Papers

In this paper, we establish general monotone quantities and sharp mass-capacity inequalities related to $p$-capacitary functions in $3$-dimensional asymptotically flat half-spaces of simple topology with nonnegative scalar curvature and nonnegative boundary mean curvature. These inequalities attain equality on a Schwarzschild half-space outside a rotationally symmetric half sphere.

In this thesis we focus on quasilinear elliptic systems driven by various nonlinear operators, such as the p-Laplacian, and nonlinear sources that are allowed to exhibit both subcritical and critical growth. We aim to establish the existence of solutions for perturbation of specific eigenvalue problems, by employing variational and topological methods. To establish existence results for autonomous systems of quasilinear PDEs in the spirit of the paper by Amann and Zehnder, we develop a local Morse theory for functional associated to quasilinear elliptic systems. By refining topological arguments introduced by Cingolani and Degiovanni in Banach product spaces, we establish the finiteness of the critical groups and we derive a Poincaré-Hopf formula in a Banach product space, in presence of both subcritical and critical nonlinear coupling. We also establish uniform boundedness results for anisotropic quasilinear systems, that are of interest within regularity theory. To show existence results for non-autonomous systems of quasilinear PDEs in the spirit of the paper of Landesman and Lazer, we consider the eigenvalue problem for quasilinear elliptic systems introduced by de Thélin. We prove the simplicity and isolation of the first eigenvalue lambda1. Furthermore, we show the existence of a sequence of eigenvalues by employing a suitable deformation lemma proved by Bonnet. Subsequently, we analyze new sufficient Landesman-Lazer type conditions within the framework of quasilinear elliptic systems. We also investigate the N-dimensional Euclidean Onofri inequality, established by Del Pino and Dolbeault for smooth functions with compact support. After extending this inequality to a suitable weighted Sobolev space, we exploit its connection with the Liouville equation on R^N to prove an equivalence with the sharp logarithmic Moser-Trudinger inequality on the unit ball of R^N.

This paper concerns a class of non-divergence nonlinear elliptic equations driven by the cone degenerate Laplacian, which is motivated by cone calculus. We establish the existence of viscosity solutions by proving the Alexandrov-Bakelman-Pucci and Hölder estimates. Furthermore, we obtain the existence of weak solutions by proving the equivalence between weak solutions and viscosity solutions.

We study the global structure of bifurcation diagrams for semilinear elliptic Dirichlet problems with supercritical nonlinearities in the unit ball. In particular, we focus on the borderline dimension $N = 10$, where the qualitative behavior of the bifurcation diagram is not determined solely by the growth rate of the nonlinearity. We show that the bifurcation curve is monotone, yielding uniqueness of classical solutions, for a class of nonlinearities including $f(u) = \exp((u+1)^p)$ with $p > 1$ and iterated exponential functions. Our approach is based on the construction of suitable singular subsolutions that satisfy a Hardy-type stability condition, avoiding the need for explicit representations of singular solutions. As a consequence, we show that, in dimension $N = 10$, these nonlinearities exhibit the same qualitative bifurcation diagram as the classical Gel'fand problem. We also characterize the monotonicity of the bifurcation curve in terms of the existence of global-in-time unbounded solutions to the associated parabolic problem.

We compare notions of admissible surfaces for Cauchy fluxes, formulated as understanding when the normal trace of the underlying stress field can be represented by a measure. If this field is unbounded, the problem of admissibility is necessitated by the fact that the normal trace need not admit a measure representation on every regular surface, but only on “almost all” such surfaces. We compare an approach based on a precise majorant introduced by Šilhavý (Arch. Ration. Mech. Anal. 116.3 (1991)) with a Minkowski-type condition introduced by Chen, Torres and the first author (Arch. Ration. Mech. Anal. 249.6 (2025)) by showing that, under mild geometric conditions, the former condition implies the latter. We also show, by means of an explicit construction, that the latter admissibility condition can allow for arbitrary measure concentrations as a normal trace.

