Sharp Onset and Exact First-Mode Classification for the Planar Lane–Emden Robin Problem

1. Introduction and Result

On the unit disk $B\subset {\mathbb{R}}^2$ consider

$$-\Delta u=u^p\mathrm{in}B,u>0,{\partial }_{\nu }u+\beta u=0\mathrm{on}\partial B,p>1,\beta >0.$$

(1)

Zeraoulia [3] established two simple first-angular bifurcation points for $p\ge 8$. That is an existence result for two points. Our contribution identifies the unique sharp onset, proves that there are exactly two simple first-mode zeros for every exponent above it, and describes the single quadratic contact at onset and the square-root separation of the two points. Thus the work is a structural classification of the radial branch, beyond a numerical improvement of the sufficient bound 8. It does not claim uniqueness among all positive solutions below the onset. Related uniqueness results near exponent one are discussed in [1].

2. Radial Parametrization and the Phase Equation

Write $P=p-1$. Let $w=w_p$ be the positive solution, up to its first zero $S_p$, of

$$w^{\prime \prime }+s^{-1}{w}^{\prime }+w^p=0,w\left(0\right)=1,w^{\prime }\left(0\right)=0.$$

Set $q=-s{w}^{\prime }/w$, $z=s^2{w}^P$. We first justify the parametrization used throughout.

Lemma 1. 

The first zero $S_p$ is finite, $w^{\prime }<0$ on $(0,S_p]$, and $q:(0,S_p)\to (0,\infty )$ is a strictly increasing bijection. For every $\beta >0$ there is exactly one positive radial solution, namely

$$U_{\beta }\left(r\right)={\rho }^{2/P}w\left(\rho r\right),\rho =q^{-1}\left(\beta \right).$$

Proof. 

Integration of ${\left(s{w}^{\prime }\right)}^{\prime }=-s{w}^p$ gives

$$-s{w}^{\prime }\left(s\right)={\int }_0^{s}tw{\left(t\right)}^{p}dt>0.$$

Direct differentiation gives $s{q}^{\prime }=q^2+z>0$. If w remained positive for all s, then $q^{\prime }>q^2/s$ would yield $1/q\left(s\right)\le 1/q\left(s_0\right)-log(s/s_0)$, impossible for large s. Thus $S_p<\infty$, and the integral formula gives $w^{\prime }\left(S_p\right)<0$. Near zero $w\left(s\right)=1-s^2/4+O\left(s^4\right)$, whence $q\left(s\right)\to 0$; as $s\uparrow S_p$, we have $w\left(s\right)\to 0$ and $w^{\prime }\left(S_p\right)<0$, whence $q\left(s\right)\to \infty$.

Every positive radial solution with central value $A>0$ equals $Aw\left(A^{P/2}r\right)$ by scaling and uniqueness of the radial initial-value problem. Writing $\rho =A^{P/2}$, positivity up to the boundary requires $\rho <S_p$, and the Robin condition becomes $q\left(\rho \right)=\beta$. Strict monotonicity proves existence and uniqueness. □

With a dot denoting differentiation with respect to $\tau =logs$, we have

$$\dot{q}=q^2+z,\dot{z}=z(2-Pq).$$

(2)

Regarding $z=Z_P\left(q\right)$ as a function of q gives

$$Z_P^{\prime }\left(q\right)=f_P(q,Z_P\left(q\right)),f_P(q,z)={\displaystyle \frac{z(2-Pq)}{q^2+z}},Z_P\left(q\right)=2q-{\displaystyle \frac{P+1}{2}}{q}^2+O\left(q^3\right).$$

(3)

The expansion follows by substituting $w\left(s\right)=1-s^2/4+p{s}^4/64+O\left(s^6\right)$ and inverting $q\left(s\right)$ near zero. It is uniform when P ranges over a compact interval. The integral form of the radial initial-value problem, followed by standard ODE dependence away from zero, shows that $Z_P\left(q\right)$ depends smoothly on $(P,q)$ for $q>0$.

Lemma 2. 

For $q>0$, $0<Z_P\left(q\right)<2q$. If $2-(P+1)q>0$, then

$$q\{2-(P+1)q\}<Z_P\left(q\right)<2q.$$

Proof. 

