 ensures that the condition (a2 ) is satisfied for all r, λ > 0 while Lemma 3.1 ensures that for each λ > 0, the critical points of the functional Iλ are weak solutions for problem (Nλ ). We first show the existence of a nontrivial solution for λ ∈]0, λ∗ [. Fix λ ∈]0, λ∗ [, and choose r = 1 to satisfy condition (a1 ) of Theorem 2.2. From (f4 ) there exists    1−  p p− 0 < ξλ < min 1, (19) ∥a∥∞ |Ω |

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such that p−



F (x, ξλ ) dx Ω p− ξλ ∥a∥∞ |Ω |

>

1 . λ

(20)

We denote by u ¯ the function of X defined by u ¯(x) = ξλ for each x ∈ Ω and we observe that Φ(¯ u) ≤

1 ∥a∥∞ |Ω |[ξλ ]p < 1 p−

(21)

and  F (x, ξλ ) dx.

Ψ (¯ u) = Ω

Condition (f2 ) implies |F (x, t)| ≤ a1 |t| +

a2 |t|q(x) q(x)

for each (x, t) ∈ Ω × R. For u ∈ Φ −1 (] − ∞, 1]) we have   a2 a2 Ψ (u) ≤ a1 |u(x)| dx + − |u(x)|q(x) dx = a1 ∥u∥L1 (Ω) + − ρq (u). q q Ω Ω Theorem 1.3 of [30] and the embeddings W 1,p(x) (Ω ) ↩→ L1 (Ω ) and W 1,p(x) (Ω ) ↩→ Lq(x) (Ω ) ensure that a1 ∥u∥L1 (Ω) +

q a2 a2  a2 q ρq (u) ≤ a1 ∥u∥L1 (Ω) + − ∥u∥Lq(x) (Ω) ≤ a1 k1 ∥u∥a + − [kq ∥u∥a ] . q− q q

(22)

Taking into account that for each u ∈ Φ −1 (] − ∞, 1]), from Proposition 2.1, one has 1

∥u∥a ≤ (p+ ) p− , and thus (20) and (22) lead to  q+ p− Ω F (x, ξλ ) dx a2 1 Ψ (¯ u) 1 q + p− sup Ψ (u) ≤ a1 k1 (p ) + − [kq ] (p ) = ∗ < < < , (23) − p q λ λ Φ(¯ u) Φ(u)≤1 ξλ ∥a∥∞ |Ω |     Φ(¯ u) Φ(¯ u) 1 ∗ and so condition (a1 ) of Theorem 2.2 is satisfied. Since λ ∈ Ψ , λ ⊆ , (¯ u) Ψ (¯ u) supΦ(u)≤1 Ψ (u) , Theorem 2.2 guarantees the existence of a local minimum point uλ for the functional Iλ such that +

1 p−

0 < Φ(uλ ) < 1, and so uλ is a non-trivial weak solution of problem (Nλ ). Now we prove that (Nλ∗ ) has a nontrivial solution. We have a sequence {uλ }, bounded in W 1,p(x) (Ω ) satisfying Iλ (uλ ) ≤ Iλ (v) for all v ∈ Φ −1 (]0, 1[). Thus, taking into account (j) of Proposition 2.1, Iλ (uλ ) ≤ Iλ (vn )

(24)

for any sequence {vn } ⊆ Φ −1 (]0, 1[), such that ∥vn ∥a → 0 . Then, from the continuity of Iλ we have Iλ (uλ ) ≤ 0

for every λ ∈]0, λ∗ [.

(25)

G. Barletta et al. / Nonlinear Analysis: Real World Applications 27 (2016) 312–325

321

¯ ∈]0, λ∗ [ and a sequence {λn }n∈N , such that 0 < λ ¯ < λn < λ∗ for any n ∈ N and λn ↗ λ∗ . Take now λ Then {uλn }n∈N is bounded and we can extract a subsequence (again call it {un } for simplicity), converging to u∗ ∈ W 1,p(x) (Ω ), weakly in W 1,p(x) (Ω ) and strongly in Lr(x) (Ω ), for any function r as in Theorem 2.1. From (9),  ⟨Φ ′ (un ), u∗ − un ⟩ = λn f (x, un (x))(u∗ − un )dx Ω

for all n ∈ N . Thus ′



∗

lim sup⟨Φ (un ), u − un ⟩ = lim sup λn n→+∞

n→+∞

f (x, un (x))(u∗ − un )dx = 0,

(26)

