Qualitative Properties of Solutions to a Class of Sixth-Order Equations

Abstract

In this paper, we present a detailed study of a class of sixth-order semilinear PDEs: existence, regularity and uniqueness. The uniqueness results are a consequence of a maximum principle called the $\mathrm{P}$-function method.

1. Introduction

The aim of this paper is to treat the existence, regularity and uniqueness of the following boundary value problem:

$$\left\{\begin{array}{c}{\mathrm{\Delta }}^{3}u-A\left(x\right){\mathrm{\Delta }}^{2}u+B\left(x\right)\mathrm{\Delta }u-C\left(x\right)u=f(x,u) \mathrm{in}\mathsf{\Omega }\hfill \\ u=\mathrm{\Delta }u={\mathrm{\Delta }}^{2}u=0\mathrm{on} \partial \mathsf{\Omega },\hfill \end{array}\right.$$

(1)

where $\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}^N$ is a bounded domain.

The treatment of sixth-order equations is motivated by the fact that such equations have arisen in a variety of contexts. We note that in the case $N=1$, such boundary value problems arise in different areas of applied mathematics and physics (see for example the Introduction Section of paper [1]).

Sixth-order PDEs arise in propeller blade design ([2]) or ulcer modelling ([3]). Applications of sixth-order problems in surface modelling and fluid flows are considered in [4] and [5]. Boundary value problems that are similar to the problems studied in this paper arise in phase field crystal models (see [6]).

First, we note that some existence results can be proven when f is sublinear, linear and superlinear (with polynomial growth) at both zero and infinity. Regularity is discussed in Section 3. For some new results concerning regularity, the reader is referred to the works [7,8,9].

Then, we deduce uniqueness results (for problem (1) with non-zero boundary data) by using a maximum principle called the $\mathrm{P}$-function method (in honor of Larry Payne, see [10]). We develop $\mathrm{P}$-functions defined on solutions of

$${\mathrm{\Delta }}^{3}u-A\left(x\right){\mathrm{\Delta }}^{2}u+B\left(x\right)\mathrm{\Delta }u-C\left(x\right)u=0\mathrm{in}\mathsf{\Omega }.$$

(2)

This technique of defining a function on the solution of an equation and deducing results about the solution of the equation is well-known. In their pioneering work [10], Larry Payne used this idea to determine gradient bounds on the solution in the torsion problem. One can find an exposition of some of these results (for second-order and fourth-order equations) and applications in the book of Sperb [11]. However, there are only a handful of papers dedicated to the study of uniqueness of sixth-order equations. We mention the papers [12,13,14] if $N>1$ and [1,15,16,17,18] in the one-dimensional case $N=1.$

Using Payne’s method, we extend a classical uniqueness result of Schaefer [14] to the non-constant coefficient case and to the higher dimensional case. Furthermore, our results complete the uniqueness results in [12,13].

In studying the uniqueness of solutions, the advantage of using a maximum principle argument over other arguments is that, by using the first argument, we directly obtain the uniqueness of classical solutions with mild regularity ($\partial \mathsf{\Omega }\in C^{2+\epsilon }$) or with no regularity on the boundary of the domain.

2. Preliminaries and Notations

We denote partial derivatives $\partial u/\partial x_i$ by $u{,}_i,$${\partial }^{2}u/\partial x_i\partial x_j$ by $u{,}_{ij}$, etc., and use the summation convention, so that, e.g., the square of the gradient of u becomes ${(\partial u/\partial x_1)}^2+{(\partial u/\partial x_2)}^2={|\mathrm{\nabla }u|}^2=u{,}_{i}u{,}_i.$ The scalar product of the gradients of two functions, say u and v, may be written as $\mathrm{\nabla }u·\mathrm{\nabla }v=u{,}_{i}v{,}_i$.

Throughout this paper, $\partial /\partial n$ and $\partial /\partial s$ denote the outward normal derivative operator and the tangential derivative operator, respectively.

The diameter of $\mathsf{\Omega }$ will be denoted by $\mathrm{diam}\mathsf{\Omega }.$

We recall the definition of weak and classical solution.

We consider the Hilbert space $H\left(\mathsf{\Omega }\right)=\{u\in H^3\left(\mathsf{\Omega }\right)\mid u=\mathrm{\Delta }u=0\mathrm{on}\partial \mathsf{\Omega }\}$, endowed with the standard inner product.

Definition 1.

A weak solution of (1) is a function $u\in H\left(\mathsf{\Omega }\right)$ such that

$${\int }_{\mathsf{\Omega }}\left(\mathrm{\nabla }\left(\mathrm{\Delta }u\right)·\mathrm{\nabla }\left(\mathrm{\Delta }v\right)+A\left(x\right)\mathrm{\Delta }u\mathrm{\Delta }v+B\left(x\right)\mathrm{\nabla }u·\mathrm{\nabla }u+C\left(x\right)uv+f(x,u)v\right)dx=0,$$

$\forall vs.\in H\left(\mathsf{\Omega }\right).$

A classical solution of (1) is a function $u\in C^6\left(\mathsf{\Omega }\right)\cap C^4\left(\overline{\mathsf{\Omega }}\right)$ that satisfies (1).

3. Existence and Regularity

We note that, under some restrictions on the coefficients and on the nonlinearity f, some existence results for weak solutions to (1) can be given by using the Mountain–Pass theorem. For example, if f is sublinear and superlinear at both zero and infinity, such results are presented in the one-dimensional case in [1]. They can easily be adapted for higher dimensions.

If f is linear at both zero and infinity, then some existence results are presented for a class of fourth-order equations in the paper [19]. We note that the results can be adapted to higher-order equations too.

Concerning the regularity of the solutions, we can prove the following:

Theorem 1.

