Countable families of solutions of a limit stationary semilinear fourth-order Cahn–Hilliard equation I. Mountain pass and Lusternik–Schnirel’man patterns in $\mathbb{R}^{N}$

The classic Cahn–Hilliard equation describes the dynamics of pattern formation in phase transition in alloys, glasses, and polymer solutions. When a binary solution is cooled sufficiently, phase separation may occur and then proceed in two ways: either nucleation, in which nuclei of the second phase appear randomly and grow, or, in the so-called spinodal decomposition, the whole solution appears to nucleate at once and then periodic or semi-periodic structures appear. Pattern formation resulting from phase transition has been observed in alloys, glasses, and polymer solutions. Parallelly, these types of equations possesses a great interest in biology and, after certain transformations (see below) as prototype for nerve conduction in the form of a FitzHugh–Nagumo system (cf. [17, 25]).

Cahn–Hilliard equations-type such as (1.1) have been studied by many authors during the last decades. Among some of the works where several aspects related to these equations are [2, 33, 34, 35, 36]. Additionally, one can check the surveys in [14, 20], where necessary aspects of global existence and blow-up of solutions for (1.1) are discussed in sufficient detail.

From the mathematical point of view, the non-stationary equation (1.1) involves a fourth order elliptic operator and it contains a negative viscosity term. In a more general (and applied) setting, the unknown function is a scalar $u=u(x,t)$, $x\in\mathbb{R}^{N}$, $t\in\mathbb{R}_{+}$, and the equation reads

Then, in general, the function $f(u)$ is a polynomial of the order $2p-1$,

In particular, the classic Cahn–Hilliard equation corresponds to the case $p=2$ and

This equation has been extensively studied in the past years but many questions still remain unanswered, especially in relation to multiplicity problems.

Thus, we concentrate on the analysis of multiplicity results for the fourth-order elliptic stationary Cahn–Hilliard equation

Note that (1.4) is not variational in $L^{2}(\mathbb{R}^{N})$, though it is variational in $H^{-1}(\mathbb{R}^{N})$. Hence, multiplying (1.4) by $(-\Delta)^{-1}$, we obtain an elliptic equation with a non-local operator of the form

Here, as customary, $(-\Delta)^{-1}u=v$, if

This yields the following $C^{1}$-functional associated with (1.5):

Solutions of the equation (1.5) are then obtained as critical points of the functional (1.6).

The non-local operator $(-\Delta)^{-1}$ is a positive linear integral operator from $L^{q}(\mathbb{R}^{N})$ to $L^{2}(\mathbb{R}^{N})$, within the Sobolev’s range, i.e., $1\leq q\leq\frac{2N}{N-2}$. Moreover, the operator $(-\Delta)^{-\frac{1}{2}}$ can be correctly defined as the square root of the operator $(-\Delta)^{-1}$ and it will also be referred to as a non-local linear operator.

Also, since the problem is set in $\mathbb{R}^{N}$ we are defining this operator in a class of exponentially decaying functions; see below the details and conditions to have the weak expression of the problem with these exponentially decay solutions.

A similar fourth-order problem was studied in [6], where further references can be found. However, in [6] the stationary equation (1.4) was considered in a bounded smooth domain $\Omega\in\mathbb{R}^{N}$, with homogeneous Navier-type boundary conditions

In particular, it was shown that this problem (1.4), (1.7) admits a countable family of solutions (critical points/values). Moreover, in [19, § 6] for a different variational fourth-order problem in $\mathbb{R}$, it was shown that a wider countable set of countable families of solutions can be expected, where only the first infinite family is the Lusternik–Schnirel’man one.

Hence, due to the analysis performed in [18, 19] we need to study, in a general multidimensional geometry, existence and multiplicity for the elliptic equation (1.4) in a class of functions properly decaying at infinity (in fact exponentially),

Consequently, to get the results obtained here one can proceed following two different approaches.