Put $H=Z_P/q$. The phase system gives $\dot{H}=H\{2-(P+1)q-H\}$. Initially $H=2-(P+1)q/2+O\left(q^2\right)<2$; a first contact at $H=2$ would have $\dot{H}=-2(P+1)q<0$, which is impossible. Also,

$$q{H}^{\prime }={\displaystyle \frac{H\{2-(P+1)q-H\}}{q+H}}.$$

Initially $H>2-(P+1)q$. At a first contact with this lower comparison curve, $H^{\prime }=0$ while the curve’s derivative equals $-(P+1)<0$; again crossing from above is impossible. □

The first-angular translation Jacobi field has boundary residual of the same sign as

$$F_P\left(q\right)=Z_P\left(q\right)-q(1-q).$$

(4)

We derive the equivalence between $F_P\left(q\right)=0$ and a first-angular zero mode in Section 4.

Theorem 1. 

There is a unique $p_{*}\in (6.39,6.40)$ with the following properties.

1. 

If $1<p<p_{*}$, the linearization at $U_{\beta }$ has trivial kernel for every $\beta >0$.

2. 

If $p=p_{*}$, the first angular eigenvalue vanishes only at

$${\beta }_{*}={\displaystyle \frac{1+\sqrt{1-4/(p_{*}-1)}}{2}}.$$

It touches zero quadratically there. The higher angular and radial sectors have no zero mode at this point.

3. 

If $p>p_{*}$, the first angular eigenvalue has exactly two simple zeros ${\beta }_{p,-}<{\beta }_{p,+}$. With $P=p-1$ and

$$a_P={\displaystyle \frac{1-\sqrt{1-4/P}}{2}},b_P={\displaystyle \frac{1+\sqrt{1-4/P}}{2}},$$

they satisfy $a_P<{\beta }_{p,-}<b_P<{\beta }_{p,+}<1$. Each is a transverse local bifurcation point of positive nonradial solutions of (1).

4. 

As $p\downarrow p_{*}$, with $P_{*}=p_{*}-1$,

$${\beta }_{p,\pm }={\beta }_{*}\pm C_{*}\sqrt{p-p_{*}}+O(p-p_{*}),C_{*}^2={\displaystyle \frac{-2{\partial }_P{F}_{P_{*}}\left({\beta }_{*}\right)}{P_{*}(2{\beta }_{*}-1)}}>0.$$

(5)

3. Zero Geometry and Parameter Ordering

The following proof uses only the scalar equation (3) and positivity of $Z_P$.

Lemma 3. 

For every fixed $q>0$, $P\mapsto Z_P\left(q\right)$ is strictly decreasing. For $P>4$, put $G\left(P\right)=F_P\left(b_P\right)=Z_P\left(b_P\right)-1/P$. Then

$$\begin{array}{cc}\hfill G\left(P\right)>0& ⟺F_P\left(q\right)>0(0<q<1),\hfill \\ \hfill G\left(P\right)=0& ⟺F_P\mathit{has}\mathit{exactly}\mathit{one}\mathit{zero},\mathit{at}{b}_P,\hfill \\ \hfill G\left(P\right)<0& ⟺F_P\mathit{has}\mathit{exactly}\mathit{two}\mathit{simple}\mathit{zeros}\mathit{in}(0,1).\hfill \end{array}$$

When $G\left(P\right)<0$, the zeros lie in $(a_P,b_P)$ and $(b_P,1)$, respectively.

Proof. 

If $P_2>P_1$, the expansion (3) makes $Z_{P_2}<Z_{P_1}$ near zero. At a hypothetical first contact $q_0>0$, their derivative difference equals

$$-{\displaystyle \frac{(P_2-P_1)q_{0}Z\left(q_0\right)}{q_0^2+Z\left(q_0\right)}}<0,$$

which precludes contact from below. Differentiation of (3) also makes strictness explicit. The sensitivity $V={\partial }_P{Z}_P$ satisfies

$$V^{\prime }={\displaystyle \frac{(2-Pq)q^2}{{(q^2+Z_P)}^2}}V-{\displaystyle \frac{q{Z}_P}{q^2+Z_P}},V\left(q\right)=-\frac{1}{2}{q}^2+O\left(q^3\right).$$

Variation of constants from a sufficiently small positive starting point, followed by a limit to zero, shows $V\left(q\right)<0$ for every $q>0$.