Let u be a weak solution to (1) and $f(x,u)=a\left(x\right)g\left(u\right).$

If one of the following assumptions hold:

1. 

Suppose that $A=B=C=0,a\in C^{0,\alpha },f\in C_{loc}^{0,\alpha }\left(\mathrm{I}\mathrm{R}\right),\partial \mathsf{\Omega }\in C^{6,\alpha }$ and there exists $C_0>0$ and $S_0>0$ such that

$$\left|f(x,s)\right|\le C_0{\left|s\right|}^{\frac{N+6}{N-6}},\forall x\in \mathsf{\Omega },\left|s\right|\ge S_0.$$

2. 

Suppose that $a\in C^3\left(\mathsf{\Omega }\right),g\in C^2\left(\mathrm{I}\mathrm{R}\right),g^{″}\in L^{\infty }\left(\mathsf{\Omega }\right),\partial \mathsf{\Omega }\in C^8.$

Then, u is a classical solution of (1).

Proof. 

1.

Follows from Lemma A.3 [20].

2.

To show the regularity, we use a “bootstrapping” argument and Theorem 2.20, p. 46, [21], which is a version of the classical result of Agmon–Douglis–Nirenberg.

Since $g\in C^2\left(\mathrm{I}\mathrm{R}\right),g^{″}\in L^{\infty }\left(\mathrm{I}\mathrm{R}\right),u\in W^{3,2}\left(\mathsf{\Omega }\right)$, it follows that $g\left(u\left(x\right)\right)\in W^{3,2}\left(\mathsf{\Omega }\right).$

Now by Theorem 2.20, [21] (take $k=8,m=3,p=2$) it follows that there exists a solution to (1) in $W^{8,2}\left(\mathsf{\Omega }\right).$

Consequently, by the Sobolev embedding theorem $u\in C^{6,1}\left(\overline{\mathsf{\Omega }}\right)$, i.e., u is a classical solution. □

4. Maximum Principles

The following four lemmas are extensions of the classical result of Schaefer (Lemma 1, [14]) to the non-constant and to the higher dimensional case $N>2.$ These results will be used to deduce corresponding uniqueness results for solutions to problem (1).

We also note that our results cannot be deduced from the results in [13] since all maximum principles/uniqueness results in the non-constant coefficient case require that at least one coefficient is greater than 1 (see Lemma 3.1 and Corollary 3.3 in [13]).

Lemma 1.

Let u be a classical solution of (1) where $\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}^2$ and suppose that

$$a>0,\mathrm{\Delta }(1/a)\le 0,$$

(3)

$$c>0,\mathrm{\Delta }(1/c)\le 0.$$

(4)

Then, the functional ${\mathrm{P}}_1$ given by

$${\mathrm{P}}_1=\frac{c\left(x\right)}{2}{u}^2+\frac{a\left(x\right)}{2}{\left(\mathrm{\Delta }u\right)}^2+{|\mathrm{\nabla }\left(\mathrm{\Delta }u\right)|}^2-\mathrm{\Delta }u{\mathrm{\Delta }}^{2}u$$

(5)

which assumes its maximum value on $\partial \mathsf{\Omega }.$

For a proof, see Lemma 2.2, [12].

Lemma 2.

Let u be a classical solution of (1) where $\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}^N,A\left(x\right)\ge 0,$ $B,C>0$ are constants and suppose that

$$\frac{2C+NC}{NB}<\frac{2N+2}{{\left(\mathit{diam}\mathsf{\Omega }\right)}^2}.$$

(6)

Then, the nonconstant function ${\mathrm{P}}_2/\mathsf{\Psi }$ attains its (non-negative) maximum value on $\partial \mathsf{\Omega },$ unless ${\mathrm{P}}_2<0.$

Here,

$${\mathrm{P}}_2=\frac{1}{2}{\left({\mathrm{\Delta }}^{2}u\right)}^2+\frac{B}{2}{\left(\mathrm{\Delta }u\right)}^2+{C\left(\right|\mathrm{\nabla }u|}^2-u\mathrm{\Delta }u)+\frac{C(2C+NC)}{NB}{u}^2$$

and

$$\mathsf{\Psi }\left(x\right)=1-\alpha (x_1^2+\dots +x_N^2)>0,\alpha >0\mathit{is}a\mathit{constant}.$$

A similar maximum principle holds for the function ${\mathrm{P}}_2/\mathsf{\Phi }$ if

$$\frac{2C+NC}{NB}<\frac{{\pi }^2}{d^2}$$

(7)

in the case when Ω lies in a slab of width $d.$ Here,

$$\mathsf{\Phi }\left(x\right)=cos\frac{\pi (2x_i-d)}{2(d+\epsilon )}\prod _{j=1}^{n}cosh\left(\epsilon x_j\right)>0\mathrm{in}\overline{\mathsf{\Omega }},$$

for some $i\in \{1,\dots ,N\}$, where $\epsilon >0$ is small. (for more details concerning the functions Ψ and Φ see [22]).

Proof. 

