We present a geometric approach to the study of quasilinear elliptic p-Laplacian problems on a ball in using techniques from dynamical systems. These techniques include a study of the invariant manifolds that arise from the union of the solutions to the elliptic PDE in phase space, as well as variational computations on two vector fields tangent to the invariant manifolds. We show that for a certain class of nonlinearities f with subcritical growth relative to the Sobolev critical exponent , there can be at most one such solution satisfying on a ball with Dirichlet boundary conditions.
In this paper, we consider solutions to the quasilinear elliptic equation
for different classes of nonlinear functions . The domain Ω is a ball about the origin of radius R in , , and . We are interested in regular solutions, meaning . The p-Laplacian is defined by
When , (1.2) is the regular uniformly-elliptic Laplacian operator. In general, solutions to (1.1) are considered in the weak sense because they belong to for some . Many of the results on the uniqueness or symmetry properties of (1.1) in the uniformly elliptic case rely on classical elliptic principles such as the maximum principle. These principles do not apply directly in (1.1) when the operator is singular () or degenerate ().
We prove the uniqueness of positive radial solutions to (1.1) on a ball with Dirichlet boundary conditions for a class of nonlinearities f that includes nonnegative functions like and the model sign-changing function , where . The proof is in the spirit of the Clemons–Jones geometric proof of the case () but must address the singularity of the operator , and includes a larger class of nonlinearities than those in .
The nonlinearity satisfies the following conditions:
, and there exists so that and on . If , then on .
The quantity is nonincreasing on and .
There is a value satisfying(1.3)
where is the Sobolev critical exponent, and(1.4)
If , then there is a value with , so that
The requirement (R) says that if f changes signs, then f behaves asymptotically like , , . With these conditions on the nonlinearity established, this paper presents the phase space of solutions to (1.1) on a ball in and proves the following theorem.
In this section we discuss previous results for (1.1) for and , where the domain is typically a ball with Dirichlet boundary conditions.
With the technique of moving parallel planes, Serrin showed in  that if Ω is a smooth bounded domain and u is a positive solution to in Ω, with on and with the outward normal vector constant on , then Ω is necessarily a ball and u is a radial function. The method of moving parallel planes was also used by Gidas, Ni and Nirenberg  to determine that in the ball , if is a positive solution of
with f of class , then u is radially symmetric and decreasing. Part of the power of this result stems from the fact that they make no assumptions on the nonlinear term except that .
In general, existence and uniqueness results for (1.1) require more restrictive conditions on . The prototypical example is the Lane-Emden equation , for . If , then is a version of the Yamabe problem from differential geometry. This particular exponent is a critical threshold for f, as demonstrated by Pohozaev , who determined that for any open, star-shaped domain about the origin, with has a positive solution only if .
The topology of the domain is important, and there may be a positive solution to on a different domain, such as an annulus. Pohozaev proved that positive solutions to must satisfy the Pohozaev identity
If the domain is star-shaped, the right-hand side of (2.2) is always positive. The left-hand side, however, is negative if . By the Sobolev embedding theorem,
is a continuous embedding if , with strict inequality resulting in a compact embedding. The lack of a compact embedding leads to nonexistence of solutions in .
Variational methods often show the existence of minimizers to certain functionals. For example, the critical exponent nonlinearity arises in the general Yamabe problem. For the semilinear case , Brezis and Nirenberg  used the energy functional
to show a solution must exist if λ is smaller than the first eigenvalue of Δ.
When satisfies , , the question of whether f is subcritical (), critical (), or supercritical () may alter not only when a solution exists but whether or not is unique. In the case , Ni and Nussbaum  determined that solutions to with are not necessarily unique in the supercritical case .
Uniqueness of positive solutions to has been addressed by many authors; the first was Coffman  for the subcritical case and . McLeod and Serrin  showed uniqueness results for for certain q, which were generalized by Kwong  to ; the method of Kwong was generalized and simplified by . Other authors who investigated uniqueness of with f subcritical, critical, or supercritical include Kwong and Zhang , who proved uniqueness in a ball by using a Sturm comparison principle.
