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In this paper, we study the existence of solution for some <i>p</i>(<i>x</i>)-polyharmonic Kirchhoff equations. The latter is allowed to vanish at the origin (degenerate case). Firstly, we study the existence of solutions of approximate equations. Secondly, we prove the existence of the solutions of the original equation. The main tool is the Schauder’s Theorem.

In this paper, we prove the existence of solution of Dirchlet problems involving the p-polyharmonic operators Δ p s . We consider

{ M ( ‖ u ‖ p ) Δ p s u + a ( x ) g ( u ) = f ( x ) in Ω , D α u ( x ) | ∂ Ω = 0 ∀ α , with | α | ≤ s − 1, (1)

where Ω ⊂ ℝ N is a bounded domain, p ≥ 2 , s = 1 , 2 , ⋯ , ‖ ⋅ ‖ is denoted in section 2, and f ( x ) ∈ L 1 ( Ω ) , 0 ≤ a ( x ) ∈ L 1 ( Ω ) . Here, the p-polyharmonic operator is defined by

Δ p s u = { − d i v Δ j − 1 ( | D Δ j − 1 u | p − 2 ) D Δ j − 1 u , s = 2 j − 1, Δ j ( | Δ j u | p − 2 Δ j u ) , s = 2 j , j = 1,2, ⋯ , (2)

which becomes the usual p-Laplacian for s = 1 . Kratochvl and Necâs introduced the p-biharmonic operator in [

{ M ( ‖ u ‖ 2 ) ( − Δ u ) s = f ( x , u ) in Ω , D α u ( x ) | ∂ Ω = 0 ∀ α , with | α | ≤ s − 1 . (3)

We introduce for s = 1 , 2 , ⋯ , the main s-order differential operator

D s u = { D Δ j − 1 u if s = 2 j − 1, Δ j u if s = 2 j j = 1,2, ⋯ . (4)

Note that D s is an n-vectorial operator when s is odd and n > 1 , while it is a scalar operator when s is even.

In our hypothesis, the Kirchhoff function M : R 0 + → R 0 + is assumed to be continuous and to verify the structural assumptions (M):

(M_{1}) M is non-decreasing;

(M_{2}) there exists a number γ ∈ [ 1, p s ) such that for all t ∈ R 0 + ;

t M ( t ) ≤ γ M ^ ( t ) , where M ^ ( t ) = ∫ 0 t M ( θ ) d θ ;

(M_{3}) for all t ≥ σ , there exists m 0 = m 0 ( σ ) > 0 such that M ( t ) ≥ m 0 for all σ ≥ 0 .

We introduce the Sobolev critical exponent p s * and the number p s defined by following

p s * = { n p n − s p if n > s p , ∞ if n ≤ s p . p s = p s * p = { n n − s p if n > s p , ∞ if n ≤ s p . (5)

A very special Kirchhoff function verifying (M) is denoted by

M ( t ) = a + b γ t γ − 1 , a , b ≥ 0 , a + b > 0 , γ { ∈ ( 1 , p s ) if b > 0 , = 1 if b = 0. (6)

when M is of the type (6) and a > 0 , b ≥ 0 , problem (1) is said to be non-degenerate, while it is called degenerate if a = 0 . Besides, problem (2) reduces to the usual well-known quasilinear elliptic equation while a > 0 , b = 0 . The existence of positive solutions of non-degenerate Kirchhoff-type problems has been proved in [

ρ ∂ 2 u ∂ t 2 − ( P 0 h + E 2 L ∫ 0 L | ∂ u ∂ x | 2 d x ) ∂ 2 u ∂ x 2 = 0 ,

as a nonlinear extension of D’Alambert’s wave equation for free vibrations for elastic strings.