A straightforward computation shows that

$$\mathrm{\Delta }\left({\left({\mathrm{\Delta }}^{2}u\right)}^2\right)/2\ge {\mathrm{\Delta }}^{2}u{\mathrm{\Delta }}^{3}u=A\left(x\right){\left({\mathrm{\Delta }}^{2}u\right)}^2-B\mathrm{\Delta }u{\mathrm{\Delta }}^{2}u+Cu{\mathrm{\Delta }}^{2}u.$$

$$\mathrm{\Delta }\left(B{\left(\mathrm{\Delta }u\right)}^2\right)/2\ge B\mathrm{\Delta }u{\mathrm{\Delta }}^{2}u.$$

$${\mathrm{\Delta }\left(C\right|\mathrm{\nabla }u|}^2-u\mathrm{\Delta }u)\ge C(2u{,}_{ij}u{,}_{ij}-{\left(\mathrm{\Delta }u\right)}^2-u{\mathrm{\Delta }}^{2}u).$$

$$\mathrm{\Delta }\left(u^2\right)\ge 2u\mathrm{\Delta }u.$$

Hence, we obtain

$$\mathrm{\Delta }{\mathrm{P}}_2\ge 2C\left(u{,}_{ij}u{,}_{ij}-\frac{1}{2}{\left(\mathrm{\Delta }u\right)}^2\right)+\frac{2C(2C+NC)}{NB}u\mathrm{\Delta }u\mathrm{in}\mathsf{\Omega }.$$

Therefore,

$$\begin{array}{ccc}\hfill \mathrm{\Delta }{\mathrm{P}}_2+\frac{2C(2C+NC)}{NB}{\mathrm{P}}_2& \ge & 2C\left(u{,}_{ij}u{,}_{ij}-\frac{1}{2}{\left(\mathrm{\Delta }u\right)}^2+\frac{2+N}{2N}{\left(\mathrm{\Delta }u\right)}^2\right)\hfill \\ & \ge & 2C\left(u{,}_{ij}u{,}_{ij}-\frac{1}{N}{\left(\mathrm{\Delta }u\right)}^2\right).\hfill \end{array}$$

Since in N dimensions the following inequality holds (see [11], Lemma 5.4, p. 73):

$$Nu{,}_{ij}u{,}_{ij}\ge {\left(\mathrm{\Delta }u\right)}^2$$

(8)

we finally obtain

$$\mathrm{\Delta }{\mathrm{P}}_2+\frac{2C(2C+NC)}{NB}{\mathrm{P}}_2\ge 0\mathrm{in}\mathsf{\Omega }.$$

Now, the proof follows from the generalized maximum principle (Theorem 3.1, [22]). □

Lemma 3.

Let u be a classical solution of (1) where $\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}^2$ and $A\left(x\right)$ are an arbitrary function. Suppose that the functions $B\left(x\right),C\left(x\right)>0$ in $\overline{\mathsf{\Omega }}$ and suppose that

$$\mathrm{\Delta }\left(\frac{C}{B}\right)\le 0,2A\ge \frac{\mathrm{\Delta }C}{B}\mathrm{in}\mathsf{\Omega }.$$

(9)

Then, the function

$${\mathrm{P}}_3=\frac{1}{C}{\left({\mathrm{\Delta }}^{2}u\right)}^2+\frac{B}{C}{\left(\mathrm{\Delta }u\right)}^2+{2\left(\right|\mathrm{\nabla }u|}^2-u\mathrm{\Delta }u)$$

attains its maximum value on $\partial \mathsf{\Omega }.$

If $B=0,A\left(x\right)\ge 0,\mathrm{\Delta }C\le 0$ in  $\mathsf{\Omega },$ $C\left(x\right)>0$ in  $\overline{\mathsf{\Omega }},$ then the function ${\mathrm{P}}_3$ may be replaced by

$${\mathrm{P}}_4=\frac{1}{C}{\left({\mathrm{\Delta }}^{2}u\right)}^2+{2\left(\right|\mathrm{\nabla }u|}^2-u\mathrm{\Delta }u).$$

Proof. 

We prove the result by showing that ${\mathrm{P}}_3$ is subharmonic in $\mathsf{\Omega }.$ First, we observe that if

$$\alpha >0,\mathrm{\Delta }(1/\alpha )\le 0$$

then

$$\mathrm{\Delta }\alpha {\beta }^2\ge 2\alpha \mathrm{\Delta }\beta .$$

It then follows that

$$\begin{array}{ccc}\hfill \mathrm{\Delta }\left(\alpha \left(x\right){\beta }^2\right)/2& \ge & \alpha \left(x\right)u\mathrm{\Delta }\beta +\sum _{i=1}^2{\left(\frac{1}{\sqrt{\alpha }}\alpha {,}_i\beta +\sqrt{\alpha }\beta {,}_i\right)}^2\hfill \\ & \ge & \alpha \mathrm{\Delta }\beta .\hfill \end{array}$$

We now calculate

$$\begin{array}{ccc}\hfill \mathrm{\Delta }{\mathrm{P}}_3& \ge & \frac{2}{C}{|\mathrm{\nabla }\left({\mathrm{\Delta }}^{2}u\right)|}^2+\frac{2}{C}{\mathrm{\Delta }}^{2}u{\mathrm{\Delta }}^{3}u-\frac{4\mathrm{\nabla }\left(C\right)}{C^2}\mathrm{\nabla }\left({\mathrm{\Delta }}^{2}u\right){\mathrm{\Delta }}^{2}u-\frac{\mathrm{\Delta }C}{C^2}{\left({\mathrm{\Delta }}^{2}u\right)}^2+\hfill \\ & & \frac{{2|\mathrm{\nabla }\left(C\right)|}^2}{C^3}{\left({\mathrm{\Delta }}^{2}u\right)}^2+2(u{,}_{ij}u{,}_{ij}-{\left(\mathrm{\Delta }u\right)}^2-u{\mathrm{\Delta }}^{2}u)+\frac{2B}{C}\mathrm{\Delta }u{\mathrm{\Delta }}^{2}u.\hfill \end{array}$$