Clemons and Jones illustrated this last uniqueness result with a geometric approach in  by recasting as a dynamical system. The union of all solution forms a two-dimensional invariant manifold; showing uniqueness of a solution to the Dirichlet equation is interpreted as showing that the rotation of the manifold can be controlled. In this paper, we will use similar geometric methods to prove a uniqueness results for the singular p-Laplacian for a large class of f. These results also extend the geometric proof of  to a larger class of nonlinearities.
2.1 Results for the singular p-Laplacian
We are interested in radially symmetric solutions to (1.1), Damascelli and Pacella ([7, 8]) considered positive solutions to (1.1) on a ball with Dirichlet boundary conditions. They determined that if , f is locally Lipschitz continuous in and either
for , or
there exists a constant and a continuous, positive (except at the origin), nondecreasing function with for all and
then u must be radially symmetric and for . The proof uses a modified moving plane method reminiscent of .
Several authors have examined existence and uniqueness questions for the p-Laplacian equation (1.1) with different choices of nonlinearity f, different domains (typically all of , a ball of radius R, or an annulus), and different boundary conditions (usually Dirichlet); we refer here to [15, 9, 14]. Guedda and Veron  determined criteria for existence of positive solutions to
in a bounded open subset of with Dirichlet boundary conditions. Their result can be seen as an extension of the Brezis and Nirenberg  result. Erbe and Tang  proved uniqueness to (1.1) with for q is subcritical. They also proved uniqueness holds for , with and , if the quantity
is positive for all , where
Recently, existence of solutions has been studied in a geometric framework, notably by Franca ([10, 11, 12]). He used an Emden–Fowler transformation to show the existence or nonexistence of ground states and singular ground states (1.1) for positive solutions to (1.1) in . The function he studied is a Pohozaev function which is related to the Hamiltonian structure that arises in phase space; we will discuss this in Section 4.4.
Adimurthi and Yadava  investigated uniqueness of
in both a ball and an annulus in by using a Pohozaev-type identity. In particular, we note that for the ball with Dirichlet boundary conditions, they established uniqueness with , , .
, with some so that in and in ,
is nonincreasing in ,
is positive for .
The prototypical nonlinearity satisfying (AP1)–(AP3) is , where . With an additional requirement on the growth of f near zero,  show that (1.1) has at most one weak radial solution if by using a variant of the maximum principle and a suitable implicit function theorem.
In this paper, we drop condition (AP3): the quantity
may change signs at some value of , and f may be nonnegative. Moreover, our proof is geometric, and both quantities (2.3) and (2.4) will emerge in a physical interpretation of nonuniqueness as quantities that determine how a particular invariant manifold bends.
3 Properties of the nonlinear function
Let us remark on a few properties of the function defined in (2.3). Under hypothesis (R),
and similarly as . By (F1), it follows that as , and if , as . Together with (F2) and (R), the following remarks on are true:
if , then for ,
if , then .
Conditions (F1) and (F2) are similar to hypotheses in . However, the case allows f to be a nonnegative function, and we do not require (AP3), as the quantity (2.4) may be positive, negative, or zero for different values of u.
Any polynomial of the form
satisfies (F1), (F2) and (R), including the representative example , , from . However, many basic nonlinearities satisfy our requirements without satisfying (AP1)–(AP3), for example
Lane–Emden positive functions ,
sign-changing functions where the slower growth term is superlinear; for example, , where p and n must be chosen to satisfy (1.3),
nonnegative functions like , where , with condition (R) satisfied for .
As we are interested in positive solutions solving the Dirichlet problem, we do not specify for . However, to show existence of solutions to (1.1) using an Emden–Fowler approach, it is often necessary for f to be odd. One way to ensure this condition holds for functions of the form (3.2) is to write f as
As the regular Laplacian operator corresponding to is well studied, we will focus on , when the p-Laplacian is singular. However, the proof in this paper covers the case for f nonnegative; a class of nonlinearities that was not addressed by .
4 The phase space
In the case , Gidas, Ni and Nirenberg  proved that if , solutions to (1.1) must be radially symmetric and monotone decreasing. This result does not extend immediately to nonuniformly elliptic case . The following theorem, a corollary to the work of Damascelli and Pacella ([7, 8]), establishes that any solution to (1.1) with in our class of nonlinearities must be radially symmetric and monotonically decreasing.
