A bifurcation phenomenon for the critical Laplace and $p$-Laplace equation in the ball

In this paper we focus on positive radial solutions for the generalized scalar curvature equation

where $\Delta_{p}u=\textrm{div}(\nabla u|\nabla u|^{p-2})$ denotes the $p$-Laplace operator, $x\in\mathbb{R}^{n}$, $2\,\frac{n}{2+n}\leq p\leq 2$,   $1<p<n$, and $q$ is the Sobolev critical exponent

The function ${\mathcal{K}}:[0,+\infty[{}\to{}]0,+\infty[{}$ is assumed to be $C^{1}$, for simplicity. We are interested in crossing solutions, which means solutions of the problem

In particular, we will show that the existence of radial solutions of (3) depends on the behavior of ${\mathcal{K}}$ in a neighborhood of zero, and on the length of the radius ${\mathcal{R}}$.

Since we exclusively concentrate our study on radial solutions, we can reduce equation (1) to the following singular ordinary differential equation

where “ $\,{}^{\prime}\,$ ” denotes the differentiation with respect to $r=|x|$, and, with a slight abuse of notation, $u(r)=u(x)$.

We say that a solution $u(r)$ of (4) is regular if and only if $\lim_{r\to 0}u(r)=d>0$ for a suitable $d>0$: in this case it will be denoted by $u(r;d)$. It is well known that $u(r;d)$ exists and it is unique for any $d>0$, cf. e.g. [25, 26, 33, 34]. Note that $u^{\prime}(0;d)=0$.

As an alternative, from a standard scaling argument, we can find a counterpart for the eigenvalue equation where we fix ${\mathcal{R}}=1$ for definiteness and we multiply the potential ${\mathcal{K}}$ by a parameter $\lambda$, i.e.

Indeed, if we set

and $\mathscr{K}(s)={\mathcal{K}}(s\lambda^{1/p})$, then $u$ solves equation (4) if and only if $w$ solves

and $w(1)=0$ if and only if $u({\mathcal{R}})=0$, where ${\mathcal{R}}:=\lambda^{1/p}$.

Hence, the number of solutions $u$ of (3) as ${\mathcal{R}}$ varies coincides with the number of solutions $u$ of (5) as $\lambda$ varies.

The scalar curvature equation (1) has been extensively studied in the literature due to its significance in a broad variety of applications, such as Riemannian geometry, astrophysics, quantum mechanic, chemistry, theory of non-Newtonian fluids and elasticity (cf. [21] for more detailed references on application of $p$-Laplace equations and e.g. [7, 8] for application to Riemannian geometry in the $p=2$ case).

In many phenomena, positivity of solutions has a physical relevance.

Non-linear eigenvalue problems similar to (5) are nowadays a classical topic, see e.g. [5] and the more recent [6, 15, 16, 28, 40] where the reaction term ${\mathcal{K}}(r)u^{q-1}$ is replaced by a sum of a linear and a term either critical or supercritical with respect to the Sobolev exponent. We address the interested reader to the introduction of [5, 15, 16] for a discussion of several possible diagrams appearing as different reaction terms are considered.

A key role in our analysis is played by the following hypothesis

$\boldsymbol{({\rm H}_{\ell})}$

There are some constants $A,B,\ell>0$ such that

$${\mathcal{K}}(r)=A+Br^{\ell}+h(r)\qquad\mbox{and}\qquad\lim_{r\to 0}\frac{|h(r)|+r|h^{\prime}(r)|}{r^{\ell}}=0\,.$$

The existence and the multiplicity of the solutions of (3) and (5) depend crucially on $\ell$, which, in literature, is often referred to as the order of flatness of the function ${\mathcal{K}}$ at $r=0$.

Problem (3) subject to condition $\boldsymbol{({\rm H}_{\ell})}$ has been already investigated in the early 1990s by Bianchi-Egnell in [3] and by Lin-Lin in [39], who determined the existence of the critical value $\ell^{*}_{2}=n-2$ in the Laplacian setting $p=2$. In fact, Bianchi-Egnell and Lin-Lin have been able to prove the following result.

In fact Bianchi and Egnell in [3] focused on the $R=1$ case. In particular, following a shooting approach based on ordinary differential equations, they constructed and glued together two regular solutions of (4), one shooting from the zero initial condition and the other shooting from infinity.