Using inequality (8) with $N=2$ and the fact that

$$\begin{array}{ccc}\hfill \frac{2}{C}{|\mathrm{\nabla }\left({\mathrm{\Delta }}^{2}u\right)|}^2& -& \frac{4\mathrm{\nabla }\left(C\right)}{C^2}\mathrm{\nabla }\left({\mathrm{\Delta }}^{2}u\right){\mathrm{\Delta }}^{2}u+\frac{{2|\mathrm{\nabla }\left(C\right)|}^2}{C^3}{\left({\mathrm{\Delta }}^{2}u\right)}^2\hfill \\ & =& |\left({\mathrm{\Delta }}^{2}u\right){,}_i-c{,}_i{\mathrm{\Delta }}^2{u/c|}^2\ge 0,\hfill \end{array}$$

we obtain

$$\begin{array}{ccc}\hfill \mathrm{\Delta }{\mathrm{P}}_3& \ge & \frac{2}{C}{\mathrm{\Delta }}^{2}u{\mathrm{\Delta }}^{3}u-\frac{\mathrm{\Delta }C}{C^2}{\left({\mathrm{\Delta }}^{2}u\right)}^2-2u{\mathrm{\Delta }}^{2}u+\frac{2B}{C}\mathrm{\Delta }u{\mathrm{\Delta }}^{2}u.\hfill \end{array}$$

Finally, since u is a solution of Equation (2), we obtain

$$\begin{array}{ccc}\hfill \mathrm{\Delta }{\mathrm{P}}_3& \ge & \frac{1}{C}\left(2A-\frac{\mathrm{\Delta }C}{C^2}\right){\left({\mathrm{\Delta }}^{2}u\right)}^2\ge 0\mathrm{in}\mathsf{\Omega }.\hfill \end{array}$$

The result can now be inferred from the classical maximum principle. □

The maximum principles presented in the works [13,14] are stated in the case when the coefficients of the sixth-order equation are positive. The next maximum principle holds without any sign restriction on the coefficients $A, B, C.$

Lemma 4.

Let u be a classical solution of (1) where $\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}^N$ and $C\ne 0$ in $\overline{\mathsf{\Omega }},$ $C\in C^2\left(\overline{\mathsf{\Omega }}\right)$.

(1). 

If

$$max\{2,\underset{\mathsf{\Omega }}{sup}(2A+{(1-B)}^2+C^2)\}<\frac{4N+4}{{\left(\mathit{diam}\mathsf{\Omega }\right)}^2}\mathrm{in}\mathsf{\Omega }.$$

(10)

Then, the function

$$\frac{\mathrm{P}}{\mathsf{\Psi }}$$

attains its maximum value on $\partial \mathsf{\Omega },$ where

$$\mathrm{P}={\left({\mathrm{\Delta }}^{2}u\right)}^2+{\left(\mathrm{\Delta }u\right)}^2+u^2.$$

Suppose that the functions $A\left(x\right), B\left(x\right), C\left(x\right)$ satisfy

$$2A+{(1-B)}^2+C^2<2\mathrm{in}\mathsf{\Omega }.$$

(11)

(2). 

If the following inequality is satisfied

$$\frac{C\mathrm{\Delta }C+{|\mathrm{\nabla }C|}^2}{C^2}+1\le 0\mathrm{in}\mathsf{\Omega }.$$

(12)

then the function

$${\mathrm{P}}_5=\frac{{\left({\mathrm{\Delta }}^{2}u\right)}^2+{\left(\mathrm{\Delta }u\right)}^2+u^2}{C^2}=\frac{\mathrm{P}}{C^2}$$

attains its maximum value on $\partial \mathsf{\Omega }.$

(3). 

Suppose that

$$\frac{C\mathrm{\Delta }C+{|\mathrm{\nabla }C|}^2}{C^2}+1<\frac{2}{e^2\mathit{diam}\mathsf{\Omega }}\mathrm{in}\mathsf{\Omega }.$$

(13)

If in addition one of the following conditions holds

(i) 

$C{,}_k/C\ge 0$ for all $k=1,\dots ,N$ in Ω; or

(ii) 

there exist(s) $i_1,\dots ,i_q(1\le q\le N-1)$ such that $C{,}_{i_1}/C,\dots ,C{,}_{i_q}/C\le 0$ in Ω and the rest of the functions $C{,}_k/C$ are nonnegative in Ω; or

(iii) 

$C{,}_k/C\le 0$ for all $k=1,\dots ,N$ in Ω,

then the function

$$\frac{{\mathrm{P}}_5}{\mathsf{\Gamma }}$$

does not attain (a nonnegative) maximum in $\mathsf{\Omega },$ unless it is constant in $\mathsf{\Omega }.$

Here, $\mathsf{\Gamma }\left(x\right)=1-b{e}^{a{x}_i}>0\mathrm{in}\overline{\mathsf{\Omega }}$ for some $i\in \{1,\dots ,N\}$, where $a,b$ are constants (For more details on the function Γ see [22]).

Proof. 

(1).

Using a calculation similar to that of Lemma 2 and a completion of the square one obtains the following identity:

$$\begin{array}{ccc}\hfill \mathrm{\Delta }\mathrm{P}& \ge & 2{\mathrm{\Delta }}^{2}u{\mathrm{\Delta }}^{3}u+2\mathrm{\Delta }u{\mathrm{\Delta }}^{2}u+2u\mathrm{\Delta }u\hfill \\ & =& -2A{\left({\mathrm{\Delta }}^{2}u\right)}^2+2(1-B)\mathrm{\Delta }u{\mathrm{\Delta }}^{2}u-2Cu{\mathrm{\Delta }}^{2}u+2u\mathrm{\Delta }u\hfill \\ & \ge & (-2A-{(1-B)}^2-C^2){\left({\mathrm{\Delta }}^{2}u\right)}^2-2{\left(\mathrm{\Delta }u\right)}^2-2u^2.\hfill \end{array}$$

Hence, $\mathrm{P}$ satisfies

$$\begin{array}{c}\hfill \mathrm{\Delta }\mathrm{P}+\gamma \mathrm{P}\ge 0\mathrm{in}\mathsf{\Omega },\end{array}$$

where $\gamma =max\{2,{sup}_{\mathsf{\Omega }}(2A+{(1-B)}^2+C^2)\}.$

The conclusion follows from the generalized maximum principle (Theorem 3.1, [22]).