Ω

and (see [19]) un → u∗ in W 1,p(x) (Ω ). Passing to the limit in  ⟨Φ ′ (un ), v⟩ = λn f (x, un (x))v(x)dx Ω

we obtain ′

∗

∗



⟨Φ (u ), v⟩ = λ

f (x, u∗ (x))v(x)dx

for every v ∈ W 1,p(x) (Ω ),

(27)

Ω

and thus u∗ is a solution to (Nλ ), with λ = λ∗ . Now we show that u∗ ̸= 0. We know that every uλ is a minimum point for Iλ and that Iλ (uλ ) ≤ Iλ (v) for all v ∈ Φ −1 (]0, 1[) and all λ ∈]0, λ∗ [. Thus, since uλ ∈ Φ −1 (]0, 1[) for every λ ∈]0, λ∗ [, Iλn (un ) ≤ Iλn (uλ¯ )

and Iλ¯ (uλ¯ ) ≤ Iλ¯ (un )

for every n ∈ N.

(28)

¯ < λn for every n ∈ N we obtain Adding the previous inequalities and considering that λ Ψ (uλ¯ ) ≤ Ψ (un )

for every n ∈ N,

whence Ψ (uλ¯ ) ≤ Ψ (u∗ ). Now, arguing by contradiction, if u∗ = 0 then Ψ (uλ¯ ) ≤ 0, hence Iλ¯ (uλ¯ ) > 0, but this contradicts (25).

(29) 

Remark 3.1. A careful reading of Lemma 3.1 shows that if we remove hypothesis (f3 ) then Theorem 3.1 guarantees the existence of a nontrivial solution to the differential inclusion  −∆p(x) u + a(x)|u|p(x)−2 u ∈ λ[f − (x, u), f + (x, u)] a.e. in Ω ˜λ ) (N  ∂u = 0 on ∂Ω . ∂ν ˜ Any solution to (Nλ ) turns into a solution to (Nλ ) only for those particular values of λ for which condition (f3 ) applies. Corollary 3.1. Let f ∈ G, satisfy f (x, 0) = 0 a.e. in Ω , (f2 ), (f4 ) and (f3′ ) for each λ > 0, for a.e. x ∈ Ω and each z ∈ Df ∩]0, +∞[ the condition λf − (x, z) ≤ a(x)|z|p(x)−2 z ≤ λf + (x, z) implies λf (x, z) = a(x)|z|p(x)−2 z. Then, with λ∗ = a1 k1

1 (p+ ) p−

1 a2 + − q

[kq ]

q

q+ (p+ ) p−

, where k1 and kq are given by (5), for each λ ∈]0, λ∗ ] prob-

lem (Nλ ) admits at least one non trivial non negative weak solution.

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Proof. We truncate f through the function f+ as follows:  0 if t < 0, x ∈ Ω f+ (x, t) = f (x, t) if t ≥ 0 x ∈ Ω . t  In a standard way, define F+ (x, t) = 0 f+ (x, τ )dτ and Ψ+ (u) = Ω F+ (x, u(x))dx. We want to apply Theorem 3.1 to the functional I+,λ (·) = Φ(·) − λΨ+ (·). Clearly I+,λ satisfies all the assumptions of Theorem 3.1. For the sake of completeness we point out only that Df+ = Df ∩]0, +∞[ and so f+ satisfies (f3 ). Theorem 3.1 guarantees the existence of a local minimum point uλ for the functional I+,λ such that 0 < Φ(uλ ) < 1 and ′



⟨Φ (uλ ), v⟩ = λ

f+ (x, uλ (x))v(x)dx

for every v ∈ W 1,p(x) (Ω ).

(30)

Ω

Acting on (30) with v = max {−uλ , 0} and arguing as in [32, Theorem 3.1], we obtain u+,λ ≥ 0 a.e. in Ω . Thus (30) is equivalent to (9) and u+,λ is a non trivial non negative weak solution of (Nλ ).  Proof of Theorem 1.1. We apply Theorem 3.1 by choosing a(x) = 1 for each x ∈ Ω . Clearly f ∈ G and conditions (f˜2 ) and (f˜4 ) establish conditions (f2 ) and (f4 ) respectively. Moreover, due to the choice of a1 and a2 , 1 ∈]0, λ∗ ]. Now, we verify that, for a.e. x ∈ Ω and each z ∈ Df , condition f − (z) ≤ |z|p(x)−2 z ≤ f + (z)

