Resonant Problems on a Unit Ball

• Resonant problems on a unit ball are characterized by PDEs where eigenvalues and explicit Bessel function representations govern solution behavior and resonance.
• Analytical techniques such as spectral decomposition, Lyapunov–Schmidt reduction, and asymptotic analysis reveal bifurcation phenomena and solution multiplicity.
• Numerical continuation methods validate theoretical predictions, illustrating how boundary conditions and space dimension influence resonance and oscillatory integrals.

A resonant problem on a unit ball refers to the study of linear or nonlinear partial differential equations (PDEs) defined on the unit ball in Euclidean space, where the spectrum of the underlying linear operator plays a central role, particularly at or near its eigenvalues. Such problems include classical spectral theory for Laplacian and biharmonic operators under various boundary conditions, as well as semilinear equations at resonance—where the linear part's spectral properties lead to nontrivial solution behavior, multiplicity, and bifurcation phenomena. The unit ball geometry greatly facilitates explicit representation of eigenfunctions and eigenvalues via special functions, especially Bessel functions, and allows detailed analysis of resonance, multiplicity, and asymptotics.

1. Spectral Problems on the Unit Ball: Linear Framework

The classical model involves the Laplacian $-\Delta$ or higher-order analogs (e.g., the biharmonic operator) acting on functions defined in the unit ball $B \subset \mathbb{R}^N$, with boundary conditions such as Dirichlet, Neumann, or Robin. The Robin eigenvalue problem is formulated as

$$\begin{cases} -\Delta u = \mu u, & x \in B, \ \partial_n u + \alpha u = 0, & x \in \partial B, \end{cases}$$

where $\alpha \in \mathbb{R}$ parametrizes the Robin boundary condition and $\partial_n$ denotes the outward normal derivative. The unit ball allows for separation of variables, with eigenfunctions expressed as products of radial factors and spherical harmonics. The radial part reduces to an ODE involving Bessel or modified Bessel functions, and the spectra are captured by transcendental equations whose roots (Bessel function zeros) determine the eigenvalues and eigenspaces (Chen et al., 30 Oct 2025).

Explicitly, for the Laplacian on the ball,

• Positive eigenvalues $\mu = k^2$ correspond to the zeros of $$F_\ell(k) = k J_{\nu+\ell+1}(k)-(\alpha+\ell) J_{\nu+\ell}(k)$$,
• Negative eigenvalues (when possible, for $\alpha \in (-1,0)$) from $\alpha I_\nu(k)+k I_{\nu+1}(k)=0$, with $\nu = N/2 - 1$ and multiplicity determined by the dimension of the $\ell$-th order spherical harmonics.

For higher-order PDEs, such as the clamped plate problem $(\Delta^2 + \nu)u = -\lambda \Delta u$ (with $u = \partial_r u = 0$ on $\partial B$), the spectral problem again admits complete Bessel-type eigenbasis, and every eigenfunction is uniquely determined up to normalization and symmetry (Coster et al., 2016).

2. Resonance: Non-Invertibility and the Fredholm Alternative

A PDE is said to be "at resonance" if the right-hand side has a nonzero component along an eigenfunction of the linear operator corresponding to a simple (or finite-multiplicity) eigenvalue—most commonly, the principal eigenvalue of $-\Delta$ under Dirichlet boundary conditions. For the Dirichlet Laplacian on the ball, this yields problems of the form

$$\Delta u + \lambda_1 u + h(u) = \mu_1 \varphi_1(x) + e(x),$$

where $\lambda_1$ is the first Dirichlet eigenvalue and $\varphi_1$ the normalized associated eigenfunction. Since the linear operator $\mathcal{L} = \Delta + \lambda_1$ is not invertible (its kernel is one-dimensional), solvability relies on the Fredholm alternative: the non-homogeneous term must be orthogonal to the kernel. Introducing $\mu_1$ ensures solvability for arbitrary data $e \perp \varphi_1$ (Korman et al., 23 Dec 2025, Korman, 21 Dec 2025).

In the linear regime, if the parameter $\lambda$ in $L[u] = \Delta^2 u + \nu u + \lambda \Delta u$ coincides with an eigenvalue, resonance leads to an infinite-dimensional solution space unless additional orthogonality conditions are satisfied. For simple eigenvalues, precisely one scalar constraint ensures solvability; for higher-multiplicity eigenvalues, $d$ orthogonality constraints are required, with $d$ the dimension of the eigenspace (Coster et al., 2016).

3. Complete Spectral Description: Robin and Biharmonic Cases

On the unit ball, all Robin eigenvalues are given explicitly via Bessel-function zeros:

• For $\alpha > 0$, all eigenvalues are positive, with $\mu_1 = k_{\nu,1}^2$ and $\mu_2 = k_{\nu+1,1}^2$ (multiplicities $1$ and $N$).
• For $-1 < \alpha < 0$, the first eigenvalue is negative ($\mu_1 = -\widehat{k}_{\nu,1}^2$), followed by the positive $\mu_2 = k_{\nu+1,1}^2$.
• For $\alpha = -1$, $\mu_1 = -\widehat{k}_{\nu,1}^2$ and $\mu_2 = 0$. The sign of the ratio $\mu_2/\mu_1$ (positive, zero, or negative) depends on the range of $\alpha$ (Chen et al., 30 Oct 2025).

Eigenfunctions for positive $\mu$ are $$u_{\ell,m}(r,\theta) = r^{-\nu} J_{\nu+\ell}(k_{\nu+\ell,m} r) Y_\ell(\theta)$$ (multiplicity $d_\ell$), and for negative $\mu$, $$u_{0,1}(r,\theta) = r^{-\nu} I_{\nu}(\widehat{k}_{\nu,1} r)$$, which is radially symmetric.