We consider an incompressible Navier–Stokes system nonlinearly coupled with a conserved Allen–Cahn equation with a singular potential (e.g., of Flory–Huggins type). This model describes a mass-conserving two-phase flow with constant density and non-constant viscosity. First, in three spatial dimensions, we prove the existence and uniqueness of local-in-time strong solutions to the associated initial–boundary value problem, subject to no-slip boundary conditions for the velocity field and homogeneous boundary conditions for the phase field. Next, by means of a relative energy approach, we establish a conditional weak–strong uniqueness principle in three dimensions, as well as unconditional uniqueness of weak solutions in two dimensions. Finally, building on recent seminal results by the first and third authors, we prove for the first time that, in both two and three dimensions and for general singular potentials, every global-in-time weak solution asymptotically separates from the pure phases and converges to a unique equilibrium. This result is obtained under minimal assumptions on the viscosity coefficient. Moreover, under additional regularity assumptions on the viscosity, we combine the asymptotic strict separation property with the conditional weak–strong uniqueness principle to show that weak solutions undergo asymptotic regularization. As a consequence, convergence to equilibrium also holds in higher-order norms.

We prove interior $C^{2}$ regularity result for convex viscosity solutions of the quadratic Hessian equation $\sigma_2(D^2u) = f(x)$, under the assumption that $f\in C^{0,1}$ with $\inf f>0$. The result is almost sharp: if $f$ are merely continuous, there exist convex viscosity solutions that fail to be $C^{1,1}$. When $f\in C^{\alpha}$ for some $\alpha\in (0,1)$, the corresponding interior regularity remains open.

We study the lowest eigenvalue of the Neumann magnetic Laplacian in a planar domain divided into two regions, with piecewise constant magnetic fields that may scale differently in the inner and outer parts. Our aim is to describe the high-flux limit and determine when the ground-state energy is eventually monotone and when it continues to oscillate. We identify several asymptotic regimes according to the relative strength of the outer field. When the outer field is fixed, the lowest eigenvalue exhibits persistent oscillations and the low-energy states localize in the outer region. When the outer field grows more slowly, the behavior depends strongly on the geometry: it is eventually monotone for non-circular domains, while oscillations may persist for disks. In the critical regime, where the two fields are comparable, geometry and flux distribution both play a decisive role. When the outer field dominates, the problem reduces asymptotically to an effective operator on the inner region. These results show how uneven magnetic scaling, topology, and geometry shape the high-flux spectral behavior.

We present a few ideas on the Riemann Hypothesis based on properties of analytic functions in the complex plane. In particular, we focus on the real and imaginary parts of the Riemann xi ($\xi$) function whose zeros coincide with those of the zeta ($\zeta$) function within the critical strip. We discuss the forms of the zero contour lines of the two conjugate harmonic functions (the real and imaginary parts of xi) and consider where their intersections could conceivably occur. Those intersections would be the roots of both $\xi$ and $\zeta$ functions. The question of whether a zero could occur away from the critical line becomes equivalent to whether the solutions of a pair of Laplace's equations with well-defined boundary conditions in some semi-infinite strip can possess zero contour lines that intersect within that strip.

The classical Liouville theorem states that every bounded harmonic function on Euclidean space is constant. On complete Riemannian manifolds, analogous conclusions hold under geometric assumptions such as nonnegative Ricci curvature. The quadratic Wasserstein space $\mathcal{P}_2(M)$ has no canonical infinite-dimensional Riemannian volume and hence no canonical Laplace–Beltrami operator. We introduce a natural finite-particle notion of harmonicity: a continuous function $u:\mathcal{P}_2(M)\to\mathbb{R}$ is called empirically harmonic if, for every $N\geq1$, its pullback under the empirical map $$ \iota_N(x_1,\ldots,x_N)=\frac1N\sum_{i=1}^N\delta_{x_i} $$ is weakly harmonic on $M^N$. We prove that if $M$ has the finite-product Liouville property, then every bounded empirically harmonic function on $\mathcal{P}_2(M)$ is constant. In particular, the result applies to $M=\mathbb{R}^d$ and to every complete connected Riemannian manifold with nonnegative Ricci curvature. We also derive a finite-particle chain rule for sufficiently regular functionals on $\mathcal{P}_2(\mathbb{R}^d)$ and show that the empirical Laplacian is exactly the Hessian trace of a discrete $N$-particle lift. Finally, if $M$ admits a nonconstant bounded harmonic function, then $\mathcal{P}_2(M)$ admits a nonconstant bounded empirically harmonic linear statistic.