The restriction $\ell<n$ has often been adopted in the Laplacian literature to ensure the existence of a solution to problem (3). For instance, it appears in [37, Theorem 0.19], where the scalar curvature ${\mathcal{K}}$ is required to be strictly decreasing in a left neighbourhood of ${\mathcal{R}}$. The upper constraint $\ell<n$ can be also found in the recent work [38], dealing with the $\sigma_{k}$-Nirenberg problem on the standard sphere $\mathbb{S}^{n}$ for $2\leq k<n/2$. Among the very few examples of existence results for the Laplacian scalar curvature problem in the absence of upper bound conditions on $\ell$, we refer to the very recent papers [10, 43], where hypothesis $\boldsymbol{({\rm H}_{\ell})}$ is combined with a (not so easily verifiable) topological global index formula on the critical points of ${\mathcal{K}}$.

Note that in the $p$-Laplace context, the condition $\ell<n$ generalizes to $\ell<\frac{n}{p-1}$, cf. [22].

We emphasize that Theorem A, besides its intrinsic interest, has been a key starting point in proving the existence of Ground States with fast decay, i.e. solutions $u(r)$ positive for any $r>0$ and decaying as $r^{-(n-2)}$ at infinity.

In fact, it can be shown that if ${\mathcal{K}}$ is increasing close to $r=0$ and decreasing close to $r=\infty$ we might expect to find Ground States with fast decay: in the 90s there was a flourishing of papers giving sufficient conditions for existence and non-existence of these solutions. A possible strategy is indeed to combine Theorem A, or similar results, with the use of Kelvin inversion, which transfers the information on regular solutions to fast decay solutions, see e.g. [3, 12]. Roughly speaking, one can expect that if ${\mathcal{K}}$ is steep enough at $0$ (i.e. $\ell<n$) and at infinity ${\mathcal{K}}(r)\sim a+br^{-{\ell}}$ with $\ell<n$ and $a>0$, $b>0$, then there is a Ground State with fast decay, while if these conditions are violated, one can construct a counterexample to Theorem A, see [3, Theorem 0.3]. For a generalization to the $p$-Laplace setting, we also refer to [22], which follows a different strategy since Kelvin inversion is not available in that context.

We think it is worthwhile to point out that if $A=0$ in $\boldsymbol{({\rm H}_{\ell})}$, then the problem becomes easier: roughly speaking, its solutions behave as the solutions of the $A>0$, $\ell$-subcritical case, i.e. (3) admits a radial solution for any ${\mathcal{R}}>0$ (or, equivalently, (5) admits a radial solution for any $\lambda>0$), for any $\ell>0$. Again, combining this result with Kelvin inversion one might obtain the existence of Ground States with fast decay. This idea was extensively used in the 90s in many papers to handle the Laplacian problem, starting, probably, from [35], [9], [44], and then it was developed and adapted to related problems, such as existence of Ground States with fast decay with a prescribed number of sign changes, see e.g. [45] and [33], dealing with the Laplacian and $p$-Laplacian setting, respectively.

The aim of this paper is to improve Theorem A in 3 main directions.

Firstly, we extend the results to the $p$-Laplace context, proving the existence of the generalized flatness order’s threshold

which coincides with $\ell^{*}_{2}=n-2$ of Theorem A for $p=2$. As far as we are aware, these are the first results in this direction, although we have to require the probably technical condition $\frac{2n}{2+n}\leq p\leq 2$. The possibility to remove this restriction will be the object of further investigations.

Secondly, we are able to remove the condition $\ell\leq n$ (i.e. $\ell\leq\frac{n}{p-1}$ in the $p$-Laplace context), by requiring that ${\mathcal{K}}(r)$ is increasing for any $r>0$.

Thirdly, and probably more importantly, we are able to prove the existence of a second solution when $\ell>n-2$ (i.e. $\ell>\ell^{*}_{p}$ in the $p$-Laplace context), thus completing the bifurcation diagram even in the $p=2$ case.

Let us state our assumptions and the main results of the paper.

$\boldsymbol{({\rm H}_{\uparrow}\!\!\!)}$

The function ${\mathcal{K}}(r)$ is increasing for any $r\geq 0$, strictly in some interval.

$\boldsymbol{({\rm W}_{s})}$

The function ${\mathcal{K}}({\rm e}^{t})$ is uniformly continuous in $[0,+\infty[{}$ and there are $\overline{K}>\underline{K}>0$ such that $\underline{K}<{\mathcal{K}}(r)<\overline{K}$ for any $r>0$.

In fact, Theorem 2 holds also if we drop the global assumption $\boldsymbol{({\rm H}_{\uparrow}\!\!\!)}$, but we strengthen the requirement on ${\mathcal{K}}^{\prime}(0)$ by introducing the upper bound on the order of flatness $\ell\leq\frac{n}{p-1}$ at zero, in the spirit of Theorem A. Unfortunately, we need to ask for some further very weak technical conditions on ${\mathcal{K}}$ for $r$ large.