At any zero of $F_P$, one has $0<q<1$ and $q^2+Z_P\left(q\right)=q$. Therefore

$$F_P^{\prime }\left(q\right)=1-Pq(1-q)\mathrm{whenever}{F}_P\left(q\right)=0.$$

(6)

At a tangential zero, differentiation of (3) yields

$$F_P^{\prime \prime }\left(q\right)=P(2q-1).$$

(7)

Near zero $F_P\left(q\right)=q+O\left(q^2\right)>0$, and $F_P\left(1\right)=Z_P\left(1\right)>0$. By (6), zeros in $(0,a_P)$ can only cross upwards, zeros in $(a_P,b_P)$ only downwards, and zeros in $(b_P,1)$ only upwards. There is no zero up to and including $a_P$: a first zero there could not be reached from positive values; equality at $a_P$ is also impossible because (7) would give a strict local maximum at level zero. There is at most one downward crossing before $b_P$ and one upward crossing after it. At $b_P$, a zero is a strict local minimum by (7). The three alternatives now follow by continuity and the endpoint signs. □

Proposition 1. 

There is a unique $P_{*}\in (5.39,5.40)$ such that $F_P$ is strictly positive on $(0,1)$ for $P<P_{*}$, has a single quadratic zero at $b_{P_{*}}$ for $P=P_{*}$, and has exactly two simple zeros for $P>P_{*}$. These two zeros obey (5).

Proof. 

Section 6 verifies $G\left(5.39\right)>0>G\left(5.40\right)$. Lemma 3 and strict monotonicity in P imply that the set of parameters for which $F_P$ takes a negative value is a nonempty upper interval. Its left endpoint $P_{*}$ belongs to $(5.39,5.40)$.

At $P_{*}$ the function is nonnegative on $(0,1)$ and must attain zero in the interior. Indeed, take negative points for $P\downarrow P_{*}$: the uniform expansion at $q=0$ excludes convergence to zero, and $Z_P\left(1\right)>0$ excludes convergence to one. An interior zero is a minimum, so (6) and (7) force it to be $b_{P_{*}}$. It is the unique zero and is a nondegenerate quadratic minimum. For $P<P_{*}$, $F_P>0$ everywhere by strict parameter ordering; for $P>P_{*}$ it is negative at the fixed point $q=b_{P_{*}}$, so Lemma 3 gives exactly two simple zeros.

At $(P_{*},{\beta }_{*})=(P_{*},b_{P_{*}})$ we have $F=F_q=0$, $F_{qq}=P_{*}(2{\beta }_{*}-1)>0$ and $F_P={\partial }_P{Z}_{P_{*}}\left({\beta }_{*}\right)<0$. Taylor’s formula gives $F(P,q)=F_P(P-P_{*})+\frac{1}{2}{F}_{qq}{(q-{\beta }_{*})}^2+O({(P-P_{*})}^2+|P-P_{*}\left|\right|q-{\beta }_{*}|+|q-{\beta }_{*}{|}^3)$. Solving at the two zeros first gives $|q-{\beta }_{*}|=O\left(\sqrt{P-P_{*}}\right)$ and then (5). □

4. Spectral Sign and Order of Contact

Fix p and write $\rho =q^{-1}\left(\beta \right)$. In angular sector $k\ge 1$, the linearized quadratic form on regular radial factors is

$$Q_k\left[\psi \right]={\int }_0^1\left\{r|{\psi }^{\prime }{|}^2+{\displaystyle \frac{k^2}{r}}{\psi }^2-pr{U}_{\beta }^{p-1}{\psi }^2\right\}dr+\beta \psi {\left(1\right)}^2.$$

Differentiating the profile equation shows that $\xi \left(r\right)=-w^{\prime }\left(\rho r\right)>0$ solves $-{\left(r{\xi }^{\prime }\right)}^{\prime }+r^{-1}\xi -pr{U}_{\beta }^{p-1}\xi =0$ and $\xi \left(r\right)\sim \rho r/2$ as $r\downarrow 0$. Writing $\psi =\xi \eta$ and integrating by parts gives

$$Q_1\left[\psi \right]={\int }_0^{1}r{\xi }^2{\left|{\eta }^{\prime }\right|}^{2}dr+\left({\displaystyle \frac{{\xi }^{\prime }\left(1\right)}{\xi \left(1\right)}}+\beta \right)\psi {\left(1\right)}^2.$$

(8)