The stochastic dynamics of dislocations on a one-dimensional periodic lattice domain are considered. Two models are studied: one without a collision rule, and one which annihilates colliding dislocations if they have opposite orientation. The behaviour of both models is investigated by means of a series of numerical simulations exploring the asymptotic behaviour of these models as the number of dislocations increases. From these simulations, evidence is obtained that the discrete model with annihilation tends to a PDE for the dislocation density that accounts for annihilation. However, the discrete model without a collision rule does not appear to exhibit consistent convergence behaviour; instead, it appears that the expected PDE with conserved dislocation density appears in the limit for some parameters, but that for other parameters the density appears to follow to the evolution of the PDE with annihilation. These findings provide evidence that a careful treatment of dislocation collisions is important in discrete dislocation dynamics models.

In this paper, we study a structural acoustic model consisting of a semilinear wave equation defined on a three-dimensional bounded domain, coupled with a Kirchhoff-Love plate equation acting on a flat portion of the boundary. Primarily for mathematical interest, we impose nonlinear damping terms and logarithmic-type supercritical source terms on the system. We investigate local and global well-posedness, energy decay rates of potential well solutions, and blow-up of solutions under different conditions on the parameters and initial data. The main novelties include the analysis of the interaction between nonlinear damping and logarithmic energy-amplifying source terms, as well as the development of techniques for handling logarithmic source terms within potential well theory. The logarithmic nonlinearities are not homogeneous under scaling, which creates difficulties in studying potential well solutions. The wave-plate coupling through the acoustic pressure also causes technical difficulties in the analysis, especially in the proof of blow-up of weak solutions.

For $(M,g)$ a compact Riemannian surface with Laplace-Beltrami operator $\Delta$, and for $\lambda,\delta \geq 0$, let $P_{\lambda,\delta}$ be the spectral projector on the frequency interval $[\lambda-\delta,\lambda+\delta]$ associated to $\sqrt{-\Delta}$. For the Euclidean disk, away from its boundary, we improve the upper bound on the $L^2\to L^4$ norm of $P_{\lambda,\delta}$ in the regime where the bandwidth $\delta$ is polynomially small compared to the target frequency $\lambda$. Decomposing on the explicit joint eigenbasis of $\left(\sqrt{-\Delta}, \frac{1}{i}\frac{\partial}{\partial \theta}\right)$ given in terms of Bessel eigenfunctions, which are well-approximated by oscillatory functions outside of their caustic set, we reduce the analysis to a number of precise quantitative estimates of nonstationary phase oscillatory integrals. We strongly use convexity phenomenon both for these estimates, and then for the summation of the contribution of all eigenfunctions through a new arithmetic estimate. The method extends to other $S^1$-symmetric surfaces satisfying similar conditions on the induced completely integrable structure.

We study the well-posedness for the intermediate nonlinear Schrödinger equation (INLS) with periodic boundary conditions. Using a gauge transform, we obtain large data local well-posedness in $H^{s}(\mathbb{T})$ for any $s\geq \frac 12$. We extend this result to global well-posedness under a small $L^2$-norm constraint by exploiting the complete integrability of the continuum Calogero-Moser equation (CCM). We also establish additional results such as the unconditional well-posedness in the energy space and the convergence of solutions to INLS to those of CCM in the infinite-depth limit.

We establish a sharp three-particle fractional Hardy inequality for the Laplacian of order $s\in(0,1)$ in dimension $d\geq 4-2s$ (Theorem 1.1), involving an explicit intrinsically three-body interaction potential $V_{s,3}$. The inequality holds with the optimal two-particle fractional Hardy constant $C_{fH}(d,s)$, which is shown to be sharp relative to the fixed potential $V_{s,3}$. This potential $V_{s,3}$ strictly dominates the standard pairwise Coulomb-type interaction and captures genuine three-body effects. As a consequence, we derive a nontrivial many-particle fractional Hardy inequality for $N\geq 3$, and, in the regime $N>d+1$, obtain an improved Coulomb-type inequality with a strictly larger constant, agreeing in spirit with the results of Hoffmann-Ostenhof et al. [M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, and J. Tidblom, "Many-particle Hardy inequalities", J. Lond. Math. Soc. 77 (2008), no. 2, 99-115] and Lundholm [D. Lundholm, "Geometric extensions of many-particle Hardy inequalities", J. Phys. A: Math. Theor. 48 (2015), no. 17, 175203]. The proof relies on a fractional ground-state representation method adapted to three-particle interactions, combined with an explicit evaluation of the resulting nonlocal interaction term. This evaluation is achieved through a new singular integral identity of Selberg-type (Theorem 1.2), extending the three-fold formula of Grafakos-Morpurgo [L. Grafakos and C. Morpurgo, "A Selberg integral formula and applications", Pacific J. Math. 191 (1999), no. 1, 85-94] beyond the radial setting. This identity provides the analytic mechanism underlying the emergence of the three-body potential and may be of independent interest in harmonic analysis.