Firstly, bearing in mind solutions satisfying (1.8), with a fast decay at infinity, one can choose a sufficiently large radius $R>0$ for the ball $B_{R}$ and consider the variational problem for the functional $\mathcal{F}(u)$ denoted by (1.6) in $W_{0}^{1,2}$ and assuming Dirichlet boundary conditions on $S_{R}=\partial B_{R}$. Thus, both spaces $L^{2}(B_{R})$ and $L^{p+1}(B_{R})$ are compactly imbedded into $W_{0}^{1,2}(B_{R})$ in the subcritical Sobolev’s range

In other words,

Note that, here, for the fourth-order elliptic operator in (1.4), $p_{\rm S}=\frac{2N}{N-2}$, with $N\geq 3$, because this operator has the representation

so that, the necessary embedding features are governed by a standard second-order one

The next step is passing to the limit as $R\to+\infty$, by using some uniform (in $R$) bounds on such families of solutions in $B_{R}$. Since the category of the functional subset increases without bound in such a limit, eventually, we then expect to arrive at, at least, a countable family of various critical values/points.

However, here we perform a variational study directly in $\mathbb{R}^{N}$, which was done previously for many fourth-order ODEs and elliptic equations; see [23, 24, 38] as key examples (though those equations, mainly, contain coercive operators, with “non-oscillatory” behaviour at infinity).

Namely, by a linearized analysis we first check that equation (1.4) provides us with a sufficient “amount” of exponentially decaying solutions at infinity. Obviously, in any bounded class of such functions, and in a natural functional viewing, since, loosely speaking, nothing happens at infinity (effectively, the solutions vanish there), the variational problem can be treated as the one in a bounded domain. So that the embedding in (1.10) comes in charge in some sense.

Let us briefly summarize what we obtain here. First, we perform an analysis based on the application of the Mountain Pass Theorem in order to ascertain the existence of one solution and, additionally, the existence of more than one for the equation (1.4). To ascertain the existence of a second solution we use the auxiliary equation

where $u^{*}$ represents a solution of the problem (1.5). Note that, since $u^{*}$ is a solution of (1.5) if $u$ is a solution of (1.13) then, $u+u^{*}$ will be also a solution of (1.5).

Thus, by construction and proving the existence of a solution for the equation (1.13), we find the existence of a second solution for (1.5) applying again a Mountain Pass argument. Indeed, performing this analysis and thanks to the equivalence between (1.4) and (1.5), we finally obtain the existence of at least a second solution for our main equation (1.4).

Thus, we state (proved in Section 3) the following.

As far as we know, to get the existence of more than one solution, this seems to be the best available approach, since, for this type of higher-order PDEs we have a big lack of classical methodology and PDE theory.

In addition, secondly, we apply a L–S-fibering approach to get a countable family of solutions (critical points), though without any detailed information about how they look like.

Therefore we state, and prove in Section 4, the following.

Finally, we apply advanced numerical methods to describe general “geometric” structure of various solutions assuming symmetric for even profiles and anti-symmetric conditions for odd profiles (see below). In particular, we introduce some chaotic patterns for equation (1.4) for $p=3$ and $p=2$, showing some profiles that become very chaotic away from the point of symmetry.

These numerical experiments suggest that the whole set of solutions, even in 1D, is much wider. However, these numerical experiments, together with some analytical approaches and estimates, will be extended and analysed with more detail in [4]. Indeed, it will be proved there that there exists, at least, a countable set of countable families of critical points, in which only the first one can be obtained by the L–S min-max approach.

Also, we observe that performing numerical experiments, shooting smoothly from $x=0$ with $u^{\prime}(0)=u^{\prime\prime}(0)=u^{\prime\prime\prime}(0)=0$ and varying $u(0)$, those chaotic patterns become more periodic when $u(0)$ increases. Indeed, this fact appears to be sooner for $p=3$ than for $p=2$.

Moreover, it should be mentioned that this kind of transition behaviour is seen in similar phase solidification fourth-order equations, such as the Kuromoto-Sivashinsky and Swift-Hohenberg equations [37], [13], as a critical order parameter increases.