(31)

implies λf (z) = |z|p(x)−2 z. Clearly, if z ∈ Df ∩] − ∞, 0] then (31) does not hold. If z ∈ Df ∩]0, +∞[ then from condition (f˜3 ) we have f − (z) > z p(x)−1 which conflicts with (31) which implies f − (z) ≤ z p(x)−1 . Thus condition (f3 ) of Theorem 3.1 holds for λ = 1. Taking into account Remark 3.1, the conclusion of Theorem 1.1 is obtained.  4. Examples Now, we present some examples of functions verifying the assumptions in Theorem 3.1. We explicitly note that all the functions below fail to satisfy the continuity assumptions required in [13]. Example 4.1. Take a function a ∈ L∞ (Ω ) satisfying a− > 0 and consider the function f : Ω ×R → R defined as   sin(t − hπ) if x ∈ Ω and t ∈]hπ, (h + 1)π] for some h ≤ 0. t − hπ f (x, t) = (32)  g(x, t) if x ∈ Ω and t > π, where g : Ω ×]π, +∞[ is continuous and satisfies the following conditions: ¯ ) with 1 < q(x) < p∗ (x) for each x ∈ Ω ¯ , such that • there exist a1 , a2 ∈ [0, +∞[ and q ∈ C(Ω |g(x, t)| ≤ a1 + a2 |t|q(x)−1 for each (x, t) ∈ Ω ×]π, +∞[; • limt→π+ g(x, t) = l(x) ≤ 0 a.e. in Ω .

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323

We observe that f ∈ G. In fact if we choose Ω0 = ∅ then Df = {kπ : k ≤ 1}, m(Df ) = 0. Further f + (x, kπ) = 1,

f − (x, kπ) = f (x, kπ) = 0,

for k ≤ 0,

(33)

+

−

f (x, π) = l(x),

f (x, π) = f (x, π) = 0.

Now, take λ ∈]0, λ∗ ], where λ∗ is defined as in Theorem 3.1, x ∈ Ω and t ∈ Df . Then, taking into account (33), the inequality (f3 ) reads as 0 ≤ a(x)|kπ|p(x)−2 kπ ≤ λ

for some k ≤ 0,

(34)

for k = 1.

(35)

or λl(x) ≤ a(x)π p(x)−1 ≤ 0

We observe that condition (34) implies k = 0. Indeed, for k < 0 the inequality cannot occur because the function in the middle is negative, while if k = 0 then we have f (x, 0) = 0 and (f3 ) holds. For k = 1 the inequality in (f3 ) turns into (35) and this cannot happen as well. Thus (f3 ) is satisfied. For (x, t) ∈ Ω ×]0, π[ we have  t sin(τ ) dτ. (36) F (x, t) = τ 0 Using De L’Hospital’s rule and the fact that p− > 1, we obtain  t F (x, t)dx |Ω | 0 sinτ τ dτ sin t Ω lim sup = lim sup = lim sup |Ω | = +∞, p p − − t t p− tp− t→0+ t→0+ t→0+

(37)

namely (f4 ). Example 4.2. Assume p− > 2 and consider a function a ∈ L∞ (Ω ) satisfying 1

a− ≥

1

.

(38)

k1 (p+ ) p− Then, for each λ ∈]0,

1

1

k1 (p+ ) p−

], the problem

 −∆p(x) u + a(x)|u|p(x)−2 u = λ(u − [u]) in Ω  ∂u = 0 on ∂Ω ∂ν admits at least one non trivial non negative weak solution. We consider the function f : R → R by f (t) = t − [t]

where [t] is the integer part of t ∈ R.

(39)

(40)

Since f (0) = 0, we apply Corollary 3.1. Clearly f satisfies (f1 ) and (f2 ) with a1 = 1, a2 = 0. Further, limt→k+ f (t) = 0 = f (k) and limt→k− f (t) = 1 for k ∈ Z. We can choose Ω0 = ∅, while Df = Z and |Df | = 0. For t ∈ (0, 1) we have F (t) =

t2 , 2

(41)

and  lim sup t→0+

Ω

F (t)dx = +∞, tp−

namely (f4 ). With our choice of f , (f3′ ) reads as for each λ > 0, and each k > 0 the condition 0 ≤ a(x)|k|p(x)−2 k ≤ λ implies 0 = a(x)|k|p(x)−2 k.