We investigate the structural and operator-theoretic foundations of indirect signal-generation mechanisms in Keller–Segel-type chemotaxis models. By analyzing a physically motivated multi-stage signaling cascade, we establish a precise mathematical dichotomy between instantaneous equilibration and transient kinetic memory. Specifically, we prove that the fully equilibrating parabolic–elliptic–elliptic (PES) cascade reduces to a static fourth-order elliptic interaction. In dimension four, an exact algebraic cancellation of the leading Newtonian singularity yields a purely logarithmic kernel, shifting the mass-critical dimension from $N=2$ to $N=4$. Through an $L^2$-gradient flow formulation, we identify the corresponding concentration-scaling candidate critical mass $M_* = 64\pi^2\tau/\chi$. In sharp contrast, we demonstrate that the mixed elliptic–parabolic (MEP) cascade retains a genuine Volterra memory effect that defies static reduction. Its interaction drift acts as a singular perturbation in time-exhibiting classical two-dimensional Keller–Segel principal order near the time diagonal, yet providing fourth-order smoothing in its frozen-time average, necessitating a mixed space-time threshold theory. These results isolate the physical origin of mass-critical dimensional shifts in multiscale biological systems and formulate the specific adapted Adams/logarithmic Hardy–Littlewood–Sobolev (log-HLS) inequalities and mixed-norm criteria required to close the threshold problems.

We exploit the connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians. First, we find a graphical relation between their smallest positive eigenvalues, which allows us to deduce a recipe for translating bounds (from above and below) from one to the other. As an application, we provide new upper and lower bounds for the eigenvalues of the quantum dot Dirac operators, which depend only on geometric quantities of the underlying domain. In particular, we obtain some Faber-Krahn type inequalities for convex thin domains.

Holographic coherent X-ray imaging enables nanoscale imaging of biological cells and tissues, rendering both phase and absorption contrast, i.e. real and imaginary parts of the refractive index. Unlike the standard model, which assumes a perfectly coherent incident beam, we consider partial coherence characterized by a known covariance operator. In addition, we assume time-resolved intensity measurements, granting access not only to expected intensities but also to their correlations. We investigate the information content of these correlations and analytically demonstrate that, under a symmetry-breaking condition on the sample and the illumination area, both phase and absorption contrast can be uniquely recovered in both the full and the linearized models. A key challenge in numerical reconstruction is the substantial increase in data dimensionality caused by computing intensity correlations during preprocessing. We propose a novel approach that leverages a low-rank assumption on the incident beam covariance operator, bypassing explicit correlation computation while still exploiting its full information. Numerical experiments demonstrate its feasibility, yielding accurate simultaneous reconstructions of phase and absorption contrast.

In this work, we study several inequalities related to a Dirichlet problem on Riemannian manifolds whose Ricci curvature is bounded from below. First, we establish inequalities involving the torsional rigidity and discuss rigidity results characterizing metric balls in this setting. Next, we derive an integral identity associated with a Dirichlet problem, which measures the spherical deficit arising in this context. In particular, we apply this identity to the setting of Einstein manifolds.

The shallow-water system is a standard model for long waves in shallow water. The system is hyperbolic and, for a large class of initial data, solutions develop steep gradients leading to shock formation in finite time. Since such singularities violate the long-wave assumptions underlying the model, their appearance limits the regime of validity of the equations. In this work, we introduce a regularized shallow-water system in which the nonlinear terms are modified by a bounded operator. This regularization removes the standard derivative-steepening mechanism responsible for shock formation in the classical system while remaining consistent with the long-wave regime. We establish local well-posedness and small-data global well-posedness in Sobolev spaces that exclude singularity formation. In addition, numerical simulations indicate that the system admits solitary-wave solutions.