(2).

It is easy to see that

$$\begin{array}{ccc}\hfill \mathrm{\Delta }{\mathrm{P}}_5& =& \mathrm{\Delta }\left(\frac{1}{C^2}\right)\mathrm{P}+\mathrm{\nabla }\left(\frac{1}{C^2}\right)\mathrm{\nabla }\mathrm{P}+\frac{1}{C^2}\mathrm{\Delta }\mathrm{P}\hfill \\ & =& -2\frac{C\mathrm{\Delta }C-{3|\mathrm{\nabla }C|}^2}{C^4}\mathrm{P}-\frac{4}{C^3}\mathrm{\nabla }C\mathrm{\nabla }\mathrm{P}+\frac{1}{C^2}\mathrm{\Delta }\mathrm{P}.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \mathrm{\Delta }{\mathrm{P}}_5+\frac{4}{C}\mathrm{\nabla }C\mathrm{\nabla }{\mathrm{P}}_5=-2\frac{C\mathrm{\Delta }C-{3|\mathrm{\nabla }C|}^2}{C^4}\mathrm{P}+\frac{1}{C^2}\mathrm{\Delta }\mathrm{P}-\frac{{8|\mathrm{\nabla }C|}^2}{C^4}\mathrm{P},\end{array}$$

which gives

$$\begin{array}{c}\hfill \mathrm{\Delta }{\mathrm{P}}_5+\frac{4}{C}\mathrm{\nabla }C\mathrm{\nabla }{\mathrm{P}}_5+\frac{2{\mathrm{P}}_5}{C^2}\left(C\mathrm{\Delta }C+{|\mathrm{\nabla }C|}^2\right)=\frac{1}{C^2}\mathrm{\Delta }\mathrm{P}.\end{array}$$

(14)

Using relation (11) and our calculation from case (1), we easily obtain

$$\begin{array}{c}\hfill \mathrm{\Delta }\mathrm{P}\ge -2({\left({\mathrm{\Delta }}^{2}u\right)}^2+{\left(\mathrm{\Delta }u\right)}^2+u^2)=-2\mathrm{P}\mathrm{in}\mathsf{\Omega }.\end{array}$$

As a consequence, we conclude from (14) that

$$\begin{array}{c}\hfill \mathrm{\Delta }{\mathrm{P}}_5+\frac{4}{C}\mathrm{\nabla }C\mathrm{\nabla }{\mathrm{P}}_5+{\mathrm{P}}_5\left(2\frac{C\mathrm{\Delta }C+{|\mathrm{\nabla }C|}^2}{C^2}+2\right)\ge 0\mathrm{in}\mathsf{\Omega }.\end{array}$$

(15)

Now, we can apply Theorem 3.2, [22] to obtain the result. □

Next, we present a maximum principle that allows us to handle the uniqueness of solutions to (2) under different boundary conditions.

Lemma 5.

Let $u\in C^7\left(\mathsf{\Omega }\right)\cap C^5(\overline{\mathsf{\Omega }}$) be a solution of

$${\mathrm{\Delta }}^{3}u-A{\mathrm{\Delta }}^{2}u+B\mathrm{\Delta }u-Cf\left(u\right)=0\mathrm{in}\mathsf{\Omega },$$

(16)

where the constants $A,B,C\ge 0.$ If $f\in C^1\left(\mathrm{I}\mathrm{R}\right)$ satisfies $f^{\prime }\ge 0$ and $N\le 3.$

Then the function

$$\begin{array}{ccc}\hfill {\mathrm{P}}_6& =& u{,}_{ijk}u{,}_{ijk}+\frac{1}{2}\mathrm{\nabla }u·\mathrm{\nabla }\left({\mathrm{\Delta }}^{2}u\right)-u{,}_{ij}\left(\mathrm{\Delta }u\right){,}_{ij}-\frac{1}{4}\mathrm{\nabla }\left(\mathrm{\Delta }u\right)·\mathrm{\nabla }\left(\mathrm{\Delta }u\right)\hfill \\ & +& \frac{A}{2}u{,}_{ij}u{,}_{ij}-\frac{A}{2}\mathrm{\nabla }u·\mathrm{\nabla }\left(\mathrm{\Delta }u\right)+\frac{B}{4}{|\mathrm{\nabla }u|}^2\hfill \end{array}$$

attains its maximum value on $\partial \mathsf{\Omega }.$

Proof. 

For a proof see Theorem 3, [23]. □

5. Uniqueness Results

We are now in a position to prove the uniqueness results.

The next uniqueness results are direct consequences of our maximum principles and are stated for classical solutions. The first two uniqueness results hold for convex domains only.

Theorem 2.

Suppose that we are under the hypotheses of Lemma 1 and that the curvature K of $\partial \mathsf{\Omega }\in C^{2+\epsilon }$ is positive. Then, there is at most one classical solution to the boundary value problem

$$\left\{\begin{array}{c}{\mathrm{\Delta }}^{3}u-A\left(x\right){\mathrm{\Delta }}^{2}u+B\left(x\right)\mathrm{\Delta }u-C\left(x\right)u=f\left(x\right)\mathrm{in}\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}^2\hfill \\ u=g\left(x\right),\mathrm{\Delta }u=h\left(x\right),{\mathrm{\Delta }}^{2}u=i\left(x\right)\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

(17)

A similar result holds if we are under the hypotheses of Lemma 3 (we also require that the curvature K of $\partial \mathsf{\Omega }$ is positive).

Proof. 