Here we study a stationary version of Kirchhoff-type problems, where u = u ( x ) is the lateral displacement at the space coordinate χ and M is typically a line with positive slope. Our result allows M to have this property. The classical Kirchhoff theory described further details and physical models, which can be found in [

We recall that study of semilinear case with datum f ( x ) ∈ L 1 ( Ω ) in [

f ( x ) , a ( x ) ∈ L 1 ( Ω ) , (7)

and there exists Q > 0 such that, for x ∈ Ω a.e.,

| f ( x ) | ≤ Q a ( x ) . (8)

There is assumption that g ( s ) is continuous function satisfies

lim s → − ∞ g ( s ) = − ∞ and lim s → + ∞ g ( s ) = ∞ . (9)

There has been an increasing interest in studying equations involving p(x)-Laplace operators over the last few decades. Motivated by theoretical research in the regularizing effect of the interaction between the coefficient of the zero order term and the datum f ( x ) ∈ L 1 ( Ω ) in some nonlinear Dirchlet problems, we pay attention to the existence of solutions for p(x)-polyharmonic Kirchhoff equations. Now we consider the problems

{ M ( φ ( u ) ) Δ p ( x ) s u + a ( x ) g ( u ) = f ( x ) in Ω , D α u ( x ) | ∂ Ω = 0 ∀ α , with | α | ≤ s − 1, (10)

where Ω ⊂ ℝ is a bounded domain Lipschitz boundary, M is a degenerate Kirchhoff function and p ∈ C ( Ω ¯ ) . More details and conditions are given in section 4. The p(x)-polyharmonic operator is given by

Δ p ( x ) s u = { − d i v Δ j − 1 ( | D Δ j − 1 u | p ( x ) − 2 ) D Δ j − 1 u , s = 2 j − 1, Δ j ( | Δ j u | p ( x ) − 2 Δ j u ) , s = 2 j , j = 1,2, ⋯ . (11)

The author exploits the symmetric mountain pass theorem to proves the multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations in [

This paper is organized as follows. In Section 2, we introduce some basic notation and properties in variable exponent Sobolev spaces. In Section 3, we prove the problem (1) ( p ≡ Const ) existing a solution u ∈ W 0 s . p ( Ω ) ∩ L ∞ ( Ω ) . In Section 4, we treat the more delicate case p = p ( x ) .

In this section, we briefly introduce some basic results and notations. Let Ω be a bounded domain in ℝ N , we denote a multi-index α = ( α 1 , α 2 , ⋯ , α n ) ∈ ℕ 0 n , with length | α | = ∑ i = 1 n α i ≤ s , such that the corresponding partial differentation:

D α = ∂ | α | ∂ x 1 α 1 ∂ x 2 α 2 ⋯ ∂ x n α n .

Write:

‖ u ‖ W s , p ( Ω ) = ( ∑ | α | ≤ s ‖ D α u ‖ p ) 1 p , (12)

where ‖ ⋅ ‖ p denotes the standard L p -norm. See [

‖ u ‖ D s , p ( Ω ) = ( ∑ | α | = s ‖ D α u ‖ p ) 1 p . (13)

By the poncaré inequality, there exists a positive constant K = K ( n , p , Ω ) , with m = s , p ≥ 1 , such that

‖ u ‖ W s , p ( Ω ) ≤ K ‖ u ‖ D s , p ( Ω ) , ∀ u ∈ W 0 s , p ( Ω ) . (14)

Hence, we obtain that the norms ‖ ⋅ ‖ W s , p ( Ω ) are equivalent, so that the two completions of C 0 ∞ ( Ω ) , with corresponding these norms, namely

W 0 s , p ( Ω ) = D s , p ( Ω ) .

We endow the vectorial space [ L p ( Ω ) ] n , with respect to the norm

‖ v ‖ p = ( ∑ i = 1 n ‖ β i ‖ p p ) 1 p , (15)

where v = ( β 1 , β 2 , ⋯ , β n ) and n > 1 , we still use the same symbol ‖ ⋅ ‖ p to denote both the standard L p -norm in the scalar space L p ( Ω ) and the norm define in (15), in the vectorial space [ L p ( Ω ) ] n .