(42)

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We observe that, if it were true, then we have a− ≤ a(x)k p(x)−1 ≤ λ ≤

1

(43)

1

k1 (p+ ) p− and this contradicts (38). The conclusion follows by Corollary 3.1 taking into account that λ∗ =

1

1

k1 (p+ ) p−

.

Acknowledgments The authors have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di a Alta Matematica (INdAM) - projects “Metodi variazionali per problemi differenziali quasi lineari in contesti non standard”, “Equazioni differenziali non lineari e applicazioni”. References [1] O. Kov´ a˘ cik, J. R´ akosn´ık, On spaces Lp(x) and W 1,p(x) , Czechoslovak Math. 41 (1991) 592–618. [2] L. Diening, P. Harjulehto, P. H¨ ast¨ o, M. Ru˘ zi˘ cka, Lebesgue and Sobolev Spaces With Variable Exponents, in: Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, 2011. [3] D.V. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkh¨ auser, 2013. [4] J. Musielak, Orlicz Spaces and Modular Spaces, in: Lecture Notes in Mathematics, vol. 1034, Springer, Berlin, 1983. [5] M. Ru˘ zi˘ cka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000. [6] K.-C.F. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981) 102–129. [7] S.A. Marano, D. Motreanu, On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems, Nonlinear Anal. 48 (2002) 37–52. [8] S.A. Marano, D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumanntype problem involving the p-Laplacian, J. Differential Equations 182 (2002) 108–120. [9] B. Ricceri, On a three critical points theorem, Arch. Math. 75 (2000) 220–226. [10] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410. [11] G. Bonanno, G. D’Agu`ı, P. Winkert, Sturm–Liouville equations involving discontinuous nonlinearities, Minimax Theory Appl. 01 (1) (2015) Copyright Heldermann Verlag 2015. [12] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012) 2992–3007. [13] G. Barletta, A. Chinn`ı, Existence of solutions for a Neumann problem involving the p(x)-Laplacian, Electron. J. Differential Equations 2013 (158) (2013) 1–12. [14] G. Bonanno, A. Chinn`ı, Existence results of infinitely many solutions for p(x)-Laplacian elliptic Dirichlet problems, Complex Var. Elliptic Equ. 57 (11) (2012) 1233–1246. [15] G. Bonanno, A. Chinn`ı, Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent, J. Math. Anal. Appl. 418 (2014) 812–827. [16] F. Cammaroto, A. Chinn`ı, B. Di Bella, Multiple solutions for a Neumann problem involving the p(x)-Laplacian, Nonlinear Anal. 71 (2009) 4486–4492. [17] Chao Ji, Remarks on the existence of three solutions for the p(x)-Laplacian equations, Nonlinear Anal. 74 (2011) 2908–2915. [18] A. Chinn`ı, R. Livrea, Multiple solutions for a Neumann-type differential inclusion problem involving the p(·)-Laplacian, Discrete Contin. Dyn. Syst. Ser. S 5 (4) (2012) 753–764. [19] X.L. Fan, S.G. Deng, Remarks on Ricceri’s variational principle and applications to the p(x)-Laplacian equations, Nonlinear Anal. 67 (2007) 3064–3075. [20] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003) 1843–1852. [21] X.L. Fan, C. Ji, Existence of infinitely many solutions for a Neumann problem involving the p(x)-Laplacian, J. Math. Anal. Appl. 334 (2007) 248–260. [22] D. Liu, Existence of multiple solutions for a p(x)-Laplace equations, Electron. J. Differential Equations 2010 (33) (2010) 1–11. [23] Q. Liu, Existence of three solutions for p(x)-Laplacian equations, Nonlinear Anal. 68 (2008) 2119–2127. [24] M. Mih˘ ailescu, Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, Nonlinear Anal. 67 (2007) 1419–1425. [25] D.S. Moschetto, A quasilinear Neumann problem involving the p(x)-Laplacian, Nonlinear Anal. 71 (2009) 2739–2743. [26] X. Ding, X. Shi, Existence and multiplicity of solutions for a general p(x)-Laplacian Neumann problem, Nonlinear Anal. 70 (2009) 3715–3720. [27] G. Bonanno, A. Chinn`ı, Multiple solutions for elliptic problems involving the p(x)-Laplacian, Matematiche 66 (2011) 105–113. [28] F.H. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976) 165–174.

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