1. Introduction
In this paper, we prove the existence of solution of Dirchlet problems involving the p-polyharmonic operators
${\Delta}_{p}^{s}$. We consider
$\{\begin{array}{l}M\left({\Vert u\Vert}^{p}\right){\Delta}_{p}^{s}u+a\left(x\right)g\left(u\right)=f\left(x\right)\text{\hspace{1em}}\text{in}\text{\hspace{0.17em}}\Omega \mathrm{,}\\ {{D}^{\alpha}u\left(x\right)|}_{\partial \Omega}=0\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}\alpha \mathrm{,}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\left|\alpha \right|\le s-\mathrm{1,}\end{array}$ (1)
where
$\Omega \subset {\mathbb{R}}^{N}$ is a bounded domain,
$p\ge 2$,
$s=1,2,\cdots $,
$\Vert \text{\hspace{0.05em}}\cdot \text{\hspace{0.05em}}\Vert $ is denoted in section 2, and
$f\left(x\right)\in {L}^{1}\left(\Omega \right)$,
$0\le a\left(x\right)\in {L}^{1}\left(\Omega \right)$. Here, the p-polyharmonic operator is defined by
${\Delta}_{p}^{s}u=\{\begin{array}{l}-div{\Delta}^{j-1}\left({\left|D{\Delta}^{j-1}u\right|}^{p-2}\right)D{\Delta}^{j-1}u,\text{\hspace{1em}}s=2j-\mathrm{1,}\\ {\Delta}^{j}\left({\left|{\Delta}^{j}u\right|}^{p-2}{\Delta}^{j}u\right),\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}s=2j\mathrm{,}\end{array}\text{\hspace{1em}}j=\mathrm{1,2,}\cdots $, (2)
which becomes the usual p-Laplacian for
$s=1$. Kratochvl and Necâs introduced the p-biharmonic operator in [1] [2] [3] to study the physical equations, the p-biharmonic operator for
$s=2$ and the polyharmonic operator for
$p=2$, which reduces to the more appoximate case
$\{\begin{array}{l}M\left({\Vert u\Vert}^{2}\right){\left(-\Delta u\right)}^{s}=f\left(x\mathrm{,}u\right)\text{\hspace{1em}}\text{in}\text{\hspace{0.17em}}\Omega \mathrm{,}\\ {{D}^{\alpha}u\left(x\right)|}_{\partial \Omega}=0\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\forall \text{\hspace{0.17em}}\alpha \mathrm{,}\text{with}\text{\hspace{0.17em}}\left|\alpha \right|\le s-1\end{array}$. (3)
We introduce for
$s=1,2,\cdots $, the main s-order differential operator
${\mathcal{D}}_{s}u=\{\begin{array}{l}D{\Delta}^{j-1}u\text{\hspace{0.05em}}\text{\hspace{1em}}\text{if}\text{\hspace{0.17em}}s=2j-\mathrm{1,}\\ {\Delta}^{j}u\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}s=2j\end{array}\text{\hspace{1em}}j=\mathrm{1,2,}\cdots $. (4)
Note that
${\mathcal{D}}_{s}$ 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
$M\mathrm{:}{R}_{0}^{+}\to {R}_{0}^{+}$ is assumed to be continuous and to verify the structural assumptions (M):
(M_{1}) M is non-decreasing;
(M_{2}) there exists a number
$\gamma \in \left[\mathrm{1,}{p}_{s}\right)$ such that for all
$t\in {R}_{0}^{+}$ ;
$tM\left(t\right)\le \gamma \stackrel{^}{M}\left(t\right)\mathrm{,}\text{\hspace{1em}}\text{where}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\stackrel{^}{M}\left(t\right)={\displaystyle {\int}_{0}^{t}}M\left(\theta \right)\text{d}\theta \mathrm{;}$
(M_{3}) for all
$t\ge \sigma $, there exists
${m}_{0}={m}_{0}\left(\sigma \right)>0$ such that
$M\left(t\right)\ge {m}_{0}$ for all
$\sigma \ge 0$.
