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Giulio Ciraolo

Publications and source records attributed to Giulio Ciraolo.

At least 19 recordsLinked to original sources
arXiv2026

The Liouville equation on Riemannian surfaces: the role of volume growth in classification and rigidity results

We study the Liouville equation $-\Delta u = e^u$ on a complete, connected, non-compact, boundaryless Riemannian surface $(M, g)$ with non-negative Ricci curvature. Assuming only some asymptotic lower bound on the solution, we establish classification results for both the solutions and the ambient manifold, discussing also their optimality. Our results reveal a close connection between the volume growth of the manifold and the classification of both the solutions and the underlying manifold.

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arXiv2026

On the anisotropic critical $p$-Laplace equation: classification, decomposition, and stability results

We investigate both qualitative and quantitative issues related to the classification of non-negative energy solutions to the anisotropic critical $p$-Laplace equation in $\mathbb{R}^n$, for $1<p<n$. Specifically, we establish an anisotropic version of Struwe's decomposition, along with the interaction estimate for the family of bubbles in this decomposition. Moreover, we provide a short proof of the classification result as well as a quantitative stability result, proving that every energy solution to a perturbation of the anisotropic critical equation must be closed to a bubble, in the absence of bubbling.

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arXiv2026

On the classification of solutions to a class of $N$-Liouville equations in $\mathbb{R}^N$

Given $N\geq 2$ and $\alpha>-1$, we consider the following weighted Liouville-type equation involving the $N$-Laplacian: \begin{equation*} \left\{ \begin{aligned} -& \Delta_N u = |x|^{N\alpha} e^u \quad \text{ in } \mathbb{R}^N && , \\ & \int_{\mathbb{R}^N} |x|^{N\alpha} e^u \, dx < + \infty\,. &&\end{aligned} \right. \end{equation*} Solutions have been completely classified when $N=2$ via complex analysis, and when $\alpha=0$ using Pohozaev identities and an isoperimetric argument. In this paper, we first devise a $P$-function approach to the classification result for all $\alpha>-1$ when $N=2$. Since it is not based on complex analysis, this alternative and more PDE-oriented approach naturally extends to $N\geq 3$ by providing the classification for any $-1<\alpha\leq 0$. In particular, the explicit radial solutions are the unique ones for $-1<\alpha\leq0$ but become degenerate for special values $\alpha_k>0$, a hint that non-radial solutions might arise for $\alpha>0$ as it happens when $N=2$.

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arXiv2026

Symmetry and rigidity results for Serrin's overdetermined type problems in weighted Riemannian manifolds

We study Serrin's overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of the ambient manifold. We further address the case of domains with non-compact closure for manifolds conformally equivalent to the Euclidean space, possibly degenerating or becoming singular at a point, where both the weight and the conformal factor are radial functions.

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arXiv2026

Some remarks on patterns for semilinear Neumann problems

We study semilinear elliptic equations \begin{equation*} \begin{cases} -\Delta u = f(u) & \text{in } \Omega, \\ \partial_\nu u = 0 & \text{on } \partial\Omega, \end{cases} \end{equation*} with homogeneous Neumann boundary conditions in bounded domains. A classical result by Casten-Holland and Matano shows that stable nonconstant solutions cannot exist in convex domains, although unstable spatial patterns may still occur. In this paper we investigate rigidity properties of classical solutions without imposing stability assumptions and aim to identify structural conditions on the nonlinearity ensuring that all solutions are constant. We prove that every classical solution of the Neumann problem is constant provided the nonlinearity satisfies a suitable `monotonicity' condition, which includes the cases where the nonlinearity has a fixed sign or changes sign in a controlled way around one of its zeros. This yields a rigidity result depending solely on the structure of the nonlinearity and does not require convexity assumptions on the domain. We also discuss the sharpness of our assumptions by constructing examples of nonlinearities for which nonconstant solutions exist. In particular, inspired by the approach of Lin-Ni-Takagi, we consider exponential-type nonlinearities in dimension $N=2$, and show that when a parameter crosses a critical threshold, the associated Neumann problem admits nontrivial and nonconstant solutions for sufficiently small diffusion.

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arXiv2025

Rigidity of weighted manifolds via classification results for semilinear equations

We study model semilinear equations on complete and non-compact weighted Riemannian manifolds with non-negative Bakry-\'Emery Ricci curvature. Our main goal is to classify positive solutions of the equation at the Sobolev-critical exponent, and furthermore to prove that the existence of such solutions implies rigidity of the manifold and triviality of the weight. This is possible when the weighted manifold has non-negative finite dimensional Bakry-\'Emery Ricci curvature, and even under the weaker condition of non-negative infinite dimensional Bakry-\'Emery Ricci curvature, up to imposing some additional conditions in the latter case. To exhibit the sharpness of these additional conditions, we construct a non-trivial positive solution of the critical problem on a weighted manifold with positive infinite dimensional curvature. We also obtain a corresponding rigidity result for solutions of the Liouville equation on weighted Riemannian surfaces. Finally, we prove some non-existence theorems when the nonlinearity is sub-critical or simply under certain volume growth conditions. In particular, the latter rules out all positive solutions on shrinking gradient Ricci solitons.