For the clamped plate operator, every eigenvalue is of the form $$\lambda_{k,\ell}(\nu) = \alpha_{k,\ell}^2 - \kappa^2/\alpha_{k,\ell}^2$$ for positive roots $\alpha_{k,\ell}$ of the transcendental $F_k(\alpha)$ constructed from $J_{\nu_k}$ and $I_{\nu_k}$ (Coster et al., 2016). The spectral structure is completely explicit: eigenfunctions possess precise nodal properties and asymptotic distribution for large indices (spacing $\sim k^2$).

4. Nonlinear Resonant Problems and Existence of Infinitely Many Solutions

The nonlinear resonant problem on the ball is typically formulated as

$$\Delta u + \lambda_1 u + h(u) = \mu_1 \varphi_1(x) + e(x), \qquad u|_{\partial B} = 0,$$

with $\int_{B} e \varphi_1 = 0$. Under mild regularity and subcriticality ($\sup_{u} h'(u) < \lambda_2 - \lambda_1$), the solution set is a global curve parameterized by the first harmonic $\xi_1 = \int_B u \varphi_1$, with a unique pair $(u_{\xi_1}, \mu_1(\xi_1))$ for each $\xi_1 \in \mathbb{R}$. If $h(u)$ is oscillatory at infinity with $\limsup |h(u)|/|u| = 0$ and sufficient oscillation in its derivative, then $\mu_1(\xi_1) \to \pm \infty$ as $|\xi_1| \to \infty$, guaranteeing infinitely many solutions for any fixed $\kappa$ in the right-hand side (Korman et al., 23 Dec 2025).

A similar conclusion holds for mean-zero periodic $g(u)$ in the resonance problem $\Delta u + \lambda_1 u + g(u) = f(r)$, but the number of solutions becomes dimension-dependent. For $1 \le n \le 3$, there are infinitely many sign-changes in $\mu_1(\xi_1)$; for $n \geq 7$ only finitely many, with $n = 4,5,6$ yielding "borderline" cases governed by vanishing of certain constants in the stationary phase expansions (Korman, 21 Dec 2025).

5. Asymptotic and Analytical Techniques: Oscillatory Integrals and Harmonics

Analysis of resonant problems leverages detailed harmonic decomposition. For radial problems, a Lyapunov–Schmidt reduction with $u = \xi_1 \varphi_1 + U$ ($U \perp \varphi_1$) recasts the PDE as a bifurcation problem for a scalar equation governing $(\xi_1, \mu_1)$. Leading asymptotics for the projection $\mu_1(\xi_1)$ are obtained via stationary phase and repeated integration by parts on oscillatory integrals of the form $$\int_0^1 h(\xi_1 \varphi_1(r)) \varphi_1(r) r^{n-1} dr$$.

For $h(u) = u^p \sin u$, the principal term for $\mu_1(\xi_1)$ in $\mathbb{R}^2$ is $$-\frac{4\pi c_0^p}{\nu_1^2} \xi_1^{p-1} \cos(c_0 \xi_1)$$ as $|\xi_1| \to \infty$, where $c_0$ describes the normalization of $\varphi_1$ and $\nu_1$ parametrizes the Bessel zero (Korman et al., 23 Dec 2025). In three dimensions, the decay is $\sim \xi_1^{-3/2} \cos(\xi_1 \sqrt{\pi/2} - \pi/4)$. For higher dimensions and mean-zero periodic nonlinearities, careful analysis of the oscillatory and endpoint terms determines whether infinitely many solutions persist (Korman, 21 Dec 2025).

6. Computational Methods and Numerical Continuation

Numerical continuation along the global solution curve in the harmonic parameter $\xi_1$ is robustly implemented by iteratively solving the linearized Dirichlet problem, projecting onto the principal eigenfunction, and updating the parameter $\mu$ to enforce the constraint $\int u \varphi_1 = \xi_1$. This approach, implemented in Mathematica using NDSolve for radial and general domains, demonstrates close agreement between computed solution curves and theoretical leading-order asymptotics for $\mu_1(\xi_1)$ in two and three dimensions (Korman et al., 23 Dec 2025). The numerical scheme proceeds as:

• Linearize around current iterate;
• Solve for Newton correction $w = \mu w_1 + w_2$ using splits $\Delta w_1 + f'(u^k) w_1 = \varphi_1$, $$\Delta w_2 + f'(u^k) w_2 = f'(u^k) u^k - f(u^k) + e$$;
• Update $\mu$ to satisfy the harmonic constraint.

7. Dimensional Dependence and Open Problems

The structure of resonant problems on the unit ball is sensitive to the space dimension:

• For $n \leq 3$, oscillatory terms persist after at most two integrations by parts, leading to infinite solution multiplicity;
• For $n \geq 7$, after three integrations the boundary contribution is constant, resulting in only finitely many solutions;
• For $n = 4,5,6$, the outcome depends on the vanishing of certain amplitude terms after stationary phase expansion—necessitating explicit calculation for the given periodic nonlinearity. An open question is whether, for $n \geq 7$ and higher vanishing of antiderivatives at the endpoints, further integration reveals oscillatory leading terms (Korman, 21 Dec 2025). This underscores the role of harmonic analysis, endpoint contributions, and space dimension in determining the global solution structure of resonant elliptic PDEs on the unit ball.

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Key References:

• "Complete spectrum of the Robin eigenvalue problem on the ball" (Chen et al., 30 Oct 2025)
• "Spectral analysis of a generalized buckling problem on a ball" (Coster et al., 2016)
• "Infinitely many solutions and asymptotics for resonant oscillatory problems" (Korman et al., 23 Dec 2025)
• "Infinitely many solutions for a class of resonant problems" (Korman, 21 Dec 2025)