Recall again that, in [6], existence and multiplicity results were obtained for the steady-states of the unstable C–H equation of the form

for a real parameter $\gamma$, where the multiplicity essentially depended on this parameter, which affected the category of the functional subset associated with the principle linear operator. The analysis was based on variational methods such as the fibering method, potential operator theory and Lusternik–Schnirel’man category-genus theory, and others, such as homotopy approaches or perturbation theory existence. Specifically, it was obtained that, depending on the parameter $\gamma$, there exists a different number of stationary solutions, i.e.,

• •

  If the parameter $\gamma\leq K\lambda_{1}$, with $K>0$ a positive constant and $\lambda_{1}>0$ the first eigenvalue of the bi-harmonic operator, i.e., $\Delta^{2}\varphi_{1}=\lambda_{1}\varphi_{1}$ with Navier boundary conditions (1.7), then there exists at least one solution for the equation (1.14), and;

• •

  When the parameter is greater than the first eigenvalue of the bi-harmonic equation $\lambda_{1}$, multiplied by the positive constant $K$, then there will not be any solution at all, if one assumes only positive solutions. However, for oscillatory solutions of changing sign the number of possible solutions increases with the value of the parameter $\gamma$. In fact, when the parameter $\gamma$ goes to infinity, one has an arbitrarily large number of distinct solutions.

Furthermore, the so-called Mountain Pass Theorem [9] has been previously applied to find a second solution. Indeed, in [41] the authors analyzed the existence of a second solution for the fourth-order elliptic problem

where $\delta_{a_{i}}$ represents the delta function supported at $a_{i}\in\mathbb{R}^{N}$ and

and with $N\geq 5$, $k>0$, $m\in{\mathbb{N}}$, $\alpha_{i}>0$, $c_{1},c_{2}>0$.

Other methods utilized to solve similar problems to (1.4) such as

might be Hamiltonian Methods or the Strong Maximum Principle. The previous equation can be written by

where $\mu_{1}$ and $\mu_{2}$ are the squares of the roots of the characteristic polynomial

Then, if $\beta\geq\sqrt{2}$ both roots are real and positive and we can write the equation (1.16) as the system

and apply the Strong Maximum Principle having a positive solution

if we assume that $u$ is a homoclinic orbit (see [38, 40] for any further details with $N=1$). Moreover, in this case one can ascertain certain qualitative properties for that homoclinic orbit, such as the existence of a precisely critical point for $u$, a priori estimates and the symmetry of $u$ with respect to that critical point. However, if $0<\beta<\sqrt{2}$, one cannot obtain such a decoupling.

Also, similar fourth-order equations to (1.16) have been analyzed in [23, 24] in the dimensional space $\mathbb{R}^{4}$ and via Hamiltonian methods obtaining the connection between the critical points of the Hamiltonian.

On the other hand, note that problem (1.4) can be written as the following elliptic system:

where $\textstyle{v:=(-\Delta)^{-1}u,}$ which gives a different perspective to the problem in hand. The solutions of the system (1.18) represent the steady-states of a reaction-diffusion system of the form

with a great interest in biology and as prototype for nerve conduction in the form of a FitzHugh–Nagumo system (cf. [17, 25]). Mathematically, we note that system (1.18) is weakly coupled. However, this coupling is non-cooperative, in other words, the coupled terms have different signs. This means that the Maximum Principle is not valid for the system (1.18). For a cooperative system where the Maximum Principle can be applied to a very similar system to (1.18) see [3].

Basically, so far, one can see that, due to the lack of comparison methods, Maximum Principle, and several others classical methods in the analysis of higher-order PDEs. Therefore, in general the methodology is very limited and restricted to very specific examples.

Our results can be applied to other C–H models. For example note that, for the “true” fourth-order semilinear operator, the critical Sobolev’s exponent is different:

Obviously, a critical Sobolev’s range as in (1.20) occurs for a different sixth-order C–H equation

However, if the unstable nonlinear diffusion operator is of fourth order, as in

we again arrive at the “second-order” Sobolev range as in (1.9).

Equations (1.21) and (1.22) can be studied in similar lines, but some aspects become more technical, though not affecting the principal conclusions and results.

The preliminary conclusions presented here formally allow us to consider our equations (1.4) in the whole $\mathbb{R}^{N}$, unlike as in [6] where the problem was assumed to be in a bounded domain $\Omega\subset\mathbb{R}^{N}$.