Any positive solution u to (1.1) with satisfying (F1), (F2) and (R) is radially symmetric and monotonically decreasing.
If f is nonnegative, then the result is automatic by (DP1). If f changes signs at , then let
where satisfy condition (R). Then is continuous with and for , and for all . As and for , there is some so that is nonincreasing on . As a result, is a nondecreasing function on .
To use (DP2), it remains to check that there is some so that
All terms in the denominator are positive for any . Moreover, if , then implies
By condition (R),
Hence for any there is a so that if , . For ,
Let so that the above inequalities are valid for all . Then
as . Hence all solutions must be radial and monotone decreasing. ∎
4.1 Dynamical system in -coordinates
Radial solutions u to (1.1) can be rewritten in terms of to obtain the following ODE:
with . Radial symmetry implies , and the boundary condition can be written simply as , where R is the radius of Ω. Setting yields a first-order ODE for ω,
and therefore the -system can be written as
This system is undefined when ; by introducing a new independent variable t and parametrizing r as , we blow up the singularity at into an invariant plane . The resulting first-order system is
In phase space, an initial condition at corresponds to the limit of a solution to (4.2)–(4.4) as . Suppose a solution satisfies the boundary condition and has an initial value , where ; for this hypothesis to be satisfied in phase space, a trajectory must have as its limit the point on the u-axis. Each such point is a fixed point of (4.2)–(4.4); linearization about yields
For and , there is one zero eigenvalue, one negative eigenvalue, and one positive eigenvalue. (We will not concern ourselves with (4.5) in the case as this is the well-understood case.) Hence each has a one-dimensional stable manifold , a one-dimensional unstable manifold , and a one-dimensional center manifold .
An eigenvector for the eigenvalue 0 is parallel to the u-axis; by invariant manifold theory, the u-axis is the (global) center manifold for each . The global stable manifold to is the vertical line in the case ; for , the stable manifold is tangent to the vertical vector at . The S-shape of the stable manifold in the plane when is due to the presence of in (4.2)–(4.4). See Figure 1 for a picture illustrating these manifolds at several values of .
Hence the initial condition a plays the role of a parameter, and we examine how behaves for different values of a. The winding behavior of the two-dimensional “center-unstable” manifold (together with the boundary formed by the r-axis)
Any point lying on is part of a solution that tends to , where , as . These solution trajectories, determined by the choice of a, foliate . Any such solution can be denoted by . To be more concise, we will occasionally write the above solution as to mean as .
4.2 Dynamical system in Emden–Fowler coordinates
The Emden–Fowler transformation for (4.1) is obtained using the relation
Let ; in terms of the original solution u, z can be written
In the uniformly elliptic case , (4.7) does not reduce to the same z in the Emden–Fowler coordinate system in the proof by Clemons and Jones in  but leads to an alternate and equally useful set of equations.
The distance r from the origin is again parametrized as to obtain
A lower bound for the Emden–Fowler parameter λ is 0; otherwise, y will diverge in (4.6) as . We will not require λ to be nonnegative, however, and will note the effect that setting has on the dynamics of (4.8)–(4.10) in Section 4.3.
Theorem 2 (Maximal Emden–Fowler parameter)
4.3 under the Emden–Fowler transformation
Let be the Emden–Fowler transformation (4.6) from to , and let . If λ satisfies , then any point on now satisfies
Thus choosing has the effect of “blowing down” the u-axis. As a consequence, it no longer makes sense to parametrize solutions via their limit as . To employ a similar notion, the notation and will mean that the solution obtained from satisfies as .
Assuming the critical exponent inequalities (1.3) are satisfied and , we describe below the behavior of the invariant manifold .
4.3.1 Invariant manifold structure if
We will never need this case in this paper as we require . As it may be interesting in future problems, should one choose , then the limit of as is undefined. However, the manifold derives from the -system and therefore exists independently of λ. For any , is a two-dimensional manifold with boundary. Selecting and defining as
yields a well-defined two-dimensional manifold with boundary in -space.
4.3.2 Invariant manifold structure if
If , then (4.11) has one negative eigenvalue and two positive eigenvalues. Under the Emden–Fowler transformation, is a two-dimensional unstable manifold of the origin.