We introduce the Sobolev critical exponent
${p}_{s}^{\mathrm{*}}$ and the number
${p}_{s}$ defined by following
${p}_{s}^{*}=\{\begin{array}{l}\frac{np}{n-sp}\text{\hspace{1em}}\text{\hspace{0.05em}}\text{if}\text{\hspace{0.17em}}n>sp,\hfill \\ \infty \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{if}\text{\hspace{0.17em}}n\le sp.\hfill \end{array}$
${p}_{s}=\frac{{p}_{s}^{*}}{p}=\{\begin{array}{l}\frac{n}{n-sp}\text{\hspace{1em}}\text{if}\text{\hspace{0.17em}}n>sp,\hfill \\ \infty \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{if}\text{\hspace{0.17em}}n\le sp.\hfill \end{array}$ (5)
A very special Kirchhoff function verifying (M) is denoted by
$M\left(t\right)=a+b\gamma {t}^{\gamma -1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}a,b\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}a+b>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\gamma \{\begin{array}{l}\in \left(1,{p}_{s}\right)\text{\hspace{1em}}\text{\hspace{0.05em}}\text{if}\text{\hspace{0.17em}}b>0,\\ =1\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{if}\text{\hspace{0.17em}}b=0.\end{array}$ (6)
when M is of the type (6) and
$a>0,\text{\hspace{0.17em}}b\ge 0$, 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
$a>0,\text{\hspace{0.17em}}b=0$. The existence of positive solutions of non-degenerate Kirchhoff-type problems has been proved in [4] [5] for
$L=1$. The novelty of this paper is to treat the degenerate case with allowing Kirchhoff function to take the zero value. Several authors have considered fourth order problems with nonlinear boundary conditions involving third order derivatives, see [6]. The classical counterpart of our problem models containning several interesting phenomena were deeply studied in physicals even in the one-dimensional case. It dates back to 1883 when Kirchhoff proposed his celebrated equation:
$\rho \frac{{\partial}^{2}u}{\partial {t}^{2}}-\left(\frac{{P}_{0}}{h}+\frac{E}{2L}{\displaystyle {\int}_{0}^{L}}{\left|\frac{\partial u}{\partial x}\right|}^{2}\text{d}x\right)\frac{{\partial}^{2}u}{\partial {x}^{2}}=0$,
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
$u=u\left(x\right)$ is the lateral displacement at the space coordinate
$\chi $ 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 [7] [8]. In the standard case
$L=2$, problem of type (2) arise in the theory of bending extensible elastic beams. There
$u=u\left(x\right)$ denotes a thin extensible elastic beam. The function f models a small changes with effect in the length of beam but acts as a force exerted on the beam. We read to [6] and the references therein for a discussion about modelling of Kirchhoff-type strings and beams. We cite the wide literature on the subject, the works [9] [10] [11] [12], where Kirchhoff-type problems new studied by exploiting different methods.
We recall that study of semilinear case with datum
$f\left(x\right)\in {L}^{1}\left(\Omega \right)$ in [13] [14] [15] [16], with respect to (1), we assume that the coefficient
$a\left(x\right)$ of the zero order term and to the datum
$f\left(x\right)$, in addition to imposing that
$f\left(x\right)\mathrm{,}\text{\hspace{0.17em}}a\left(x\right)\in {L}^{1}\left(\Omega \right)$, (7)
and there exists
$Q>0$ such that, for
$x\in \Omega $ a.e.,
$\left|f\left(x\right)\right|\le Qa\left(x\right)\mathrm{.}$ (8)
There is assumption that
$g\left(s\right)$ is continuous function satisfies
$\underset{s\to -\infty}{\mathrm{lim}}g\left(s\right)=-\infty \text{\hspace{1em}}\text{and}\text{\hspace{1em}}\underset{s\to +\infty}{\mathrm{lim}}g\left(s\right)=\infty \mathrm{.}$ (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
$f\left(x\right)\in {L}^{1}\left(\Omega \right)$ in some nonlinear Dirchlet problems, we pay attention to the existence of solutions for p(x)-polyharmonic Kirchhoff equations. Now we consider the problems
$\{\begin{array}{l}M\left(\phi \left(u\right)\right){\Delta}_{p\left(x\right)}^{s}u+a\left(x\right)g\left(u\right)=f\left(x\right)\text{\hspace{1em}}\text{in}\text{\hspace{0.17em}}\Omega \mathrm{,}\\ {{D}^{\alpha}u\left(x\right)|}_{\partial \Omega}=0\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}\alpha \mathrm{,}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\left|\alpha \right|\le s-\mathrm{1,}\end{array}$ (10)
where
$\Omega \subset \mathbb{R}$ is a bounded domain Lipschitz boundary, M is a degenerate Kirchhoff function and
$p\in C\left(\stackrel{\xaf}{\Omega}\right)$. More details and conditions are given in section 4. The p(x)-polyharmonic operator is given by
${\Delta}_{p\left(x\right)}^{s}u=\{\begin{array}{l}-div{\Delta}^{j-1}\left({\left|D{\Delta}^{j-1}u\right|}^{p\left(x\right)-2}\right)D{\Delta}^{j-1}u,\text{\hspace{1em}}s=2j-\mathrm{1,}\\ {\Delta}^{j}\left({\left|{\Delta}^{j}u\right|}^{p\left(x\right)-2}{\Delta}^{j}u\right),\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.05em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}s=2j\mathrm{,}\end{array}\text{\hspace{1em}}j=\mathrm{1,2,}\cdots $. (11)
The author exploits the symmetric mountain pass theorem to proves the multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations in [17]. In contrast, in this paper, the keystone of the proofs them is the deduction, by condition (7), (8), of the
${L}^{\infty}$ -estimate of the approximated solutions, we prove the problem existing a solution
$u\in {W}_{0}^{s\mathrm{.}p(\cdot )}\left(\Omega \right)\cap {L}^{\infty}\left(\Omega \right)$.