Using the change of variable (6), we can rewrite Theorems 1, 2, and 3 as follows.

Again, according to Theorem 3, we can drop the global condition $\boldsymbol{({\rm H}_{\uparrow}\!\!\!)}$ and get the same result by restricting the interval in which $\ell$ varies and by imposing some further weak asymptotic conditions on ${\mathcal{K}}(r)$ for $r$ large.

In fact via Theorem 3 and Corollary 6 we are also able to extend Theorem A to the case where $\ell=\frac{n}{p-1}$, which was not covered by [3, 39], with the addition of a very weak technical condition which, roughly speaking, is satisfied unless ${\mathcal{K}}(r)$ is subject to wild oscillations for $r$ large, or converges to $0$ for $r$ large.

Moreover, our methods are considerably different from those of Lin-Lin in [39]. Our proofs are based on the Fowler transformation, which discloses the geometrical aspect of our problem by converting the singular ordinary differential equation (4) into an equivalent dynamical system, cf. (11), following the way paved by [32, 31, 30] and later on by [1, 2, 23]. Then, we develop a detailed phase plane analysis, involving invariant manifold theory for non-autonomous systems, energy estimates, comparisons of the non-autonomous planar system with suitable autonomous ones and a Grönwall’s argument.

Let us introduce a change of variables known as Fowler transformation, which allows to transform (4) into a two-dimensional dynamical system. We define

This change of variable is known from the ’30s, see [17], and it has been generalized to the $p$-Laplacian case by Bidaut-Véron [4] and some years later (independently) by Franca, see e.g. [18, 19, 20, 22, 23].
According to (2), we can rewrite (4) as the following dynamical system:

where $K(t)={\mathcal{K}}({\rm e}^{t})$, $q=p^{*}=\frac{np}{n-p}$ as in (2), and “$\cdot$” denotes the differentiation with respect to $t$.

Given the initial data $\tau\in{\mathbb{R}}$ and ${\boldsymbol{Q}}\in{\mathbb{R}}^{2}$, we will denote by

the trajectory of (11) such that $\boldsymbol{\phi}(\tau;\tau,{\boldsymbol{Q}})={\boldsymbol{Q}}$.

We also need to consider the autonomous system obtained by freezing the $t$-dependence of $K(\cdot)$. In particular, fixed $\tau\in{\mathbb{R}}\cup\{\pm\infty\}$, we consider the frozen autonomous system:

where ${\mathcal{K}}$ is assumed to have finite limit, whenever $\tau=\pm\infty$.

If we differentiate the energy $\mathcal{H}(\cdot;\tau)$ of system (15) along a solution $(x(t),y(t))$ of the dynamical system (11), we get

Let us fix $\tau\in{\mathbb{R}}\cup\{\pm\infty\}$ in (15), let $\tau_{0}\in{\mathbb{R}}$ and ${\boldsymbol{Q}}\in{\mathbb{R}}^{2}$; we denote by

the trajectory of (15), such that $\boldsymbol{\phi_{\tau}}(\tau_{0};\tau_{0},{\boldsymbol{Q}})={\boldsymbol{Q}}$.

System (15) exhibits an equilibrium point in $\{x>0\,,y<0\}$, of coordinates

It is also well known (see, among others, [22]) that, for any fixed $\tau\in{\mathbb{R}}\cup\{\pm\infty\}$, system (15) admits a homoclinic orbit (see Figure 3)

recalling that the case $\tau=+\infty$ requires a boundedness restriction on ${\mathcal{K}}$. Taking into account that the flows of (11) and (15) are ruled by their linear part near the origin, we easily deduce that the origin is a saddle-type critical point for (11) and (15). Finally, notice that ${\boldsymbol{E}}(\tau)$ is a centre and it lies in the interior of the region enclosed by $\boldsymbol{\Gamma_{\tau}}$.

According to [23, 27], we also know the exact expression of the homoclinic trajectories $\boldsymbol{\phi^{*}}=(\boldsymbol{x^{*}},\boldsymbol{y^{*}})$ of (15). In particular, the one corresponding to the regular Ground State $u^{*}$ such that $u^{*}(0)=d$ satisfies

where $C(d)>0$ is a computable constant, see [27].
Moreover, fixed $\boldsymbol{P}\in\boldsymbol{\Gamma_{-\infty}}$ there exists a constant $c_{\boldsymbol{P}}>0$ such that

where $\boldsymbol{\widetilde{\Gamma}_{-\infty}^{P}}:=\boldsymbol{\phi_{-\infty}}({}]-\infty,0];0,\boldsymbol{P})$.

Let us now list some immediate consequences of assumption $\boldsymbol{({\rm H}_{\uparrow}\!\!\!)}$.