4.3.3 Invariant manifold structure if
If , then the Emden–Fowler transformation is simply . Hence is identical to , a two-dimensional center-unstable manifold.
4.3.4 Invariant manifold structure if
If , then there is one positive eigenvalue of (4.11) and two negative eigenvalues, . Thus all trajectories on the plane tend to the origin as . In this case, transforms to a two-dimensional stable-unstable manifold composed of the unstable manifold of the origin and the one-dimensional stable of the origin in associated with the eigenvalue λ and tangent to the y-axis at the origin.
4.4 Existence of solutions
At this point, we have not specified any particular λ; the choice we make to demonstrate the existence of solutions is the upper bound . This selection characterizes as described above in Section 4.3.2, and moreover, this choice is ideal as (4.8)–(4.10) simplifies to
which in the invariant plane reduces to
If it were the case that , then this system would be Hamiltonian in the plane with
However, whenever ,
The resulting behavior of in system (4.13)–(4.14) produces a “bowtie” as seen in Figure 3. Existence of solutions to problem (1.1) follows whenever the structure of the stable and unstable manifolds is in the configuration of Figure 3 (c): the stable manifold is trapped inside of the curve while the unstable manifold appears to spiral outwards. It is a result of Franca  that this occurs for a large class of nonlinearities.
Figure 3 illustrates precisely why (1.3) must be satisfied for existence. The different dynamics corresponding to different values of , with , in the -plane are pictured. The switching of roles between the stable and unstable manifolds of as is varied to be below, in and above the inequalities in (1.3).
As Theorem 1 is concerned with uniqueness, rather than existence, we will not explore existence further here.
5 Variational equations
For any time , we define the intersection curve by
For any chosen u with initial condition , let be the truncated intersection curve defined by
These curves lie in the -plane. The curve defined by
which we call a solution trajectory, tends to as and intersects at . Examples of both of these curves are sketched in Figure 4.
5.1 Variational equations
For any , the curve from (5.1) can be parametrized by initial conditions a via
Taking the derivative along with respect to a yields a family of tangent vectors:
In the -plane, the center manifold is parametrized by
In particular, the tangent vectors in (5.2) satisfy
We define two curves in the tangent bundle to as follows: for any point , the tangent vectors from (5.2) form the following curve:
Similarly, for each point along a single solution trajectory , the tangent vectors defined by (5.2) form the curve
Figure 4 illustrates the tangent vectors that define these curves.
5.2 Variational equations under the Emden–Fowler transformation
Recalling that T is the Emden–Fowler transformation, the intersection curve in Emden–Fowler coordinates is . By construction, is a curve lying in in the -plane. As before, this is parametrized by a. Differentiating with respect to initial condition yields
which defines the tangent vectors
Under the Emden–Fowler transformation, the variational equations for the -system are given by
if , then ,
if , then ,
if , then the limit of is undefined.
In cases (1) and (2), the relation implies that as . In case (3), although is undefined in the limit (more precisely, ), the tangent vector field exists independently of λ, and for any , is a well-defined vector field.
5.3 Winding of admissible curves
We define a continuous angle measure so that is on the appropriate branch of , see Figure 5. For , along the invariant line , the first component has by (5.3). Hence along , and therefore (5.11) below is defined for every τ. Thus
(The case is done in .)
The “winding number” of along the intersection curve is defined by
the symbol denotes the greatest integer function. This quantity counts the number of net crossings (with clockwise about the origin crossings positive and counterclockwise about the origin crossings negative) of the -axis in the -plane. See Figure 5 for a demonstration.
We use the word homotopic for curves to refer to the notion of being path-homotopic (i.e. homotopic preserving endpoints) in the punctured plane . The winding number I is then invariant for homotopic curves. Let us consider the piecewise-defined curves
As they form the boundary of the region
there is a piecewise smooth path homotopy between these two curves. Thus the winding number along them must be the same. However, along both pieces and . Thus any winding behavior happens along and and .
With this construction, we can now state a result connecting the algebraic winding number of and the number of zeros of along . The following lemma is similar to [16, Proposition 3.5].
For any trajectory at time , is the exact number of zeros of for .
This is not immediate as the winding number is a lower bound on the number of times . To prove this lemma, therefore, we must show that along , the winding curve can only cross the axis in one direction, namely in a manner clockwise about the origin.
