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Countable families of solutions of a limit stationary semilinear fourthorder CahnHilliardtype equation I. Mountain pass and LusternikSchnirel’man patterns in \(\mathbb{R}^{N}\)
 P ÁlvarezCaudevilla^{1, 2}Email authorView ORCID ID profile,
 JD Evans^{3} and
 VA Galaktionov^{3}
https://doi.org/10.1186/s1366101606775
© ÁlvarezCaudevilla et al. 2016
 Received: 16 March 2016
 Accepted: 12 September 2016
 Published: 22 September 2016
Abstract
However, through numerical methods it is shown that the whole set of solutions, even in 1D, is much wider. This suggests that, actually, there exists, at least, a countable set of countable families of solutions, in which only the first one can be obtained by the LS minmax approach.
Keywords
 stationary CahnHilliard equation
 variational setting
 nonunique oscillatory solutions
 countable family of critical points
MSC
 35G20
 35K52
1 Introduction and motivation for main problems
1.1 Models and preliminaries
Paralelly, these types of equations possess a great interest in biology and, after certain transformations (see below) as prototype for nerve conduction in the form of a FitzHughNagumo system which is represented by a noncooperative system (cf. [9, 10]). This is actually a great motivation for equation (1.1) presented here because the direct connection between equation (1.1) and a class of noncooperative systems; see details and further discussion below.
From the mathematical point of view, the nonstationary equation (1.1) involves a fourthorder elliptic operator and it contains a negative viscosity term and that is the reason because we say that (1.1) is a CahnHilliard equationstype. Hence, although both equations have a nonlinearity depending on a secondorder operator here we consider solutions of changing sign.
Many questions still remain unanswered for equations of this type, as well as noncooperative systems, especially in relation to multiplicity problems (see details below and references). Here we show some existence and multiplicity results for equation (1.1), which might be applied for other models such as the classical CahnHilliard equation (1.3). Additionally we have found numerically some surprising chaotic oscillatory patterns.
1.2 Variational approach and main results
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.
However, here we perform a variational study directly in \(\mathbb{R}^{N}\), which was done previously for many fourthorder ODEs and elliptic equations; see [14–16] as key examples (though those equations, mainly, contain coercive operators, with ‘nonoscillatory’ behaviour at infinity).
Namely, by a linearised 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 (1.11) comes in charge in some sense.
Remark 1.1
Let us briefly summarise 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 radial solution and, eventually, through a LusternikSchnirel’man argument the existence of more than one for equation (1.4).
Thus, we state (proved in Section 3) the following.
Theorem 1.1
Suppose \(N\geq2\). Then the nonlocal equation (1.5) possesses at least one solution in the space H, denoted by (1.7), with exponential decay and for exponents p’s in the Sobolev range (1.10).
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 higherorder PDEs, we have a big lack of classical methodology and PDE theory.
In addition, secondly, we apply a LSfibering approach to get a countable family of solutions (critical points), though without any detailed information as regards how they look like. Indeed, we cannot be assured if the solutions obtained using the LS theory are radially symmetric or not.
Therefore we state and prove in Section 4 the following.
Theorem 1.2
Suppose \(N\geq2\). Then there is a countable family of solutions for the nonlocal equation (1.5) of the LS type.
Finally, we apply advanced numerical methods to describe general ‘geometric’ structure of various solutions assuming symmetric for even profiles and antisymmetric 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 in more detail in [19]. 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 LS minmax approach. Note that we have used the numerical analysis in 1D just to show the existence of other families of solutions. Something that has not been proved here analytically.
Also, we observe that performing numerical experiments, shooting smoothly from \(x=0\) with \(u'(0)=u''(0)=u'''(0)=0\) 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 fourthorder equations, such as the KuromotoSivashinsky and SwiftHohenberg equations [20, 21], as the critical order parameter increases.
1.3 Previous related results

If the parameter \(\gamma\leq K \lambda_{1}\), with \(K>0\) a positive constant and \(\lambda_{1}>0\) the first eigenvalue of the biharmonic operator, i.e., \(\Delta^{2} \varphi _{1}=\lambda_{1} \varphi_{1}\) with Navier boundary conditions (1.8), then there exists at least one solution for the equation (1.14).