We analyze how the interaction between local and nonlocal dispersions, combined with different types of nonlinearities, influences the smoothing effects of solutions. To achieve this goal, we consider a model that generalizes the KdV and Benjamin equations and demonstrate that its solutions exhibit Kato's smoothing effect and satisfy the propagation of regularity principle. As a result, we confirm that the higher-order dispersive term determines the local gain of fractional regularity of solutions. Our results are general; they not only recover known results for the KdV and Benjamin equations, but also provide new insights for a broader family of models of physical and mathematical interest with polynomial dispersions of arbitrary order.

We study positive solutions of the Dirichlet problem $-\Delta u = u^p$ in a uniformly convex domain $\Omega \subset \mathbb S^2$, $u= 0$ on $\partial\Omega.$ For $p=1$, we assume that the right-hand side is replaced by $\lambda_1 u$, where $\lambda_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ with zero Dirichlet boundary condition. We prove that for $0 \leq p < 1$ the unique positive solution $u$ is such that $u^{\frac{1-p}{2}}$ is strictly concave in $\Omega$, while for $1 < p \leq 3$ every positive solution $u$ is such that $u^{\frac{1-p}{2}}$ is strictly convex in $\Omega.$ For $p=0,$ our result gives the strict $1/2-$concavity of the torsion function in $\Omega.$ For $p=1,$ a result due to Lee and Wang gives the strict log-concavity of the first eigenfunction in $\Omega.$ As a consequence, for each $0 \leq p \leq 3,$ any positive solution has strictly convex superlevel sets and a unique nondegenerate maximum.

In this paper, we consider the hyperdissipative Navier-Stokes equations with fractional dissipation $(-\Delta)^{\beta}$ with $\beta>1$. We prove that smooth solutions of the hyperdissipative Navier-Stokes equations are non-unique with arbitrarily small initial data in ${B}^{-\beta-\alpha}_{\infty,1}(\mathbb{T}^d)$ for any $\alpha>0$. Moreover, we show the existence of a solution with arbitrarily small initial data in ${B}^{-\beta-\alpha}_{\infty,1}(\mathbb{T}^d)$ ($\alpha>0$) that grows arbitrarily large in $\dot{B}^{-s}_{\infty,\infty}(\mathbb{T}^d)$ for all $s\in\mathbb{R}$ in arbitrarily small time. It is worth pointing out that ${B}^{-\beta-\alpha}_{\infty,1}(\mathbb{T}^d)$ lies in the subcritical regime when $0<\alpha<\beta-1$. To the best of our knowledge, this is the first non-uniqueness result of the Navier-Stokes equations with initial data at the subcritical regularity. To show the sharpness of the above results, we establish the local well-posedness of the hyperdissipative Navier-Stokes equations with initial data in $\dot{B}^{-\beta-\alpha}_{\infty,\infty}(\mathbb{T}^d)$ with $\alpha< 0$.

We establish the existence of solutions of the 2D incompressible non-homogeneous Euler equations with $C^{0}_{t}C^{1,\,\sqrt{\frac{4}{3}}-1-\varepsilon}_{x}\cap C^{0}_{t}L^{2}_{x}$ source terms that develop a singularity in finite time. In order to achieve this, we adapt the Boussinesq blow-up we set up in arXiv:2505.20988 to the non-homogeneous Euler setting. Furthermore, we bring the potential existence of two different types of singularities of the forced system to light.

The term Gibbons conjecture is widely used in connection with symmetry results for the Allen-Cahn equation. However, its origin is less transparent than its frequent citation suggests. In this note, we revisit its emergence, tracing it to a 1995 paper by Carbou and to subsequent developments in the literature. We argue that the attribution likely arose from informal communication rather than from a formally stated conjecture, illustrating how mathematical terminology may develop through transmission and collective usage.

We prove the WALA conjecture of G. David and S. Semmes: every AD-regular Radon measure for which all real-valued Lipschitz functions satisfy the weak approximation by affine functions condition is uniformly rectifiable. In order to prove the conjecture, we identify the correct quantitative analogue of the decomposability bundle and establish several structural results for general AD-regular measures.