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) (
$p\equiv \text{Const}$ ) existing a solution
$u\in {W}_{0}^{s\mathrm{.}p}\left(\Omega \right)\cap {L}^{\infty}\left(\Omega \right)$. In Section 4, we treat the more delicate case
$p=p\left(x\right)$.
2. Notations and Preliminaries
In this section, we briefly introduce some basic results and notations. Let
$\Omega $ be a bounded domain in
${\mathbb{R}}^{N}$, we denote a multi-index
$\alpha =\left({\alpha}_{1}\mathrm{,}{\alpha}_{2}\mathrm{,}\cdots \mathrm{,}{\alpha}_{n}\right)\in {\mathbb{N}}_{0}^{n}$, with length
$\left|\alpha \right|={\displaystyle {\sum}_{i=1}^{n}}\text{\hspace{0.05em}}{\alpha}_{i}\le s$, such that the corresponding partial differentation:
${D}^{\alpha}=\frac{{\partial}^{\left|\alpha \right|}}{\partial {x}_{1}^{{\alpha}_{1}}\partial {x}_{2}^{{\alpha}_{2}}\cdots \partial {x}_{n}^{{\alpha}_{n}}}$.
Write:
${\Vert u\Vert}_{{W}^{s\mathrm{,}p}\left(\Omega \right)}={\left({\displaystyle \underset{\left|\alpha \right|\le s}{\sum}}{\Vert {D}^{\alpha}u\Vert}^{p}\right)}^{\frac{1}{p}}\mathrm{,}$ (12)
where
${\Vert \text{\hspace{0.05em}}\cdot \text{\hspace{0.05em}}\Vert}_{p}$ denotes the standard
${L}^{p}$ -norm. See [18], we denote the space
${W}_{0}^{s\mathrm{,}p}\left(\Omega \right)$ is the completion of
${C}_{0}^{\infty}\left(\Omega \right)$ with respect to the standard norm of
${W}^{s\mathrm{,}p}\left(\Omega \right)$. Moreover, denote
${\mathcal{D}}^{s\mathrm{,}p}\left(\Omega \right)$ be the completion of
${C}_{0}^{\infty}\left(\Omega \right)$, with respect to the norm:
${\Vert u\Vert}_{{\mathcal{D}}^{s\mathrm{,}p}\left(\Omega \right)}={\left({\displaystyle \underset{\left|\alpha \right|=s}{\sum}}{\Vert {D}^{\alpha}u\Vert}^{p}\right)}^{\frac{1}{p}}$. (13)
By the poncaré inequality, there exists a positive constant
$\mathcal{K}=\mathcal{K}\left(n\mathrm{,}p\mathrm{,}\Omega \right)$, with
$m=s,p\ge 1$, such that
${\Vert u\Vert}_{{W}^{s\mathrm{,}p}\left(\Omega \right)}\le \mathcal{K}{\Vert u\Vert}_{{D}^{s\mathrm{,}p}\left(\Omega \right)}\mathrm{,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall u\in {W}_{0}^{s\mathrm{,}p}\left(\Omega \right)\mathrm{.}$ (14)
Hence, we obtain that the norms
${\Vert \text{\hspace{0.05em}}\cdot \text{\hspace{0.05em}}\Vert}_{{W}^{s\mathrm{,}p}\left(\Omega \right)}$ are equivalent, so that the two completions of
${C}_{0}^{\infty}\left(\Omega \right)$, with corresponding these norms, namely
${W}_{0}^{s\mathrm{,}p}\left(\Omega \right)={\mathcal{D}}^{s\mathrm{,}p}\left(\Omega \right).$
We endow the vectorial space
${\left[{L}^{p}\left(\Omega \right)\right]}^{n}$, with respect to the norm