As an immediate consequence we have the following.

The next part of the section is devoted to explore the dynamical system (11) through an invariant manifold approach. From [23, 29, 31], and [11, §13.4], we know that the existence of the unstable manifold is ensured by the following condition:

$\boldsymbol{({\rm W}_{u})}$

The function $K$ is bounded and uniformly continuous in ${}]-\infty,0]$ and $\displaystyle{{\lim_{t\to-\infty}}K(t)=K(-\infty)\geq 0}$ is finite.

Notice that $\boldsymbol{({\rm W}_{u})}$ is always satisfied when $\boldsymbol{({\rm H}_{\ell})}$ holds.

Correspondingly, the existence of the stable manifold is guaranteed by $\boldsymbol{({\rm W}_{s})}$. The unstable manifold will play a key role in this paper, since we will see via Remark 17 that regular solutions $u(r;d)$ correspond to trajectories of (11) converging to the origin as $t\to-\infty$, i.e. leaving from the unstable manifold.

Let $B(\boldsymbol{0},\delta)$ denote the open ball of radius $\delta$, centered at the origin. Following [31, Theorem 2.1], which is in fact a rewording of [29, Theorem 2.25], and its reformulation in[24, Appendix], we find the next result.

This result may also be obtained by using the simpler approach developed in [11, §13.4], with the exception of the part concerning the uniform continuity of $\boldsymbol{Q^{u}}(\cdot)$ and $\boldsymbol{Q^{s}}(\cdot)$. Actually, [31, Theorem 2.1] ensures that $\boldsymbol{Q^{u}}(\tau)$ and $\boldsymbol{Q^{s}}(\tau)$ are $C^{1}$ if $K(t)$ is $C^{1}$.

We notice that $M^{u}_{\textrm{loc}}(\tau)$ is split by the origin in two connected components. Since we are just interested in positive solutions, we denote by $W^{u}_{\textrm{loc}}(\tau)$ and $W^{u,-}_{\textrm{loc}}(\tau)$ the components lying respectively in $x>0$ and $x<0$. Similarly, $M^{s}_{\textrm{loc}}(\tau)$ is split by the origin in two connected components, say $W^{s}_{\textrm{loc}}(\tau)$ and $W^{s,-}_{\textrm{loc}}(\tau)$, lying in $x>0$ and $x<0$, respectively.
Following again either [29, Theorem 2.25] or [11, §13.4] and in particular Theorem 4.5, we deduce some crucial properties concerning the uniform exponential asymptotic behaviour of the trajectories intersecting the manifolds $W^{u}_{\textrm{loc}}(\tau)$ and $W^{s}_{\textrm{loc}}(\tau)$.

Using the flow of (11), with a standard argument, we pass from the local manifolds $W^{u}_{\textrm{loc}}(\tau_{1})$ and $W^{s}_{\textrm{loc}}(\tau_{2})$, defined for $\tau_{1}\leq 0\leq\tau_{2}$, to the global manifolds $W^{u}(\tau)$ and $W^{s}(\tau)$ defined for any $\tau\in{\mathbb{R}}$. In fact, rephrasing [24, Appendix], we set

We observe that $W^{u}(\tau)$ and $W^{s}(\tau)$, the unstable and stable leaves respectively, are $C^{1}$ immersed manifolds which can be characterized as follows

By construction, if ${\boldsymbol{Q}}\in W^{u}(\tau)$, then $\boldsymbol{\phi}(t;\tau,{\boldsymbol{Q}})\in W^{u}(t)$ for any $t\in{\mathbb{R}}$.

From Theorem 11 and the smooth dependence of the flow of (11) on initial data, we get the smoothness property of the unstable and stable manifold.

Let us denote by $W^{u}(-\infty)$ the unstable manifold $\boldsymbol{\Gamma_{-\infty}}$, defined in (19).
According to [1] and [24, §2.2], the smoothness property of $W^{u}(\tau)$ observed in Remark 13 can be extended to $\tau_{0}=-\infty$.

Let ${\mathcal{L}}$ be a segment transversal to $\boldsymbol{\Gamma_{-\infty}}$, take $\tau\leq-N$ and denote by $\tilde{W}^{u}_{{\mathcal{L}}}(\tau)$ the compact and connected branch of $W^{u}(\tau)$ between the origin and ${\boldsymbol{Q}}_{{\mathcal{L}}}(\tau)$. Analogously, denote by $\tilde{W}^{u}_{{\mathcal{L}}}(-\infty)$ the compact connected branch of $\boldsymbol{\Gamma_{-\infty}}=W^{u}(-\infty)$ between the origin and ${\boldsymbol{Q}}_{{\mathcal{L}}}(-\infty)$.