In this work, we analyze a diffuse-interface model for tumor growth, subject to multiplicative white noises, posed on a bounded domain $\mathcal{O} \subset \mathbb{R}^d$, $d=2,3$. The model couples a stochastic incompressible convective Brinkman-Forchheimer (CBF) equation or Navier-Stokes equation with damping $\eta|v|^{r-1}v $ for the averaged velocity field $v$, to a Cahn-Hilliard (CH) equation for the phase field variable $\phi$ and to a stochastic reaction-diffusion equation governing the nutrient concentration $\sigma$. We establish the existence of local strong solutions , for $ r \geq 1 $ in $d=2$ and $ r \in [1,3] $ in $d=3$. We prove the weak-strong uniqueness holds in both $d = 2, 3$. In addition, for $d = 2$, the uniqueness of weak solutions is obtained for all $\eta,\nu > 0$, and $r \geq 1$, while it holds in $d = 3$ for $r \geq 3$ and $\eta \nu \geq 1$ when $r = 3$ under an assumption on $\sigma$. Moreover, for $d=2$ and $r \in [1,3]$, we obtain that the strong solution exists globally in time.

We consider a smooth compact manifold with boundary, $M$, embedded in a smooth manifold of the same dimension on which an amenable group $\Gamma$ acts by isometries. We do not assume $M$ to be invariant under $\Gamma$. This results in a partial action of $\Gamma$ on $M^\circ$: For $g\in \Gamma$ we let $M^\circ_g = g(M^\circ)\cap M^\circ$ and obtain diffeomorphisms $g:M^\circ_{g^{-1}} \to M^\circ_g$. We assume that any two images of $\partial M$ under $ \Gamma$ either coincide or are disjoint and that only finitely many lie in $M$. The spherical blow-up of these images of $\partial M$ in $M$ yields a manifold $Y$ with boundary consisting of finitely many components. Moreover, $Y$ inherits a partial action of~$\Gamma$. We can then define the $C^*$-algebra $\mathcal A=\overline{\Psi_\Gamma(Y,\partial Y)}$ of operators on $L^2(Y)\oplus L^2(\partial Y)$, generated by the algebra $\Psi(Y,\partial Y)$ of operators of order and type zero in Boutet de Monvel's calculus on $Y$ and partial isometries associated with the partial action. Denote by $\Sigma=\overline{\Psi(Y,\partial Y)}/\mathbb K$ its symbol space. If the partial action of $\Gamma$ on Prim$(\Sigma)$ is topologically free, we find a criterion for the Fredholm property of the operators in $\overline{\Psi_\Gamma(Y,\partial Y)}$. Moreover, we obtain the classification of the elliptic elements in $\overline{\Psi_\Gamma(Y,\partial Y)}$ modulo stable homotopies: For $\mathcal A_0= C(Y\sqcup \partial Y)\rtimes\Gamma$ $${\rm Ell}(\mathcal A_0,\mathcal A)\cong K_0(C_0(T^*Y^\circ)\rtimes\Gamma)\oplus K_0(C(\partial Y)\rtimes \Gamma).$$ If $\Gamma$ is finitely generated and of polynomial growth, then the elements associated with the second summand do not contribute to the index.

We consider the weighted $p$-Laplacian associated with a measure $\mu$ that is absolutely continuous with respect to the Lebesgue measure on an open connected subset $X\subset\mathbb{R}^N$. We prove that Talenti's weighted Pólya–Szegő inequality – originally established for Lipschitz functions on $X$ – extends to Sobolev functions with zero boundary trace on arbitrary Borel subsets $\Omega\subset X$. This yields Faber–Krahn-type inequalities for the first $(p,q)$-eigenvalue of the weighted Dirichlet $p$-Laplacian. We present several examples fitting this abstract framework, including classical Euclidean and Gaussian cases alongside new results for homogeneous weights in convex cones, anisotropic Gaussians, and log-concave Gaussian perturbations.

We investigate critical polyharmonic equations of the type: $$ Lu = |u|^{2^\sharp-2} u \quad \text{ in } \Omega $$ with Dirichlet boundary conditions, in a smooth bounded domain $\Omega$ of $\mathbb{R}^n$. Here $L$ is an elliptic differential operator of integer order $2 \le 2k < n$ whose leading order term is $(-\Delta)^k$ and $2^\sharp = \frac{2n}{n-2k}$ is the critical Sobolev exponent. Our main result establishes, in large dimensions, uniform a priori bounds in $C^{2k}(\overline{\Omega})$ for bounded-energy solutions of this problem, that only depend on an upper bound on the energy. We prove this under a coercivity assumption of sorts on the lower-order terms of $L$. Our results are sharp, at least when $k=1$. Our approach uses asymptotic analysis techniques and in the course of the proof we obtain in particular a new global pointwise description of bounded-energy blowing-up solutions for this problem, which is of independent interest.