${\Vert v\Vert}_{p}={\left({\displaystyle \underset{i=1}{\overset{n}{\sum}}}{\Vert {\beta}_{i}\Vert}_{p}^{p}\right)}^{\frac{1}{p}},$ (15)
where
$v=\left({\beta}_{1},{\beta}_{2},\cdots ,{\beta}_{n}\right)$ and
$n>1$, we still use the same symbol
${\Vert \text{\hspace{0.05em}}\cdot \text{\hspace{0.05em}}\Vert}_{p}$ to denote both the standard
${L}^{p}$ -norm in the scalar space
${L}^{p}\left(\Omega \right)$ and the norm define in (15), in the vectorial space
${\left[{L}^{p}\left(\Omega \right)\right]}^{n}$.
For
$s=2,\text{\hspace{0.17em}}1<p<\infty $, by the Caldéron-Zygmund inequality, see details in [19] [20], there exists a constant
${k}_{2}={k}_{2}\left(n,p\right)>0$ such that:
${\Vert u\Vert}_{{\mathcal{D}}^{\mathrm{2,}p}\left(\Omega \right)}\le {k}_{2}{\Vert {\mathcal{D}}_{2}u\Vert}_{p},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall u\in {W}_{0}^{\mathrm{2,}p}\left(\Omega \right)\mathrm{.}$ (16)
Proposition 2.1. If
$p\in \left(\mathrm{1,}\infty \right)$ and
$s=1,2,\cdots $, then there exists a positive constant
${k}_{s}={k}_{s}\left(n,p\right)$ such that:
${\Vert u\Vert}_{{\mathcal{D}}^{s\mathrm{,}p}\left(\Omega \right)}\le {k}_{s}{\Vert {\mathcal{D}}_{s}u\Vert}_{p},\text{\hspace{1em}}\forall u\in {W}_{0}^{s\mathrm{,}p}\left(\Omega \right)\mathrm{.}$ (17)
where
${\mathcal{D}}_{s}$ is denoted in (4), see also in [17].
Hence, from now on we endow
${W}_{0}^{s\mathrm{,}p}\left(\Omega \right)$ with the norm
$\Vert \text{\hspace{0.05em}}\cdot \text{\hspace{0.05em}}\Vert ={\Vert {\mathcal{D}}_{s}\cdot \text{\hspace{0.05em}}\Vert}_{p}$, which is equivalent to the standard Sobolev norm.
Remark 2.1. For all
$s=1,2,\cdots $,
$1<p<\infty $.
${W}_{0}^{s,p}\left(\Omega \right)$ is a separable, uniformly convex, reflexive, real Banach space.
Note that, when
$p=2$, this norm is introduced by the inner product
$\langle u\mathrm{,}v\rangle ={\displaystyle {\int}_{\Omega}}\text{\hspace{0.05em}}{\mathcal{D}}_{s}u{\mathcal{D}}_{s}v\text{d}x\mathrm{,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall u\mathrm{,}v\in {H}_{0}^{s}\left(\Omega \right),$ (18)
when s is even the operation between
${\mathcal{D}}_{s}u$ and
${\mathcal{D}}_{s}v$ is scalar multiplication, while s is odd, it is the n-Euclidean scalar product.
Lemma 2.1. See [21] (Schuader’s theorem) Let F be a completely continuous map and let K be a convex, bounded, closed and invariant subset of X. Then F has a fixed point in K.
F is completely continuous map:
1) F is continuous.
2) For every B is bounded subset of X, then
$\stackrel{\xaf}{F\left(B\right)}$ is compact.
