On the existence of a weak solution for some singular p ⁢ ( x ) -biharmonic equation with Navier boundary conditions

1 Introduction

Let Ω be a smooth bounded domain in ℝ N ( N ≥ 3 ) with C 2 boundary condition. In this paper, we are dealing with the following problem:

where λ is a positive parameter, γ ⁢ ( x ) ∈ C ⁢ ( Ω ¯ ) satisfying 0 < γ - = inf x ∈ Ω ⁡ γ ⁢ ( x ) ≤ γ + = sup x ∈ Ω ⁡ γ ⁢ ( x ) < 1 , p ∈ C ⁢ ( Ω ¯ ) with 1 < p - := inf x ∈ Ω ⁡ p ⁢ ( x ) ≤ p + := sup x ∈ Ω ⁡ p ⁢ ( x ) < N 2 , as usual, p * ⁢ ( x ) = N ⁢ p ⁢ ( x ) N - 2 ⁢ p ⁢ ( x ) , and

and is almost everywhere positive in Ω. In the sequel, X will denote the Sobolev space W 2 , p ⁢ ( x ) ⁢ ( Ω ) ∩ W 0 1 , p ⁢ ( x ) ⁢ ( Ω ) . Associated to problem (P-+λ), we have the singular functional I ∓ λ : X → ℝ given by

where

and

where F ⁢ ( x , t ) = ∫ 0 t f ⁢ ( x , s ) ⁢ d s .

The operator Δ p ⁢ ( x ) 2 ⁢ u := Δ ⁢ ( | Δ ⁢ u | p ⁢ ( x ) - 2 ⁢ Δ ⁢ u ) is called the p ⁢ ( x ) -biharmonic operator of fourth order, where p is a continuous non-constant function. This differential operator is a natural generalization of the p-biharmonic operator Δ p 2 ⁢ u := Δ ⁢ ( | Δ ⁢ u | p - 2 ⁢ Δ ⁢ u ) , where p > 1 is a real constant. However, the p ⁢ ( x ) -biharmonic operator possesses more complicated non-linearity than the p-biharmonic operator, due to the fact that Δ p ⁢ ( x ) 2 is not homogeneous. This fact implies some difficulties; for example, we can not use the Lagrange multiplier theorem in many problems involving this operator.

The study of this kind of operators occurs in interesting areas such as electrorheological fluids (see [19]), elastic mechanics (see [25]), stationary thermo-rheological viscous flows of non-Newtonian fluids, image processing (see [6]) and the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium (see [1]).

Problem (P-+λ) is a new variant of p ⁢ ( x ) -biharmonic equations due to the singular term and the indefinite one. Note that results for p ⁢ ( x ) -Laplace equations with singular non-linearity are rare. Meanwhile, elliptic and singular elliptic equations involving the p ⁢ ( x ) -Laplace and the p ⁢ ( x ) -biharmonic operators can be found in [1, 2, 5, 8, 9, 10, 13, 14, 17, 18, 21, 20, 22, 23].

Recently, Ayoujil and Amrouss [4] studied the following problem:

In the case when max x ∈ Ω ⁡ q ⁢ ( x ) < min x ∈ Ω ⁡ p ⁢ ( x ) , they proved that the energy functional associated to problem (P) has a nontrivial minimum for any positive λ; see [4, Theorem 3.1]. In the case when min x ∈ Ω ⁡ q ⁢ ( x ) < min x ∈ Ω ⁡ p ⁢ ( x ) and q ⁢ ( x ) has a subcritical growth, they used Ekeland’s variational principle in order to prove the existence of a continuous family of eigenvalues which lies in a neighborhood of the origin. Finally, when

they showed (see [4, Theorem 3.8]) that for every Λ > 0 the energy functional Φ λ corresponding to (P) has a Mountain Pass-type critical point which is nontrivial and nonnegative, and hence Λ = ( 0 , + ∞ ) , where Λ is the set of the eigenvalues. The same problem, for p ⁢ ( x ) = q ⁢ ( x ) , is studied by Ayoujil and Amrouss in [3]. They established the existence of infinitely many eigenvalues for problem (P) by using an argument based on the Ljusternik–Schnirelmann critical point theory. They showed that sup ⁡ Λ = + ∞ , and they pointed out that only under special conditions, which are somehow connected with a kind of monotony of the function p ⁢ ( x ) , one has inf ⁡ Λ > 0 (this is in contrast with the case when p ⁢ ( x ) is a constant, where one always has inf ⁡ Λ > 0 ).

Later, Ge, Zhou and Wu [10] considered the following problem:

where V is an indefinite weight and λ is a positive real number. They considered different situations concerning the growth rates, and they proved, using the mountain pass lemma and Ekeland’s principle, the existence of a continuous family of eigenvalues. A recent paper concerning this type of problems is [12].

Inspired by the above-mentioned papers, we study problem (P-+λ), which contains a singular term and indefinite many more general terms than the one studied in [10]. In this new situation, we will show the existence of a weak solution for problem (P-+λ). The paper is organized as follows: In Section 2, we recall some definitions concerning variable exponent Lebesgue spaces, L p ⁢ ( x ) ⁢ ( Ω ) , as well as Sobolev spaces, W k , p ⁢ ( x ) ⁢ ( Ω ) . Moreover, some properties of these spaces will also be exhibited to be used later. Our main results are stated in Section 3. The proofs of our results will be presented in Section 4 and Section 5.