Let $u_1$ and $u_2$ be two solutions of (17) and let $v=u_1-u_2$. The function v satisfies (2) with zero boundary data:

$$v=\mathrm{\Delta }v={\mathrm{\Delta }}^{2}v=0on\partial \mathsf{\Omega }.$$

By introducing normal coordinates in the neighborhood of the boundary, we can write (see [11], p. 46)

$$\mathrm{\Delta }w=\frac{{\partial }^{2}w}{\partial n^2}+\frac{{\partial }^{2}w}{\partial s^2}+K\frac{\partial w}{\partial n}.$$

Hence, we obtain

$${\mathrm{\Delta }}^{2}v=\frac{{\partial }^2\left(\mathrm{\Delta }v\right)}{\partial n^2}+\frac{{\partial }^2\left(\mathrm{\Delta }v\right)}{\partial s^2}+K\frac{\partial \left(\mathrm{\Delta }v\right)}{\partial n}\mathrm{on}\partial \mathsf{\Omega }.$$

(18)

Using the boundary data, we obtain from relation (18) that

$$\frac{{\partial }^2\left(\mathrm{\Delta }v\right)}{\partial n^2}=-K\frac{\partial \left(\mathrm{\Delta }v\right)}{\partial n}\mathrm{on}\partial \mathsf{\Omega }.$$

(19)

Furthermore, since $\mathrm{\Delta }v=0$ on $\partial \mathsf{\Omega }$, we obtain

$${|\mathrm{\nabla }\left(\mathrm{\Delta }v\right)|}^2={\left(\frac{\partial \left(\mathrm{\Delta }v\right)}{\partial n}\right)}^2\mathrm{on}\partial \mathsf{\Omega }.$$

(20)

A computation using (19) and (20) gives

$$\begin{array}{c}\hfill \frac{\partial {\mathrm{P}}_1}{\partial n}=-2K{\left(\frac{\partial \left(\mathrm{\Delta }v\right)}{\partial n}\right)}^2<0\mathrm{on}\partial \mathsf{\Omega },\end{array}$$

(21)

which contradicts the maximum principle of Hopf (Theorem 7, p. 65, [24]), unless ${\mathrm{P}}_1$ is a constant function.

Hence, it follows that the smooth function ${\mathrm{P}}_1$ is a constant function in $\overline{\mathsf{\Omega }}.$

As a consequence, we obtain

$$\frac{\partial {\mathrm{P}}_1}{\partial n}=0\mathrm{on}\partial \mathsf{\Omega },$$

which is contradictory to (21). Hence, ${\mathrm{P}}_1\equiv 0\mathrm{in}\overline{\mathsf{\Omega }}$ which implies that

$${|\mathrm{\nabla }\left(\mathrm{\Delta }v\right)|}^2=0\mathrm{in}\overline{\mathsf{\Omega }}.$$

Furthermore, we obtain $\mathrm{\Delta }v\equiv \mathrm{constant}\mathrm{in}\overline{\mathsf{\Omega }}.$ Since $v=0\mathrm{on}\partial \mathsf{\Omega }$, we finally obtain $v\equiv 0\mathrm{in}\overline{\mathsf{\Omega }}$ and the uniqueness follows. □

The following result is a N-dimensional version of Theorem 1 [14] and Theorem 2.

Theorem 3.

Suppose that we are under the hypotheses of Lemma 2 and that the mean curvature H of $\partial \mathsf{\Omega }\in C^{2+\epsilon }$ is positive. Then, there is at most one classical solution of the boundary value problem (17) in $\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}^N.$

Proof. 

We can suppose without loss of generality that $\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}_{+}^N=\{x\in {\mathrm{I}\mathrm{R}}^n|x_N>0\}.$ As in the proof of Theorem 2, we consider $u_1$ and $u_2$ as two solutions of (17) and let $v=u_1-u_2$.

A computation shows that

$$\begin{array}{ccc}\hfill \frac{\partial {\mathrm{P}}_2}{\partial n}& =& {\mathrm{\Delta }}^{2}v\frac{\partial \left({\mathrm{\Delta }}^{2}v\right)}{\partial n}+B\mathrm{\Delta }v\frac{\partial \left(\mathrm{\Delta }v\right)}{\partial n}+C\left(2\frac{\partial v}{\partial n}\frac{{\partial }^{2}v}{\partial n^2}-\frac{\partial v}{\partial n}\mathrm{\Delta }v-v\frac{\partial \left(\mathrm{\Delta }v\right)}{\partial n}\right)\hfill \\ & & +2\frac{C(2C+NC)}{NB}v\frac{\partial v}{\partial n}\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}$$

Using the boundary conditions and relation (4.68), [11], p. 63, we obtain

$$\frac{\partial {\mathrm{P}}_2}{\partial n}=-2C(N-1)H{\left(\frac{\partial v}{\partial n}\right)}^2<0\mathrm{on}\partial \mathsf{\Omega }.$$

(22)

By (22) we obtain

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial n}\left(\frac{{\mathrm{P}}_2}{\mathsf{\Psi }}\right)& =& \frac{1}{{\mathsf{\Psi }}^2}\left(\frac{\partial {\mathrm{P}}_2}{\partial n}\mathsf{\Psi }-{\mathrm{P}}_2\frac{\partial \mathsf{\Psi }}{\partial n}\right)\hfill \\ \hfill & & =\frac{1}{{\mathsf{\Psi }}^2}{\left(\frac{\partial v}{\partial n}\right)}^2\left(-2C(N-1)H\mathsf{\Psi }-C\frac{\partial \mathsf{\Psi }}{\partial n}\right)\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}$$

(23)