Indeed, for the functional (1.6) we deal with the integrals over $\mathbb{R}^{N}$ and, actually, with the functional setting over a certain weighted Sobolev space¹¹1This is just a characteristic of a functional class: surely, we cannot use any weighted metric, where any potential approaches are lost., instead of $W_{0}^{2,2}(\Omega)$ as assumed in [6]. Such a functional setting of the problem in $\mathbb{R}^{N}$ is key in what follows. In fact, a proper functional setting assumes certain admissible asymptotic decay of solutions at infinity, which, for (1.4), is governed by the corresponding linearized operator.

Thus, considering (1.4) in the radial geometry, with $u=u(r)$ and $r=|x|\geq 0$, we then obtain

Next, as usual, calculating the admissible decaying asymptotics from (2.1), using a two scale WKBJ-type asymptotics, as a first approximation (sufficient for our purposes), we use an exponentially pattern of the form $u(r)\approx r^{\delta}\,{\mathrm{e}}^{ar}$ (as $r\to\infty$) in (2.1) leading easily to the following characteristic equation:

To be precise, note that the first equation in (2.2) comes from the homogeneity of the leading terms in (2.1). The second equality in (2.2) comes from a similar argument after evaluating the next leading terms on the left-hand side in (2.1). This yields a two-dimensional exponential bundle:

where $C_{1,2}\in\mathbb{R}$ are two arbitrary parameters of this linearized bundle.

Through this asymptotic analysis we shall be able to show some patterns after performing a shooting problem in Section 5.

Let us now perform a preliminary analysis. In this radial geometry, any regular bounded solution of (1.4) must satisfy two boundary conditions at the origin (making the bi-Laplacian non-singular)

Hence, using a standard shooting strategy from $r=+\infty$, algebraically, at least two parameters are needed to satisfy both (2.4).

Looking again at (2.3), where there exist two parameters $C_{1,2}\in\mathbb{R}$, we observe that matching with two symmetry boundary conditions (2.4) yields a well-posed and well-balanced algebraic “2D–2D shooting problem”.

To justify existence of such solutions (critical points), we then need carefully apply different variational techniques. Of course, for the elliptic problem in $\mathbb{R}^{N}$, one needs a more technical and delicate “asymptotic separation” analysis. Namely, one needs to resolve the following “separation” procedure (as is well-known, for the bi-Laplacian, a standard separation of variables is not available):

where $\Delta_{\sigma}$ is the Laplace–Beltrami operator on $S^{N-1}$. Indeed, using the representation

with $R(r)$ obtained above, it then leads to a complicated equations on $Y(\sigma)$. In general, such an asymptotic separation procedure is expected to determine a sufficiently wide infinite-dimensional asymptotic bundle of solutions with an exponential decay at infinity.

However, we do not need such a full and rather technical analysis. We must admit that we still do not know that whether a countable family of L–S critical points are radially symmetric solutions or not. If the former is true, then the above radial analysis is sufficient. In general, using our previous experience, we expect that min-max critical points are not all radially symmetric, but cannot prove that.

Finally, we note that, obviously, this is not the case in 1D. Indeed, in addition to the symmetric (even) profiles satisfying (2.4), there exist others dipole-like/anti-symmetric (odd) solutions such that

We can also check existence of such solutions numerically; see Section 5.

Now, we consider the linear spectral problem for the corresponding linearized non-local equation (1.5), i.e.,

which was analyzed, however, in a bounded domain, in full detailed in [17]. In problem (2.8) $\lambda_{\beta}$ represents the $\beta$th-eigenvalue associated with the eigenfunction $\psi_{\beta}$.

In the following we show and prove several properties of the spectrum of the linear eigenvalue problem (2.8). Subsequently, we will apply them, in particular, in order to get estimations of the category; see section 4.

As usual, we begin checking that the linearized equation (2.8) admits solutions with a proper exponential decay at infinity. Indeed, writing it down in the equivalent form

we have that a proper exponential decay at infinity holds (here, as before, $r=|x|\gg 1$, so we are again restricted to the radial setting of this linear spectral problem).