Proposition 2.2. See [18], for
$1\le h<{p}_{s}^{*}$, the embedding
${W}_{0}^{s\mathrm{,}p}\left(\Omega \right)\mathrm{,}{L}^{h}\left(\Omega \right)$ is compact and continuous, there exists
${\delta}_{h}={\delta}_{h}\left(n,p,s,\Omega \right)>0$, such that:
${\Vert u\Vert}_{h}\le {\delta}_{h}\Vert u\Vert \mathrm{,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall u\in {W}_{0}^{s\mathrm{,}p}\left(\Omega \right)\mathrm{.}$ (19)
We study problem (1) for a solution, we understand:
$\{\begin{array}{l}u\in {W}_{0}^{s\mathrm{,}p}\left(\Omega \right)\cap {L}^{\infty}\left(\Omega \right)\mathrm{,}\\ M\left({\Vert u\Vert}^{p}\right){\displaystyle {\int}_{\Omega}}{\left|{\mathcal{D}}_{s}u\right|}^{p-2}{\mathcal{D}}_{s}u{\mathcal{D}}_{s}v\text{d}x+{\displaystyle {\int}_{\Omega}}\text{\hspace{0.05em}}a\left(x\right)g\left(u\right)\phi ={\displaystyle {\int}_{\Omega}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}f\left(x\right)\phi \mathrm{,}\\ \phi \in {W}_{0}^{s\mathrm{,}p}\left(\Omega \right)\cap {L}^{\infty}\left(\Omega \right)\mathrm{.}\end{array}$ (20)
where
${\mathcal{D}}_{s}$ is the operator in (4) and
${\int}_{\Omega}}{\left|{\mathcal{D}}_{s}u\right|}^{p-2}{\mathcal{D}}_{s}u{\mathcal{D}}_{s}v\phi \text{d}x$ is the p-polyharmonic operator
${\Delta}_{p}^{s}$ in weak sense.
3. Existence and Uniqueness of Solution for (1)
In order to study the solution of problem (1), we consider problems:
$\{\begin{array}{l}M\left({\Vert {u}_{n}\Vert}^{p}\right){\Delta}_{p}^{s}{u}_{n}+{a}_{n}\left(x\right)g\left({u}_{n}\right)={f}_{n}\text{\hspace{1em}}\text{in}\text{\hspace{0.17em}}\Omega \mathrm{,}\\ {{D}^{\alpha}{u}_{n}\left(x\right)|}_{\partial \Omega}=0\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}\alpha \mathrm{,}\text{with}\text{\hspace{0.17em}}\left|\alpha \right|\le s-1.\end{array}$ (21)
where
$\Omega $ is a bounded domain in
${\mathbb{R}}^{N}$,
$p\in \left[\mathrm{2,}\infty \right)$, and
$s=1,2,\cdots $. Indeed, suppose
$f\left(x\right)\in {L}^{1}\left(\Omega \right)$,
$0\le a\left(x\right)\le {L}^{1}\left(\Omega \right)$, and exist
$h\left(x\right)\in {L}^{q\text{'}}\left(\Omega \right)$, then
$\left|g\left(s\right)\right|\le h\left(x\right)$.
Let us define:
${a}_{n}\left(x\right)=\frac{a\left(x\right)}{1+\frac{Q}{n}\left|a\left(x\right)\right|},\text{\hspace{1em}}{f}_{n}\left(x\right)=\frac{f\left(x\right)}{1+\frac{1}{n}\left|f\left(x\right)\right|}.$ (22)
and that we choose
${k}_{0}>0$, such that
$g\left(t\right)t\ge \mathrm{0,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left|g\left(t\right)\right|\ge Q\mathrm{.}$ (23)
for every
$t\ge {k}_{0}$.
Theorem 3.1. There is a solution
${u}_{n}\in {W}_{0}^{s\mathrm{,}p}\left(\Omega \right)$ to the problem (21).