By the maximum principle Lemma 2, for some $\alpha >0$, the function ${\mathrm{P}}_2/\mathsf{\Psi }$ takes its nonnegative maximum on the boundary $\partial \mathsf{\Omega }$ at a point, say $x_0=(x_0^1,\dots ,x_0^N),$ unless, ${\mathrm{P}}_2/\mathsf{\Psi }\equiv \mathrm{constant}\mathrm{in}\overline{\mathsf{\Omega }}.$

We now choose at $x_0$ a principal coordinate system. The outer unit normal at $x_0$ is $n\left(x_0\right)=(0,\dots ,-1).$

Hence, $\partial \mathsf{\Psi }/\partial n=2\alpha x_N>0$ and using (23) we obtain

$$\begin{array}{c}\hfill \frac{\partial }{\partial n}\left(\frac{{\mathrm{P}}_2}{\mathsf{\Psi }}\right)\left(x_0\right)<0,\end{array}$$

which contradicts the generalized maximum principle of Hopf (Theorem 10, p. 73, [24]).

As a consequence, ${\mathrm{P}}_2/\mathsf{\Psi }\equiv \mathrm{constant}=\delta \ge 0\mathrm{in}\overline{\mathsf{\Omega }}.$

If $\delta >0$ again a contradiction occurs at $x_0$ since

$$0<\delta \frac{\partial \mathsf{\Psi }}{\partial n}=\frac{\partial {\mathrm{P}}_2}{\partial n}<0$$

Hence, $\delta =0$ which implies ${\mathrm{P}}_2\equiv 0\mathrm{in}\overline{\mathsf{\Omega }},$ from which follows $v\equiv 0\mathrm{in}\overline{\mathsf{\Omega }}.$

We are left to verify the case ${\mathrm{P}}_2<0,$ i.e.,

$${|\mathrm{\nabla }v|}^2-v\mathrm{\Delta }v<0\mathrm{in}\mathsf{\Omega }.$$

We can now use the same arguments as in the proof of Corollary 10.1, [11], p. 177, to obtain $v\equiv 0$ in $\overline{\mathsf{\Omega }}$. □

Theorem 4.

Suppose that we are under one of hypotheses of Lemma 4. Then, there is at most one classical solution to the boundary value problem (17).

Proof. 

As before, we consider $u_1$ and $u_2$ two solutions of (17) and let $v=u_1-u_2$.

According to Lemma 4

$$\underset{\overline{\mathsf{\Omega }}}{max}{\mathrm{P}}_5=\underset{\partial \mathsf{\Omega }}{max}{\mathrm{P}}_5=0,\underset{\overline{\mathsf{\Omega }}}{max}\frac{\mathrm{P}}{\mathsf{\Psi }}=\underset{\partial \mathsf{\Omega }}{max}\frac{\mathrm{P}}{\mathsf{\Psi }}=0,\mathrm{or}\underset{\overline{\mathsf{\Omega }}}{max}\frac{{\mathrm{P}}_5}{\mathsf{\Gamma }}=\underset{\partial \mathsf{\Omega }}{max}\frac{{\mathrm{P}}_5}{\mathsf{\Gamma }}=0.$$

Hence, $v\equiv 0$ in $\overline{\mathsf{\Omega }}.$ □

The last two uniqueness results are given for a different sixth-order boundary value problem.

Theorem 5.

Suppose that we are under the hypotheses of Lemma 3, where $B=0$ and $\partial \mathsf{\Omega }\in C^1$. Then, there is at most one classical solution to the boundary value problem

$$\left\{\begin{array}{c}{\mathrm{\Delta }}^{3}u-A\left(x\right){\mathrm{\Delta }}^{2}u-C\left(x\right)u=f\left(x\right)\mathrm{in}\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}^2\hfill \\ u=g\left(x\right),\partial u/\partial n=h\left(x\right),{\mathrm{\Delta }}^{2}u=i\left(x\right)\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

(24)

Proof. 

By Lemma 3, the function ${\mathrm{P}}_4$ attains its maximum value on $\partial \mathsf{\Omega }.$ Hence,

$$\underset{\overline{\mathsf{\Omega }}}{max}{\mathrm{P}}_4=\underset{\partial \mathsf{\Omega }}{max}{\mathrm{P}}_4=0,$$

i.e.,

$${|\mathrm{\nabla }v|}^2-v\mathrm{\Delta }v<0\mathrm{in}\mathsf{\Omega }.$$

After integrating the last inequality over $\mathsf{\Omega }$ we obtain

$${\int }_{\mathsf{\Omega }}{|\mathrm{\nabla }v|}^{2}dx<0,$$

which is impossible.

Hence, $v\equiv 0$ in $\overline{\mathsf{\Omega }}$ could be the only solution. □

Remark 1.

It can easily be checked that the result in Theorem 5 can be extended to the semilinear problem

$$\left\{\begin{array}{c}{\mathrm{\Delta }}^{3}u-A\left(x\right){\mathrm{\Delta }}^{2}u-C\left(x\right)\phi \left(u\right)=f\left(x\right)\mathrm{in}\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}^2\hfill \\ u=g\left(x\right),\partial u/\partial n=h\left(x\right),{\mathrm{\Delta }}^{2}u=i\left(x\right)\mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

where φ is a $C^1$ function for which $C{\phi }^{\prime }>0$ in Ω.

Similarly, Theorems 2–4 can be extended as well to the problem

$$\left\{\begin{array}{c}{\mathrm{\Delta }}^{3}u-A\left(x\right){\mathrm{\Delta }}^{2}u+B\left(x\right)\mathrm{\Delta }u-C\left(x\right)\phi \left(u\right)=f\left(x\right)\mathrm{in}\mathsf{\Omega }\subset {\mathrm{I}\mathrm{R}}^2\hfill \\ u=g\left(x\right),\mathrm{\Delta }u=h\left(x\right),{\mathrm{\Delta }}^{2}u=i\left(x\right)\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

Theorem 6.