For s = 2 , 1 < p < ∞ , by the Caldéron-Zygmund inequality, see details in [

‖ u ‖ D 2, p ( Ω ) ≤ k 2 ‖ D 2 u ‖ p , ∀ u ∈ W 0 2, p ( Ω ) . (16)

Proposition 2.1. If p ∈ ( 1, ∞ ) and s = 1 , 2 , ⋯ , then there exists a positive constant k s = k s ( n , p ) such that:

‖ u ‖ D s , p ( Ω ) ≤ k s ‖ D s u ‖ p , ∀ u ∈ W 0 s , p ( Ω ) . (17)

where D s is denoted in (4), see also in [

Hence, from now on we endow W 0 s , p ( Ω ) with the norm ‖ ⋅ ‖ = ‖ D s ⋅ ‖ p , which is equivalent to the standard Sobolev norm.

Remark 2.1. For all s = 1 , 2 , ⋯ , 1 < p < ∞ . W 0 s , p ( Ω ) is a separable, uniformly convex, reflexive, real Banach space.

Note that, when p = 2 , this norm is introduced by the inner product

〈 u , v 〉 = ∫ Ω D s u D s v d x , ∀ u , v ∈ H 0 s ( Ω ) , (18)

when s is even the operation between D s u and D s v is scalar multiplication, while s is odd, it is the n-Euclidean scalar product.

Lemma 2.1. See [

F is completely continuous map:

1) F is continuous.

2) For every B is bounded subset of X, then F ( B ) ¯ is compact.

Proposition 2.2. See [

‖ u ‖ h ≤ δ h ‖ u ‖ , ∀ u ∈ W 0 s , p ( Ω ) . (19)

We study problem (1) for a solution, we understand:

{ u ∈ W 0 s , p ( Ω ) ∩ L ∞ ( Ω ) , M ( ‖ u ‖ p ) ∫ Ω | D s u | p − 2 D s u D s v d x + ∫ Ω a ( x ) g ( u ) φ = ∫ Ω f ( x ) φ , φ ∈ W 0 s , p ( Ω ) ∩ L ∞ ( Ω ) . (20)

where D s is the operator in (4) and ∫ Ω | D s u | p − 2 D s u D s v φ d x is the p-polyharmonic operator Δ p s in weak sense.

In order to study the solution of problem (1), we consider problems:

{ M ( ‖ u n ‖ p ) Δ p s u n + a n ( x ) g ( u n ) = f n in Ω , D α u n ( x ) | ∂ Ω = 0 ∀ α , with | α | ≤ s − 1. (21)

where Ω is a bounded domain in ℝ N , p ∈ [ 2, ∞ ) , and s = 1 , 2 , ⋯ . Indeed, suppose f ( x ) ∈ L 1 ( Ω ) , 0 ≤ a ( x ) ≤ L 1 ( Ω ) , and exist h ( x ) ∈ L q ' ( Ω ) , then | g ( s ) | ≤ h ( x ) .

Let us define:

a n ( x ) = a ( x ) 1 + Q n | a ( x ) | , f n ( x ) = f ( x ) 1 + 1 n | f ( x ) | . (22)

and that we choose k 0 > 0 , such that

g ( t ) t ≥ 0, | g ( t ) | ≥ Q . (23)

for every t ≥ k 0 .

Theorem 3.1. There is a solution u n ∈ W 0 s , p ( Ω ) to the problem (21).

Proof. Since φ = s ( 1 + s n ) − 1 is increasing, we deduced by (8) that,

| f n ( x ) | = | f ( x ) | 1 + 1 n | f ( x ) | ≤ Q a ( x ) 1 + Q n a ( x ) = Q a n ( x ) . (24)

We define:

J ( ω ) = 1 p M ^ ( ‖ ω ‖ p ) + ∫ Ω a n ( x ) g ( v ) ω − ∫ Ω f n ( x ) ω ,

where v ∈ W 0 s , p ( Ω ) , by M_{2} and (24), we can get

J ( ω ) ≥ b p ‖ ω ‖ p γ − ∫ Ω a n ( x ) | g ( v ) − Q | ω ,

by the Hölder’s inequality and the Poincaré equality, then

J ( ω ) ≥ b p ‖ ω ‖ p γ − C a n ( x ) ‖ g ( v ) − Q ‖ L q ′ ‖ ω ‖ ,

since p ≥ 2 and γ ∈ ( 1, p s ) , then J ( ω ) is bounded, coercive and weakly lower semicontinuous, such that J ( ω ) has a minimizer and the Euler equation is:

M ( ‖ ω ‖ p ) Δ p s ω + a n ( x ) g ( v ) = f n .

Moreover, such a minimizer is unique, by the strict convexity of J.

Fixed n ∈ N , let v ∈ W 0 s , p ( Ω ) , define ω = S ( v ) to be the unique solution of the problem:

{ M ( ‖ ω ‖ p ) Δ p s ω + a n ( x ) g ( v ) = f n in Ω , D α u ( x ) | ∂ Ω = 0. ∀ α , with | α | ≤ s − 1. (25)

We will use the

by (M_{1}, M_{3}), (22), (24) and Hölder’s inequality, we obtain

where

We take

under s in

Step 1: We prove the continuity. In order to do this, we define

Since the convergence of

In fact, let

To this end, by choosing

by the inequality

as

where

Step 2. We prove S is compact, first we take a sequence

Since S is continuous,

with S is a positive constant, independent of k, such that,

Because of the continuity of S, necessarily

as

and therefore S is compact.

Hence, by lemma 2.1, there exists a solution

We will use the following function defined for

We use

by (M_{1}, M_{3}), and (24) we obtain:

by (9) and (23), this means:

which by (33) implies that

Next, we use

by

then,

and we obtain that

Moreover, using that

In this section, we begin by recalling some basic results on the variable exponent Lebesgue and Sobolev spaces, see details in [

As before, we define:

where

Let p be a fixed function in

by variable exponent Lebesgue space, it is a separable, reflexive Banach space. For

Note that, by Hölder-type inequality is valid:

with

For

and endow the standard norm:

We point out that the nonstandard growth condition of

Lemma 4.1. (Therorems 1.3 of [

and

From now on we also assume that

With

Lemma 4.2.

By the Poincaré inequality, see [

under this assumption, when

We recall that the operator

where

Proposition 4.1. See [

We endow the space

Let

If

Moreover, for

Consider problem (10) with

We denote the Dirchlet function

where

We study problem (10) for a solution we understand:

where

In order to study the solvability of problem (10), we will analyze the associated approximate problem.

We recall some basic conditions and hypothesis by (22)-(24), and exist

Theorem 4.1. There is a solution

Proof. We define

Observe that, by (46), and lemma 4.1 we get:

we take_{1}, M_{3}), there exists

by M_{2} and (49), we can get:

by the Hölder’s inequality and the Poincaré equality, then

since

Moreover, such a minimizer is unique, by the strict convexity of J.

Fix

We will use the

by (22), (24), (50) and Hölder’s inequality, thus,

1) If

2) If

We take

ball of radius

Step 1: We prove the continuity. In order to do this, we define

Since the convergence of

In fact, let

To this end, by choosing

by the inequality

as

where

hence s is continuous from

Step 2. We prove S is compact, first we take a sequence

Since S is continuous,

with C is a positive constant, independent of k, such that,

Because of the continuity of S, necessarily

as

therefore, S is compact.

Given these conditions on S, Schauder’s Fixed Point Theorem provides the existence of

By Section 3, we also use

Thus, we can learn some Kirchhoff equations by the above method.

We closely thank the following instructions. It will definitely save a lot of time and expedite the process of your paper’s publication.

The author declares no conflicts of interest regarding the publication of this paper.

Ge, Y.J. (2019) Existence of Solutions for Some p(x)-polyharmonic Elliptic Kirchhoff Equations. Advances in Pure Mathematics, 9, 863-878. https://doi.org/10.4236/apm.2019.910043