2 Notations and preliminaries

To study p ⁢ ( x ) -biharmonic problems, we need some results on the spaces L p ⁢ ( x ) ⁢ ( Ω ) , W 1 , p ⁢ ( x ) ⁢ ( Ω ) and W k , p ⁢ ( x ) ⁢ ( Ω ) (for details, see [18, 9]) and some properties of the p ⁢ ( x ) -biharmonic operator, which will be needed later. Set

Let p be a Lipschitz continuous function on Ω ¯ . We set 1 < p - := min x ∈ Ω ¯ ⁡ p ⁢ ( x ) ≤ p + = max x ∈ Ω ¯ ⁡ p ⁢ ( x ) < ∞ and

We recall the following so-called Luxemburg norm on this space defined by the formula

Clearly, when p ⁢ ( x ) = p , a positive constant, the space L p ⁢ ( x ) ⁢ ( Ω ) reduces to the classical Lebesgue space L p ⁢ ( Ω ) , and the norm | u | p ⁢ ( x ) reduces to the standard norm

For any positive integer k, as in the constant exponent case, let

where α = ( α 1 , … , α N ) is a multi-index, | α | = ∑ i = 1 N α i and

Then W k , p ⁢ ( x ) ⁢ ( Ω ) is a separable and reflexive Banach space equipped with the norm

Furthermore, W 0 k , p ⁢ ( x ) ⁢ ( Ω ) is the closure of C 0 ∞ ⁢ ( Ω ) in W k , p ⁢ ( x ) ⁢ ( Ω ) . Let L p ′ ⁢ ( x ) ⁢ ( Ω ) be the conjugate space of L p ⁢ ( x ) ⁢ ( Ω ) with 1 p + 1 p ′ = 1 . Then the Hölder-type inequality

holds.

The modular on the space L p ⁢ ( x ) ⁢ ( Ω ) is the map ρ p ⁢ ( x ) : L p ⁢ ( x ) ⁢ ( Ω ) → ℝ defined by

and it satisfies the following propositions.

For more details concerning the modular, see [9, 16].

In order to discuss problems (P-+λ), we need some theories on the space X := W 0 1 , p ⁢ ( x ) ⁢ ( Ω ) ∩ W 2 , p ⁢ ( x ) ⁢ ( Ω ) . From Definition 2.3 we know that for any u ∈ X ,

and thus

Zang and Fu [24], proved the equivalence of the norms, and they even proved that the norm | Δ ⁢ u | p ⁢ ( x ) is equivalent to the norm ∥ u ∥ (see [24, Theorem 4.4]). Let us choose on X the norm defined by ∥ u ∥ = | Δ ⁢ u | p ⁢ ( x ) . Note that ( X , ∥ ⋅ ∥ ) is also a separable and reflexive Banach space and that the modular is defined as ρ p ⁢ ( x ) : X → ℝ by ρ p ⁢ ( x ) ⁢ ( Δ ⁢ u ) = ∫ Ω | Δ ⁢ u | ⁢ d x and satisfies the same properties as in Proposition 2.1. Hereafter, let

Throughout this paper, the letters k , c , C , C i , i = 1 , 2 , …  , denote positive constants which may change from line to line.

3 Hypotheses and main results

Let us impose the following hypotheses on the non-linearity f : Ω ¯ × ℝ → ℝ :

1. f is a C 1 function such that f ⁢ ( x , 0 ) = 0 .

2. There exists Ω 1 ⋐ Ω with | Ω 1 | > 0 , and a nonnegative function h 1 on Ω 1 such that h 1 ∈ L s 1 ⁢ ( x ) ⁢ ( Ω ) with

   lim | t | → 0 ⁡ f ⁢ ( x , t ) h 1 ⁢ ( x ) ⁢ | t | r 1 ⁢ ( x ) - 1 = 0   for  ⁢ x ∈ Ω ⁢  uniformly .

3. There exists a positive function h on Ω such that h ∈ L s ⁢ ( x ) ⁢ ( Ω ) and

   lim | t | → + ∞ ⁡ f ⁢ ( x , t ) h ⁢ ( x ) ⁢ | t | r ⁢ ( x ) - 1 = 0   for  ⁢ x ∈ Ω ⁢  uniformly ,

   where s , s 1 , r and r 1 ∈ C ⁢ ( Ω ¯ ) are such that 1 < max ⁡ { r ⁢ ( x ) , r 1 ⁢ ( x ) } < p ⁢ ( x ) < N 2 < min ⁡ ( s ⁢ ( x ) , s 1 ⁢ ( x ) ) for all x ∈ Ω .

4. There exists A > 0 such that

   ∫ Ω F ⁢ ( x , t ) ⁢ d x > 0   for all  ⁢ t > A .

5. f ⁢ ( x , t ) ≤ C ⁢ h ⁢ ( x ) ⁢ | t | r ⁢ ( x ) - 2 ⁢ t for all t ∈ ℝ and all x ∈ Ω ¯ , where C is a positive constant, h ∈ L s ⁢ ( x ) ⁢ ( Ω ) and s , r ∈ C ⁢ ( Ω ¯ ) are such that for all x ∈ Ω ¯ we have 1 < r ⁢ ( x ) < p ⁢ ( x ) < N 2 < s ⁢ ( x ) .

6. There exists Ω 1 ⋐ Ω with | Ω 1 | > 0 such that f ⁢ ( x , t ) , h ⁢ ( x ) > 0 in Ω 1 .

Some remarks regarding the hypotheses are in order.

Moreover, under conditions (f5) and (f6), we remark the following.

Here we state our main results asserted in the following two theorems.

4 Proof of Theorem 3.3

The study of the existence of solutions to problem (P - λ ) is done by looking for critical points to the functional I - λ : X → ℝ defined by

in the Sobolev space X. The proof of the Theorem 3.3 is organized in several lemmas. Firstly, under Remark 3.1 one has

and

Now, we are in a position to show that I - λ possesses a nontrivial global minimum point in X.

In the sequel, we put m λ = inf u λ ∈ X ⁡ I - λ ⁢ ( u λ ) . Then we have the following lemma.