When the parameter is greater than the first eigenvalue of the biharmonic 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 γ. In fact, when the parameter γ goes to infinity, one has an arbitrarily large number of distinct solutions.
Remark 1.2
Note that the previous distinction in terms of the number of solutions is related to pattern formation of problems such as the KuramotoSivashinsky equation; see [20, 22] for further references and details.
Also, similar fourthorder equations to (1.16) have been analysed in [14, 15] in the dimensional space \(\mathbb{R}^{4}\) and via Hamiltonian methods obtaining the connection between the critical points of the Hamiltonian.
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 higherorder PDEs different approaches must be followed in order to get new results for this type of higherorder PDEs. Therefore, in general the methodology is very limited and restricted to very specific examples.
1.4 Further extensions
Equations (1.21) and (1.22) can be studied along similar lines, but some aspects become more technical, though not affecting the principal conclusions and results.
2 Preliminary results: exponentially decaying patterns in \(\mathbb{R}^{N}\)
2.1 Exponentially decaying patterns in \(\mathbb{R}^{N}\)
The preliminary conclusions presented here formally allow us to consider our equations (1.4) in the whole \(\mathbb{R}^{N}\), unlike as in [1] 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 space,^{1} instead of \(W_{0}^{2,2}(\Omega)\) as assumed in [1]. Such a functional setting of the problem in \(\mathbb{R}^{N}\) is key in the following. In fact, a proper functional setting assumes certain admissible asymptotic decay of solutions at infinity, which, for (1.4), is governed by the corresponding linearised operator.
Through this asymptotic analysis we shall be able to show some patterns after performing a shooting problem in Section 5.
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 wellposed and wellbalanced algebraic ‘2D2D shooting problem’.
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 LS 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 minmax critical points are not all radially symmetric, but cannot prove that.
2.2 Spectral theory in \(\mathbb{R}^{N}\)
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.
Proposition 2.1
The operator L admits a discrete set of eigenvalues that tend to +∞ and there exists at least a solution \(\psi\in W_{\mathrm{rad}}^{1,2}(\mathbb{R}^{N})\) for problem (2.9).
Remarks

According to our analysis above, to get exponentially decaying solutions, the real eigenvalues in (2.8) must satisfyNote that problem (2.9) admits a positive first eigenvalue \(\lambda _{1}\) characterised by the Raileigh quotient$$ \lambda_{\beta}>0 \quad \mbox{for any }\beta. $$(2.10)$$ \lambda_{1}:=\inf_{u\in W_{\mathrm{rad}}^{2,2}(\mathbb{R}^{N})} \frac{ \int _{\mathbb{R} ^{N}} \Delta u^{2}+\int_{\mathbb{R}^{N}} u^{2}}{\int_{\mathbb{R}^{N}} \nabla u^{2}}. $$(2.11)

Furthermore, the strict positivity of eigenvalues, for eigenfunctions with an exponential decay at infinity follows directly from the equalityobtained after multiplying 2.9 by \(\psi_{\beta}\) and integrating in \(L^{2}\).$$ \int\Delta\psi_{\beta}^{2} + \int\psi_{\beta}^{2}= \lambda_{\beta}\int\nabla\psi _{\beta}^{2}, $$(2.12)