Proof. Since
$\phi =s{\left(1+\frac{s}{n}\right)}^{-1}$ is increasing, we deduced by (8) that,
$\left|{f}_{n}\left(x\right)\right|=\frac{\left|f\left(x\right)\right|}{1+\frac{1}{n}\left|f\left(x\right)\right|}\le \frac{Qa\left(x\right)}{1+\frac{Q}{n}a\left(x\right)}=Q{a}_{n}\left(x\right).$ (24)
We define:
$J\left(\omega \right)=\frac{1}{p}\stackrel{^}{M}\left({\Vert \omega \Vert}^{p}\right)+{\displaystyle {\int}_{\Omega}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{a}_{n}\left(x\right)g\left(v\right)\omega -{\displaystyle {\int}_{\Omega}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{f}_{n}\left(x\right)\omega \mathrm{,}$
where
$v\in {W}_{0}^{s\mathrm{,}p}\left(\Omega \right)$, by M_{2} and (24), we can get
$J\left(\omega \right)\ge \frac{b}{p}{\Vert \omega \Vert}^{p\gamma}-{\displaystyle {\int}_{\Omega}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{a}_{n}\left(x\right)\left|g\left(v\right)-Q\right|\omega \mathrm{,}$
by the Hölder’s inequality and the Poincaré equality, then
$J\left(\omega \right)\ge \frac{b}{p}{\Vert \omega \Vert}^{p\gamma}-C{a}_{n}\left(x\right){\Vert g\left(v\right)-Q\Vert}_{{L}^{{q}^{\prime}}}\Vert \omega \Vert \mathrm{,}$
since
$p\ge 2$ and
$\gamma \in \left(\mathrm{1,}{p}_{s}\right)$, then
$J\left(\omega \right)$ is bounded, coercive and weakly lower semicontinuous, such that
$J\left(\omega \right)$ has a minimizer and the Euler equation is:
$M\left({\Vert \omega \Vert}^{p}\right){\Delta}_{p}^{s}\omega +{a}_{n}\left(x\right)g\left(v\right)={f}_{n}\mathrm{.}$
Moreover, such a minimizer is unique, by the strict convexity of J.
Fixed
$n\in N$, let
$v\in {W}_{0}^{s\mathrm{,}p}\left(\Omega \right)$, define
$\omega =S\left(v\right)$ to be the unique solution of the problem:
$\{\begin{array}{l}M\left({\Vert \omega \Vert}^{p}\right){\Delta}_{p}^{s}\omega +{a}_{n}\left(x\right)g\left(v\right)={f}_{n}\text{\hspace{1em}}\text{in}\text{\hspace{0.17em}}\Omega \mathrm{,}\\ {{D}^{\alpha}u\left(x\right)|}_{\partial \Omega}=0.\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\forall \alpha \mathrm{,}\text{with}\text{\hspace{0.17em}}\left|\alpha \right|\le s-1.\end{array}$ (25)
We will use the as a test in (25), we get
by (M_{1}, M_{3}), (22), (24) and Hölder’s inequality, we obtain
where, and is the conjugate exponent of q, by (23) and the Poincaré equality, it follows that
We take, so that the ball of radius is invariant
under s in. In order to apply the Schauder’s Fixed Point Theorem, apart from the invariant, we need to check the continuity and compactness of s as an operator from to. So, the proof will be divided into two steps.
Step 1: We prove the continuity. In order to do this, we define and then:
(26)
Since the convergence of in, by (26) we obtain:
(27)
In fact, let, be a sequence in converging to.
To this end, by choosing as a test function, we have
by the inequality, for, and, by Hölder’s inequality, we obtain
(28)
as,
where is bounded, then by (27), and, there is, hence S is continuous from to.
Step 2. We prove S is compact, first we take a sequence that, therefore by Rellich-Kondrachov Theorem, we obtain
(29)
Since S is continuous,.
with S is a positive constant, independent of k, such that,
(30)
Because of the continuity of S, necessarily, so that proceeding as in (28), we can get:
as, the second term of left hand is vanished by (29) and (30). we can conclude
and therefore S is compact.
Hence, by lemma 2.1, there exists a solution of problem (21), next we will prove the problem (1).
We will use the following function defined for, by
(31)
We use as a test function in approximate problem (21), then
by (M_{1}, M_{3}), and (24) we obtain:
(32)
by (9) and (23), this means:
(33)
which by (33) implies that and the sequence is bounded in.
Next, we use as a test function to deduce, such that,
by and (23) we can get
(34)
then,
(35)
and we obtain that is bounded in and a sequence, still denoted, which converges weakly in and a.e. to u with.
Moreover, using that, we obtain by the dominated convergence theorem, the convergence of the sequence to, which together with the convergence of to, we pass to the limit in the problem (21), we prove that u satisfies (1), with.
4. A p(x)-polyharmonic Kirchhoff Equation
In this section, we begin by recalling some basic results on the variable exponent Lebesgue and Sobolev spaces, see details in [22] [23].