The origin contribution vanishes for smooth regular factors and then by density. The profile equation and $\beta =q\left(\rho \right)$ give

$${\xi }^{\prime }\left(1\right)+\beta \xi \left(1\right)={\displaystyle \frac{w\left(\rho \right)}{\rho }}\{Z_P\left(\beta \right)-\beta (1-\beta )\}={\displaystyle \frac{w\left(\rho \right)}{\rho }}{F}_P\left(\beta \right).$$

(9)

For $F_P>0$, (8) is positive definite; for $F_P=0$ its kernel is spanned by $\xi$; for $F_P<0$, testing on $\xi$ gives a negative Rayleigh quotient. Thus the principal $k=1$ eigenvalue has precisely the sign of $F_P$. Moreover $Q_k=Q_1+(k^2-1){\int }_0^1{\psi }^2/rdr$, so all $k\ge 2$ sectors are positive whenever $F_P\ge 0$.

The radial sector cannot have a zero mode at any $\beta >0$. Every regular radial solution of the linearized interior equation is a multiple of ${\partial }_{\rho }\left[{\rho }^{2/P}w\left(\rho r\right)\right]$, by initial-value uniqueness at the origin. If $U_{\rho }\left(r\right)={\rho }^{2/P}w\left(\rho r\right)$, its Robin residual is $U_{\rho }^{\prime }\left(1\right)+\beta U_{\rho }\left(1\right)=U_{\rho }\left(1\right)\{\beta -q\left(\rho \right)\}$. Differentiating with $\beta$ held fixed at $\beta =q\left(\rho \right)$ yields

$${\left({\partial }_{\rho }{U}_{\rho }\right)}^{\prime }\left(1\right)+\beta {\partial }_{\rho }{U}_{\rho }\left(1\right)=-q^{\prime }\left(\rho \right)U_{\rho }\left(1\right)\ne 0.$$

To determine the order of eigenvalue contact, normalize the regular $k=1$ eigenfunction as $y(r;\rho ,\lambda )=rv(r;\rho ,\lambda )$, $v\left(0\right)=1$. Its Volterra equation is

$$v\left(r\right)=1-{\int }_0^r{t}^{-3}{\int }_0^t{s}^3\{p{U}_{\rho }{\left(s\right)}^{p-1}+\lambda \}v\left(s\right)dsdt.$$

Set $D(\rho ,\lambda )=y^{\prime }(1;\rho ,\lambda )+q\left(\rho \right)y(1;\rho ,\lambda )$. As $y(r;\rho ,0)=2\xi \left(r\right)/\rho$, we have

$$D(\rho ,0)={\displaystyle \frac{2w\left(\rho \right)}{{\rho }^2}}{F}_P\left(q\left(\rho \right)\right).$$

(10)

At a zero of $F_P$, differentiation of the eigenvalue equation in $\lambda$ and integration against y give

$$-y\left(1\right)D_{\lambda }(\rho ,0)={\int }_0^{1}ry{\left(r\right)}^{2}dr>0.$$

(11)

Indeed, with $\dot{y}={\partial }_{\lambda }y$, subtraction gives ${\left[r(y{\dot{y}}^{\prime }-y^{\prime }\dot{y})\right]}^{\prime }=-r{y}^2$; the origin term vanishes. The implicit eigenvalue ${\lambda }_1\left(\beta \right)$ therefore has the same order of vanishing as $F_P\left(\beta \right)$, with positive leading factor. At $p_{*}$ its contact is quadratic; for $p>p_{*}$ both crossings are transverse.

5. Local Symmetry Breaking

At each simple first-mode zero, the higher modes are positive and the radial sector has no zero mode. In the subspace even under $\theta \mapsto -\theta$, the kernel is spanned by $\Phi (r,\theta )=\xi \left(r\right)cos\theta$. Fix $0<\alpha <1$ and let $X=C_{\mathrm{even}}^{2,\alpha }\left(\overline{B}\right)$ and $Y=C_{\mathrm{even}}^{0,\alpha }\left(\overline{B}\right)\times C_{\mathrm{even}}^{1,\alpha }(\partial B)$. Put $g\left(t\right)={\left|t\right|}^{p-1}t$ and

$$\mathcal{G}(\beta ,v)=\left(-\Delta (U_{\beta }+v)-g(U_{\beta }+v),{\partial }_{\nu }(U_{\beta }+v)+\beta (U_{\beta }+v)\right).$$