Suppose that we are under the hypotheses of Lemma 5, where $\partial \mathsf{\Omega }\in C^3$. Consider the boundary value problem

$$\left\{\begin{array}{c}{\mathrm{\Delta }}^{3}u-A\left(x\right){\mathrm{\Delta }}^{2}u+B\mathrm{\Delta }u-C\left(x\right)f\left(u\right)=0\mathrm{in}\mathsf{\Omega }\hfill \\ \partial u/\partial n=g\left(x\right),{\partial }^{2}u/\partial n^2=h\left(x\right),{\partial }^{3}u/\partial n^3=i\left(x\right)\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

(25)

(1). 

If $A\ge 0,B>0$, then (25) has no classical solution except the constant solution.

(2). 

If either $A>0,B\ge 0$ or $A\ge 0,B\ge 0$, then (25) has a unique solution up to an additive constant.

Proof. 

According to Lemma 5, the function ${\mathrm{P}}_6$ attains its maximum value on $\partial \mathsf{\Omega },$ i.e.,

$$\underset{\overline{\mathsf{\Omega }}}{max}{\mathrm{P}}_6=\underset{\partial \mathsf{\Omega }}{max}{\mathrm{P}}_6.$$

Since

$$\frac{{\partial }^{2}u}{\partial n^2}=u{,}_{ij}{n}^i{n}^j,\frac{{\partial }^{3}u}{\partial n^3}=u{,}_{ijk}{n}^i{n}^j{n}^k$$

and $n^i{n}^j{n}^i{n}^j=1,n^i{n}^j{n}^k{n}^i{n}^j{n}^k=1,$ where $n=(n^1,n^2)$ or $n=(n^1,n^2,n^3),$ we obtain

$${\left(\frac{{\partial }^{2}u}{\partial n^2}\right)}^2=u{,}_{ij}u{,}_{ij},{\left(\frac{{\partial }^{3}u}{\partial n^3}\right)}^2=u{,}_{ijk}u{,}_{ijk}\mathrm{on}\partial \mathsf{\Omega }.$$

Hence, by the last relations and the zero boundary data, we obtain

$${\mathrm{P}}_6\le 0\mathrm{in}\mathsf{\Omega }.$$

Integrating by parts the last inequality over $\mathsf{\Omega }$ and using that

$$\begin{array}{ccc}\hfill {\int }_{\mathsf{\Omega }}(u{,}_{ijk}u{,}_{ijk}& +& \frac{1}{2}\mathrm{\nabla }u·\mathrm{\nabla }\left({\mathrm{\Delta }}^{2}u\right)-u{,}_{ij}\left(\mathrm{\Delta }u\right){,}_{ij}-\frac{1}{4}\mathrm{\nabla }\left(\mathrm{\Delta }u\right)·\mathrm{\nabla }\left(\mathrm{\Delta }u\right))dx\hfill \\ & & =\frac{9}{4}{\int }_{\mathsf{\Omega }}{|\mathrm{\nabla }\left(\mathrm{\Delta }u\right)|}^{2}dx\ge 0,\hfill \end{array}$$

$$\begin{array}{c}\hfill -{\int }_{\mathsf{\Omega }}\mathrm{\nabla }u·\mathrm{\nabla }\left(\mathrm{\Delta }u\right)dx={\int }_{\mathsf{\Omega }}{\left(\mathrm{\Delta }u\right)}^{2}dx\ge 0,\end{array}$$

we obtain (case 1) that $u\equiv \mathrm{constant}$ in $\mathsf{\Omega }.$

If we are under the hypotheses of case 2, then it follows that

$$|\mathrm{\nabla }(\mathrm{\Delta }u\left)\right|=0,$$

i.e., u satisfies the Neumann problem

$$\left\{\begin{array}{c}\mathrm{\Delta }u\equiv \mathrm{constant}\mathrm{in}\mathsf{\Omega }\hfill \\ \partial u/\partial n=0\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

Using Theorem 9, [24], p.70, we get the required result. □

Remark 2.

The following example shows that if we do not impose some restrictions on the coefficients $A,B,$ and C, then the uniqueness results presented in this section may be violated.

The boundary value problem

$$\left\{\begin{array}{cc}{\mathrm{\Delta }}^{3}u+\mathrm{\Delta }u+10u=0\hfill & \mathrm{in}\mathsf{\Omega }=(0,\pi )\times (0,\pi )\hfill \\ u=\mathrm{\Delta }u={\mathrm{\Delta }}^{2}u=0\hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

has (at least) two solutions $u_1(x,y)\equiv 0$ and $u_2(x,y)=sinxsiny$ in $\mathsf{\Omega }.$

6. Discussion

In this paper, we treat a sixth-order boundary value problem in detail. We note that there are only a few works concerning such boundary value problems (see [12,13,14]) in the higher-dimensional case, which only address the uniqueness. We present here some new results concerning existence and regularity. Furthermore, using some maximum principles, we extend a classical uniqueness result of Schaefer [14] to the non-constant coefficient case and to the higher dimensional case. Furthermore, our results complete the uniqueness results in [12,13]. The intention of the author is to extend the study to the case of p-Laplacian as well as to a general class of higher-order equations.

7. Conclusions

This paper addresses a sixth-order boundary value problem from the qualitative point of view: existence, regularity and uniqueness. Uniqueness is given for classical solutions and follows as a consequence of maximum principles for a functional which is defined on solutions of the sixth-order equation. We note that the existence and regularity results can be directly generalized to higher-order equations, while our $\mathrm{P}$-functions will not satisfy a maximum principle if they are defined on solutions to a higher-order equation. We mention that there is no method to find such $\mathrm{P}$-functions.