As before, we define:
(36)
where is a bounded domain, and,. Let h be the function in, an important role in manipulating the generalized Lebesgue-Sobolev spaces is played by spaces, which is the convex function: defined by:
(37)
Let p be a fixed function in. We endow the Luxemburg norm:
(38)
by variable exponent Lebesgue space, it is a separable, reflexive Banach space. For, in, then the embedding is continuous and the norm of the embedding operator does not exceed. see [23].
be the function obtained by conjugating the exponent p pointwise, so that for all, the belongs to.
Note that, by Hölder-type inequality is valid:
(39)
with as proved in [23].
For, we introduced the variable exponent Sobolev space defined by:
(40)
and endow the standard norm:
We point out that the nonstandard growth condition of type.
Lemma 4.1. (Therorems 1.3 of [24] ) If, with , then the following relations hold:
and in measure in and . In particular, is continuous in.
From now on we also assume that, where is the space of all the functions of, which are logarithmic Hölder continuous, there exists, such that:
With, the space denotes the completion of with respect to the norm.
Lemma 4.2. is a separable, uniformly convex, Banach space, see details in [22].
By the Poincaré inequality, see [25] [26], the equivalent norm for the space is given by:
(41)
under this assumption, when, as a consequence for the main Coldéron-Zygmund results, there exists a constant such that:
(42)
We recall that the operator is defined in (4) is vectorial, when s is odd, we endow space with the norm
(43)
where with abuse of the notation we use the same symbol to denote both the standard Luxemburg norm in the scalar space and the norm defined in (43) for the vectorial space.
Proposition 4.1. See [17] for all there exists, such that
(44)
We endow the space with the norm, such that
Let denote the critical variable exponent related to p defined for all, by the pointwise relation:
(45)
If for all, the Sobolev embedding is continuous and compact. If and, the embedding is continuous whenever for all, there exists such that:
Moreover, for for all (or equivalent), then is compactly embedded in, see details in [17] [22] [24] [27].
Consider problem (10) with and, such that either or. The Kirchhoff function is assumed to be continuous and to verify condition (M) given in introduction, where.
We denote the Dirchlet function:
(46)
where is given by (4).
We study problem (10) for a solution we understand:
(47)
where is the operator in (4), and is the -polyharmonic operator, in weak sense.
In order to study the solvability of problem (10), we will analyze the associated approximate problem.
(48)
We recall some basic conditions and hypothesis by (22)-(24), and exist, then.
Theorem 4.1. There is a solution to the problem (48).
Proof. We define
Observe that, by (46), and lemma 4.1 we get:
(49)
we take, so that by (M_{1}, M_{3}), there exists,
(50)
by M_{2} and (49), we can get:
by the Hölder’s inequality and the Poincaré equality, then
since and, then is bounded, coercive and weakly lower semicontuous, such that has a minimizer and the Euler equation is
Moreover, such a minimizer is unique, by the strict convexity of J.
Fix, let, define to be the unique solution of the problem:
(51)
We will use the as a test in (51) we get:
by (22), (24), (50) and Hölder’s inequality, thus,
1) If, such that
2) If, such that
We take, so that the
ball of radius is invariant under s in. In order to apply the Schauder’s Fixed Point Theorem, apart from the invariant, we need to check the continuity and compactness of s as an operator from to. So, the proof will be divided into two steps.
Step 1: We prove the continuity. In order to do this, we define and then:
(52)
Since the convergence of in, by (52) we obtain:
(53)
In fact, let, be a sequence in converging to.
To this end, by choosing as a test function, we have
by the inequality for, , (22), (38) and Hölder’s inequality, we obtain:
(54)
as,
where is bounded, since (53) and g is continuous, such that
hence s is continuous from to.
Step 2. We prove S is compact, first we take a sequence that, therefore by Rellich-Kondrachov Theorem, we obtain
(55)
Since S is continuous,.
with C is a positive constant, independent of k, such that,
(56)
Because of the continuity of S, necessarily, so that proceeding as in (54), we can get
as, the first term of the left hand is vanished, then by (55) and (56)
therefore, S is compact.
Given these conditions on S, Schauder’s Fixed Point Theorem provides the existence of, such that, i.e., solves:
(57)
By Section 3, we also use as a test function, then we can obtain the sequence is bounded in, next we will use as a test function, we can get a sequence, which converges weakly in and a.e. to u with. Finally, by the dominated convergence theorem and convergence, we prove that u satisfies (10) with.
Thus, we can learn some Kirchhoff equations by the above method.
Acknowledgements
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