Here $\mathcal{G}(\beta ,0)=0$. Since $p>p_{*}>6$, this map is $C^2$. Its derivative $A={\mathcal{G}}_v({\beta }_0,0)$ is Fredholm of index zero: the Robin Laplacian with an added positive constant is an isomorphism onto Y, and the potential term is compact. Green’s identity identifies its range as the kernel of

$$\ell (f,h)={\int }_B\Phi fdx+{\int }_{\partial B}\Phi hd\sigma .$$

Differentiating the normalized eigenpair of $L_{\beta }$ at ${\beta }_0$, including its boundary condition, and pairing with $\Phi$ gives

$$\ell \left({\mathcal{G}}_{\beta v}({\beta }_0,0)\Phi \right)={\lambda }_1^{\prime }\left({\beta }_0\right){\int }_B{\Phi }^{2}dx\ne 0.$$

The Crandall–Rabinowitz theorem [2] provides a smooth local curve with $\beta =\beta \left(s\right)$ and $v\left(s\right)=s\Phi +o\left(s\right)$ of nonradial solutions. Positivity follows for sufficiently small s from the positive minimum of $U_{{\beta }_0}$ and continuity in $C^0$.

Proof of Theorem 1. 

Take $p_{*}=P_{*}+1$ from Proposition 1. The spectral conclusions follow from Section 4; when $p<p_{*}$, $F_P>0$ also makes all angular forms positive. The contact and crossing statements follow from (10)–(11). The bifurcation curves follow from the preceding argument, and the asymptotic formula is (5). □

6. A Rational Interval Certificate

We describe an independently reproducible certificate. The accompanying Python script performs all evolving endpoint arithmetic on integers scaled by ${10}^{20}$, rounding each multiplication and division outward. No floating-point operation enters the sign checks.

Set $q_0={10}^{-4}$ and $h={10}^{-6}$. For $H=Z_P/q$, the equation implies $q{H}^{\prime }=H[2-(P+1)q-H]/(q+H)$. Lemma 2 gives the analytic initial enclosure

$$q_0\{2-(P+1)q_0\}<Z_P\left(q_0\right)<2q_0.$$

(12)

On $0<q<0.7546$ for $P=5.39,5.40$, $|Z_P^{\prime }\left(q\right)|<3$. Given an enclosure $[{\ell }_i,u_i]$ at $q_i$, the solution throughout a step of length $h_i\le h$ lies in

$$R_i=[q_i,q_i+h_i]\times [{\ell }_i-3h,u_i+3h].$$

The code verifies that its second coordinate stays positive and that

$${\partial }_q{f}_P(q,z)=-{\displaystyle \frac{z(Pz+4q-P{q}^2)}{{(q^2+z)}^2}}<0\mathrm{on}\mathrm{every}{R}_i.$$

For fixed q, $f_P(q,z)$ is monotone in z, with direction determined by $2-Pq$. Consequently the minimum on $R_i$ is the smaller of $f_P(q_i+h_i,{\ell }_i-3h)$ and $f_P(q_i+h_i,u_i+3h)$, and the maximum is the larger of the corresponding two values at $q_i$. We update ${\ell }_{i+1}={\ell }_i+h_i{min}_{R_i}{f}_P$ and $u_{i+1}=u_i+h_i{max}_{R_i}{f}_P$, with outward rounding. The endpoint $b_P$ is enclosed by integer square root, and the final enclosure is widened by 3 times its endpoint uncertainty.

The resulting conservative rational bounds are

P	enclosure for $G\left(P\right)$	certified sign
$5.39$	$(0.000144,0.000153)$	positive
$5.40$	$(-0.001011,-0.001002)$	negative

The inequalities in the table are deliberately weaker than the program’s exact rational endpoints. They suffice to establish $6.39<p_{*}<6.40$ without interpreting a plotted or floating-point root as a proof.

The accompanying script certifiedthreshold.py uses only the Python standard library (version 3.8 or later). Run python3 certifiedthreshold.py to reproduce the certified bounds; READMEthreshold.md documents the checks and expected output.

Remark 1. 

This threshold classifies thefirst angular mode; the result does not classify possible higher-mode degeneracies for $p>p_{*}$. The local bifurcation assertion does not place a nonradial solution at the exact limiting parameter. The bound on $p_{*}$ is computer assisted, while the existence, uniqueness, crossing count and fold argument are analytic once the two certified signs are given.