Moreover, the corresponding associated family of eigenfunctions \(\{\psi_{\beta}\}\) is a complete orthogonal set in \(W^{2,2}\).
Also, we find the following simple observation:
Proposition 2.2
Proof
3 Mountain pass theorem and existence of at least one solution
In this section, we apply the celebrated mountain pass theorem (cf. [23, 29] for details of this highly cited theorem) to ascertain the existence of a solution for problem (1.4) in \(\mathbb{R}^{N}\).
Recall that, in \(\mathbb{R}^{N}\), with any \(N \ge2\), we are restricted to a class of radially symmetric solutions that belong to the space H denoted by (1.7) and for which we know their exponential decay at infinity. For \(N=1\), we can deal with both even and odd patterns. However, in general, restrictions to both symmetries or not, makes no difference in the variational analysis. Nevertheless, in order to have the compact Sobolev’s embedding (1.12) in the subcritical range (1.10) our results in this section are restricted to \(N\geq2\). Note that for \(N=1\) one could argue to the possibility of getting that compact with some extra conditions in the cone of positive radially decreasing functions, however, here we cannot be assured of the positivity of the solutions.
3.1 Mountain pass theorem to ascertain the existence of a solution for problem (1.4)
The main ingredient we are applying in getting the existence of a solution is the celebrated mountain pass theorem due to AmbrosettiRabinowitz [23]. Before applying the mountain pass theorem, we show a sufficient condition in order to find a critical point via the mountain pass theorem. This is the socalled PalaisSmale condition.
3.2 PalaisSmale condition (PS)
Remark 3.1
Since the functional \(\mathcal{F}\) is \(C^{1}\) it is easily proved that if there exists a minimizing sequence \(\{u_{n}\}\) for the functional \(\mathcal{F}\) weakly convergent in \(H_{{\mathrm{rad}}}^{1}(\mathbb{R}^{N})\) to certain \(u_{0}\in H_{{\mathrm{rad}}}^{1}(\mathbb{R}^{N})\) and such that \(\mathcal{F}'(u_{n}) \to0\) then we can be assured that \(u_{0}\) is a critical point, i.e., \(\mathcal{F}'(u_{0})=0\).
Remark 3.2
As AmbrosettiRabinowitz mentioned, on a heuristic level, the mountain pass theorem says that, if a pair of points in the graph of \(\mathcal{F}\) are separated by a mountain range there must be a mountain pass containing a critical point between them. Also, although the statement of the theorem does not imply it, normally in the applications the origin \(u=0\) is a local minimum for the functional \(\mathcal{F}\). As we will see below that is our case.
Most of the critical points will be maxima or minima. However, we cannot be assured directly that those critical points are global maxima or minima, we shall need to work a bit harder to obtain that.
Thus, we first show that for our problem the trivial solution \(u=0\) is a local minimum.
Lemma 3.1
The functional \(\mathcal{F}(u)\) defined by (1.6) possesses a local minimum at \(u=0\).
Proof
As a first step we prove that the (PS) condition is satisfied by the functional \(\mathcal{F}\) (1.6).
Lemma 3.2
The functional \(\mathcal{F}\) denoted by (1.6) satisfies the PalaisSmale condition.
Proof
Finally, we apply the mountain pass theorem in order to obtain the existence of a solution for equation (1.5). Thus, we state the following result.
Lemma 3.3
 (a)there exist \(\rho, \alpha>0\) such thatwhere \(B(0,\rho)\) represents the ball centered at the origin and of radius \(\rho>0\);$$\mathcal{F}_{\partial B(0,\rho)} \geq\alpha, $$
 (b)
also, there exists \(e\in\mathbf{H} \setminus B(0,\rho)\) such that \(\mathcal{F}(e)\leq0\).
Proof
These results provide us with the existence of at least a solution for the nonlocal equation (1.5) and, hence, the existence of at least a stationary solution for the CahnHilliard equation (1.4).
4 Towards a first countable family of LS critical points
In order to estimate the number of critical points of a functional, we shall apply LusternikSchnirel’man’s (LS) classic theory of calculus of variations; see [1] and Chapters 8, 9, 10 of [29] for further details. Thus, the number of critical points of the functional (1.6) will also depend on the category of a functional subset (see below some details).
Moreover, an estimate of the number of critical points of a functional is at the same time an estimate of the number of eigenvectors of the gradient functional (in Krasnosel’skii’s terms) and, hence, of the number of solutions of the associated nonlinear equation.
Lemma 4.1
Proof
Note that \(\rho(\mathcal{R}_{0})\) measures, at least, a lower bound of the total of number of LS critical points. Moreover, thanks to the spectral theory shown above in Section 2 we have the sufficient spectral information as regards the eigenvalue problem (4.4) to obtain a sharp estimate of the category (4.3).
4.1 LS sequence of critical points
Note that just applying the definition of those critical points (4.8) we have the next result.
Lemma 4.2
(Monotonicity property of the genus)
Proof
Additionally, for this functional we show the following particular result (see Chapter 9 of [29] for any further details), which provides us with a countable family of critical points for the functional \(\mathcal{F}\) (1.6) in the spirit of the mountain pass theorem.
Theorem 4.1
Let \(\mathcal{F} \in C^{1}(\mathbf{H},\mathbb{R})\) be the functional defined by (1.6) with p an odd number, \(C^{1}\) and \(\mathcal{F}(0)=0\), such that the conditions of the mountain pass theorem proved in Lemma 3.3 are satisfied. Then the functional \(\mathcal{F}\) possesses a countable number of critical values.
Remark 4.1
Thus, we find a countable family of critical points of the functional \(\mathcal{F}\) defined by (4.8) such that \(c_{\beta}=\mathcal{F}(u_{\beta})\) with \(u_{\beta}\) a weak solution of problem (1.5). However, we cannot be assured how many exactly since so far we only acknowledge the existence of a solution obtained through the mountain pass arguments performed in the previous section.
5 Numerical analysis in 1D and in the radial geometry
Using the farfield behaviour (5.2), a 2D shooting problem may be formulated where the parameters \(k_{1,2}\) are determined by satisfying the symmetry condition (2.4) at the origin for the even profiles and antisymmetry condition (2.7) for the odd profiles. Matlab’s ode15s solver is used with tight error tolerances (\(\mathrm{RelTol}=\mathrm{AbsTol}=10^{13}\)).
We complete this numerical introduction with illustration of a different class of solutions to (5.1). We may perform numerical experiments, shooting smoothly from \(x=0\) with \(u'(0)=u''(0)=u'''(0)=0\) and varying \(u(0)\).
This emergence appears sooner for \(p=3\) than \(p=2\) when increasing the size of \(u(0)\). Such a transition behaviour is seen in similar phase solidification fourthorder equations, such as the KuromotoSivashinsky and SwiftHohenberg equations [20, 21], as a critical order parameter increases.
This is just a characteristic of a functional class: surely, we cannot use any weighted metric, where any potential approaches are lost.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees by their valuable suggestions, helpful comments which further improved the content and presentation of the paper. Also, this work has been partially supported by the Ministry of Economy and Competitiveness of Spain under research project MTM201233258. The first author has also been supported by the Ramón y Cajal project RYC201415284 of the Ministry of Economy and Competitiveness.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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