When examining Figures 4–7 and 10, it is important to remember that is differentiation of with respect to time, and not with respect to the initial condition, a. Therefore, it is generally not possible to determine when examining an constant plane.
As and is invariant, then whenever , the relation
implies that any time crosses the -axis, then . Hence the curve must be perpendicular to the -axis at any such crossing. Furthermore, as the sign of and must be the same, then must be increasing in the first and second quadrants, and decreasing in the third and fourth quadrants. Thus if it crosses the -axis with , it must be crossing from the fourth quadrant to the third quadrant, and if it crosses the -axis with , it must be crossing from the second quadrant to the first quadrant. Therefore, the winding is clockwise about the origin and the winding number is equal to the exact number of zeros of . An example of an that follows these guidelines is pictured in Figure 6. ∎
5.4 Two vector fields
Following the methodology in , a normal vector in the dual space to is
We are solely interested in the third component, , whose sign indicates whether the normal vector points forward toward increasing r, or backwards toward the origin. The third component of the normal vector has derivative
Let us now define a second vector field that does not appear in :
Analogous to (5.13), we are particularly interested in the third component
The choice of λ for will depend on cases; for we set the Emden–Fowler parameter λ to be . Omitting in the notation below the dependence on time and the initial condition , we obtain
which is also a linear differential equation.
To derive integral expressions for and , we first show that as , the quantity tends to 0 for all .
The vector component has an integral expression for all values of the Emden–Fowler parameter .
As the choice of the parameter λ can result in Emden–Fowler coordinates which are not necessarily defined in the limit, the proof of Lemma 2 requires a careful examination of u, y, and λ, done in full detail below.
Employing the expression to write , and recalling that and , we obtain
As , , , and (and is independent of λ), the first term vanishes in the limit for all . For the second term, , , and . Hence
independently of the chosen parameter . ∎
In the linear differential equation for in (5.14), let
By Lemma 2, along a given trajectory the normal vector component for that trajectory is given by the integral
As and , this can be written as
After the conversion from y back to u, the Emden–Fowler transformation only manifests itself in (5.15) in the λ-dependence of .
For , observe
The quantity is positive by (1.3); thus . Moreover, , , , and , and consequently
In terms of from hypothesis (F2), the above is equivalent to
For both vector components, once a particular trajectory has been identified, we will frequently suppress the dependence on the initial condition in (5.15) and write to mean .
6 The geometry of uniqueness
For a given radius , let be the initial condition of the first positive solution on the intersection curve , , to vanish on the boundary of :
There fails to be a unique solution to (1.1) if contains a second point with initial condition
6.1 Transversality of the first intersection
By construction we have , as implies that is not the minimal initial condition for . Thus there are three possible configurations: either , or with either or . Therefore the truncated intersection curve must be homotopic to one of the three possible configurations pictured in Figure 7.
By Lemma 1, the corresponding number of zeros for is either 1 (Figure 7 (a) and (b)) or 2 (Figure 7 (c)). Figure 7 (a) depicts a transversal intersection; in Figure 7 (b) and (c) the intersection is tangent with either a winding number of 1, which we refer to as “underrotation” as the manifold appears to fold back on itself, or 2, which we call “overrotation” as the manifold is curling inward. With Lemmas 1 and 2, we prove that must be homotopy equivalent to Figure 7 (a).Figure 7
In the underrotation configuration, the single zero for on occurs when ; thus for all . The following lemma proves that this configuration is impossible.
There is some so that .
By the remarks on following (3.1), the quantity is positive for and negative for , hence
(This statement holds for f nonnegative by setting .) In underrotation, for all ; thus the integrand of (6.3), and consequently , is negative. However, in Emden–Fowler coordinates the underrotation configuration in Figure 7 (b) indicates that
as y is decreasing and by hypothesis. Thus, must vanish before . ∎
As a consequence of Lemma 1, exactly once before ; denote this time by . Let , which must be positive.
If can be chosen to force to be zero at when , then would be explicitly given by
The denominator is (AP3). Unlike , we will not require it to be nonzero or have a particular sign; if is defined, however, then must be nonzero. As (6.4) depends on , and is not easily determined, let
so that and Λ is continuous at even if . An example of the graph of this function is given in Figure 8.