Moreover, due to the positivity of the operator ${\bf L}=\Delta^{2}+{\rm Id}$ on the left-hand side of (2.9), $\lambda_{\beta}=0$ is not an eigenvalue, so that $\lambda_{\beta}>0$, for any $\beta$. Secondly, owing to [11, Theorem $VI.8$] and the compactness of the inverse integral operator the spectrum might contain either infinitely many isolated real eigenvalues or a finite number of isolated eigenvalues, formed by a monotone sequence of eigenvalues

In other words, for a sufficiently large $\alpha$, the resolvent of the operator ${\bf L}+\alpha{\rm I}$ is compact and, hence, the spectrum is discrete and, of course, real, by symmetry (self-adjoint). Note as well that when the operator is self-adjoint the method shown in [21] can be used to prove that there are infinitely many eigenvalues. Thus, we have the following result:

Moreover, we also find the following simple observation:

Furthermore, the natural space for the eigenfunctions of problem (2.9) is $W_{\rm rad}^{2,2}(\mathbb{R}^{N})$, i.e., the closure of $C_{0}^{\infty}$-functions with respect to the norm

and the associated inner product

Indeed, the space $W_{\rm rad}^{2,2}(\mathbb{R}^{N})$ is a reflexive Banach space.

Under those assumptions we have the following variational expression of the problem (2.9):

Thus, $\psi_{\beta}\in W_{\rm rad}^{2,2}(\mathbb{R}^{N})\setminus\{0\}$ is an eigenfunction of the problem (2.9) associated with the eigenvalue $\lambda_{\beta}$. Moreover, a weak formulation of the problem requires the following result:

Note that, since, for $\lambda_{\beta}>0$ for any $\beta$, we always deal with solutions with exponential decay at infinity (see above), the previous assumptions are, hence, achievable.

To obtain the existence of solutions for the stationary Cahn–Hilliard equation (1.4) we apply the Mountain Pass Theorem to the equation (1.5). Hence, we look for critical points of the functional $\mathcal{F}(u)$ (1.6) which correspond to weak solutions of the equation (1.5), i.e.,

for any $\varphi\in W_{{\rm rad}}^{1,2}(\mathbb{R}^{N})\equiv H_{{\rm rad}}^{1}(\mathbb{R}^{N})$ (or $C_{0}^{\infty}(\mathbb{R}^{N})$) a radial function. However, by elliptic regularity, those weak solutions $u$ are also strong solutions of the equation (1.5); see [10].

Then, we look for the existence of critical points for the functional $\mathcal{F}(u)$ (1.6). As previously proved in [6] this functional is weakly lower semicontinuous and its Fréchet derivative has the expression

such that the directional derivative (Gateaux’s derivative) of the functional (1.6) is the following

Moreover, we know that the functional is $C^{1}$ and the critical points of the functional (1.6) denoted by

are weak solutions in $H_{{\rm rad}}^{1}(\mathbb{R}^{N})$ for the equation (3.10), i.e.,

Thus, $u\in\mathcal{C}$ if and only if

where $\mathcal{F}^{\prime}$ is called the gradient of $\mathcal{F}$ at $u$. Again, by classic elliptic regularity (Schauder’s theory; see [10] for further details) we will then always obtain classical solutions for such equations.

The main ingredient we are applying in getting the existence of a solution is the celebrated Mountain Pass Theorem due to Ambrosetti–Rabinowitz [9]. Before stating the Mountain Pass Theorem, we show a very important and necessary condition in order to satisfy the theorem. This is the so-called Palais–Smale condition.

Let $E$ be a Banach space and $\{u_{n}\}\subset E$ a sequence such that

then $\{u_{n}\}$ is pre-compact, i.e., $\{u_{n}\}$ has a convergence subsequence. In particular,

The (PS) condition is a convenient way of imposing some kind of compactness into the functional $\mathcal{F}$. Indeed, this (PS) condition implies that the set of critical points at the level value $c$

is compact for any $c\in\mathbb{R}$.

Thus, we first show that for our problem the trivial solution $u=0$ is a local minimum.

As a first step we prove that the (PS) condition is satisfied by the functional $\mathcal{F}$ (1.6).