The winding number of the truncated intersection curve must be 1.
Lemma 2 implies that the first solution cannot exhibit overrotation. To prove this lemma, there are three cases to consider:
These three possibilities are highlighted in Figure 8. Case (3) is equivalent to ; since , then remarks (i)–(iii) on imply that . Therefore, for any nonlinearity satisfying (F1), (F2) and (R), lies on the u-axis to right of the vertical asymptote. Consequently, the case f nonnegative falls under Case (1) with .
The Case (1) region, as indicated by Figure 8, satisfies . This fact is the subject of the next lemma.
If , then .
We note that if and only if . By (F2), is nonincreasing on , and so is nondecreasing. Thus for all . By remark (i), for all ; thus
6.1.3 Proof for Case (1)
Whenever is defined and bounded above by , simplifies to the following expression:
At first glance, appears to change signs twice if . If , however, by (F2) the expression is negative for and positive for ; see Figure 9. If , then is positive for and negative for . Hence changes signs exactly once, at ; see Figure 9 (b).
If f is a nonnegative nonlinearity, then is either zero (e.g. for ), or satisfies the same statement as above (e.g. for ).Figure 9
To prove Case (1) is impossible, we consider the behavior of the vector component . Omitting the initial condition , we find
as y is decreasing and by assumption.
Suppose first that f changes signs at . If , then by Lemma 3, satisfies , with if and only if . Hence in (6.6) changes sign once from positive to negative as u decreases through . As a result, the integral expression for ,
is positive, which contradicts (6.7).
Now suppose in the Case (1) region satisfies . Again selecting to be whatever value of λ forces to change signs at . Since and is a monotone increasing function, .
Although the integral definition (6.8) of contains u and , which are independent of λ, we take extra care to calculate if because the dynamics passed through the Emden–Fowler transformation to arrive at (5.15). This step reconciles the two different ways to calculate : the first way is simply , which exists for all and all . However, is also defined by the intersection curve on , which does not exist for in the limit , as described in Section 4.3.4.
By Lemma 2, the integral definition in (6.8) solves (5.14) for any . Let be sufficiently small so that the first zero of occurs at . Then the manifold described in Section 4.3.1 captures the sign-switching behavior at . However, for any , we can just as easily construct so that . Thus
as for all , and and implies must be positive for all . Therefore,
However, as and both change signs from positive to negative at , we conclude that the integrand must always be nonnegative. Hence . To verify that this conclusion contradicts (6.7), we ensure that (6.7) is true for by converting to an expression containing u, ω, , and rather than y and z. The cross product at yields
Thus Case (1) is impossible for sign-changing f.
If is not identically zero, then the nonincreasing behavior of implies
Hence the sign of
is determined by ; consequently, . However, this result contradicts (6.7). Thus Case (1) is impossible.
6.1.4 Proof for Case (2)
This case is the interesting case frequently omitted in the literature, as . We first verify that this region does lie to the left of , as seen in Figure 8. Notice
and therefore exactly one of and is negative. If , then both are positive. Consequently, . As , it follows that ; moreover, as u is monotone decreasing and K is nonincreasing (or nondecreasing as u decreases), the expression
for all .
To rule out overrotation in this case, we use (with ) to write
We can compute directly:
which must be negative as and at .
Thus Case (2) is impossible.
6.1.5 Proof for Case (3) (the asymptote case)
In the asymptote case, we consider
(If , then .)
For this particular region, we write
The last integral is negative, as for and (6.9) is true. As in Case (2), . However, this is a contradiction, as must be zero.
6.2 Proof of Theorem 1
Let ; for each , , see Figure 10. By the Intermediate Value Theorem, for each there is some time such that . Let be the map that sends each to the first time such that . This map is well-defined and continuous, and as I is compact, is compact. Therefore, attains its minimum; let , and let denote a solution trajectory that satisfies with
There is a neighborhood about so that ; thus
as u has a local minimum at . Figure 10 illustrates this for the overrotation case. However, by construction is in fact the first solution on ; in other words, with . By Section 6.1, the first intersection solution must intersect transversally, a contradiction. ∎
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-0940363
Funding statement: This work was supported by the National Science Foundation under grant DMS-0940363.
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