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arXiv2025

On the stability of the critical $p$-Laplace equation

For $1<p<n$, it is well-known that non-negative, energy weak solutions to $\Delta_p u + u^{p^{\ast}-1} =0$ in $\mathbb{R}^n$ are completely classified. Moreover, due to a fundamental result by Struwe and its extensions, this classification is stable up to bubbling. In the present work, we investigate the stability of perturbations of the critical $p$-Laplace equation for any $1<p<n$, under a condition that prevents bubbling. In particular, we show that any solution $u \in \mathcal{D}^{1,p}(\mathbb{R}^n)$ to such a perturbed equation must be quantitatively close to a bubble. This result generalizes a recent work by the first author, together with Figalli and Maggi (Int. Math. Res. Not. IMRN 2018 (2018), no. 21, 6780-6797), in which a sharp quantitative estimate was established for $p=2$. However, our analysis differs completely from theirs and is based on a quantitative $P$-function approach.

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arXiv2025

Stability for the Sobolev inequality in cones

We prove a quantitative Sobolev inequality in cones of Bianchi-Egnell type, which implies a stability property. Our result holds for any cone as long as the minimizers of the Sobolev quotient are nondegenerate, which is the case of most cones. When the minimizers are the classical bubbles we have more precise results. Finally, we show that local estimates are not enough to get the optimal constant for the quantitative Sobolev inequality.

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arXiv2025

A quantitative study of radial symmetry for solutions to semilinear equations in $\mathbb{R}^n$

A celebrated result by Gidas, Ni & Nirenberg asserts that positive classical solutions, decaying at infinity, to semilinear equations $Δu +f(u)=0$ in $\mathbb{R}^n$ must be radial and radially decreasing. In this paper, we consider both energy solutions in $\mathcal{D}^{1,2}(\mathbb{R}^n)$ and non-energy local weak solutions to small perturbations of these equations, and study its quantitative stability counterpart. To the best of our knowledge, the present work provides the first quantitative stability result for non-energy solutions to semilinear equations involving the Laplacian, even for the critical nonlinearity.

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arXiv2025

Second order regularity for degenerate p-Laplace type equations with log-concave weights

We consider weighted p-Laplace type equations with homogeneous Neumann boundary conditions in convex domains, where the weight is a log-concave function which may degenerate at the boundary. In the case of bounded domains, we provide sharp global second-order estimates. For unbounded domains, we prove local estimates at the boundary. The results are new even for the case p = 2.

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arXiv2024

On the classification of extremals of Caffarelli-Kohn-Nirenberg inequalities

We consider a family of critical elliptic equations which arise as the Euler-Lagrange equation of Caffarelli-Kohn-Nirenberg inequalities, possibly in convex cones in $\mathbb{R}^d$, with $d\geq 2$. We classify positive solutions without assuming that the solution has finite energy and when the intrinsic dimension $n \in (\frac{3}{2},5]$.

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arXiv2024

A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities

In this paper, we study positive solutions $u$ of the homogeneous Dirichlet problem for the $p$-Laplace equation $-Δ_p \,u=f(u)$ in a bounded domain $Ω\subset\mathbb{R}^N$, where $N\ge 2$, $1<p<+\infty$ and $f$ is a discontinuous function. We address the quantitative stability of a Gidas-Ni-Nirenberg type symmetry result for $u$, which was established by Lions and Serra when $Ω$ is a ball. By exploiting a quantitative version of the Pólya-Szegö principle, we prove that the deviation of $u$ from its Schwarz symmetrization can be estimated in terms of the isoperimetric deficit of $Ω$.

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arXiv2024

A quantitative version of the Gidas-Ni-Nirenberg Theorem

A celebrated result by Gidas, Ni & Nirenberg asserts that classical positive solutions to semilinear equations $- Δu = f(u)$ in a ball vanishing at the boundary must be radial and radially decreasing. In this paper we consider small perturbations of this equation and study the quantitative stability counterpart of this result.

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arXiv2024

Classification results, rigidity theorems and semilinear PDEs on Riemannian manifolds: a P-function approach

We consider solutions to some semilinear elliptic equations on complete noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is non-negative, we prove both the classification of positive solutions to the critical equation and the rigidity for the ambient manifold. The same results are obtained when we consider solutions to the Liouville equation on Riemannian surfaces. The results are obtained via a suitable P-function whose constancy implies the classification of both the solutions and the underlying manifold. The analysis carried out on the P-function also makes it possible to classify non-negative solutions for subcritical equations on manifolds enjoying a Sobolev inequality and satisfying an integrability condition on the negative part of the Ricci curvature. Some of our results are new even in the Euclidean case.

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arXiv2024

Classification of solutions to the anisotropic $N$-Liouville equation in $\mathbb{R}^N$

Given $N\geq 2$, we completely classify the solutions of the anisotropic $N$-Liouville equation $$-Δ_N^H\,u=e^u \quad\text{in }\mathbb{R}^N,$$ under the finite mass condition $\int_{\mathbb{R}^N} e^u\,dx<+\infty$. Here $Δ_N^H$ is the so-called Finsler $N$-Laplacian induced by a positively homogeneous function $H$. As a consequence for $N=2$, we give an affirmative answer to a conjecture made in [G. Wang and C. Xia, J. Differential Equations 252 (2012) 1668--1700].

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arXiv2023

Global second-order estimates in anisotropic elliptic problems

We deal with boundary value problems for second-order nonlinear elliptic equations in divergence form, which emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions. Integrands with non polynomial growth are included in our discussion. The $W^{1,2}$-regularity of the stress-field associated with solutions, namely the nonlinear expression of the gradient subject to the divergence operator, is established under the weakest possible assumption that the datum on the right-hand side of the equation is a merely $L^2$-function. Global regularity estimates are offered in domains enjoying minimal assumptions on the boundary. They depend on the weak curvatures of the boundary via either their degree of integrability or an isocapacitary inequality. By contrast, none of these assumptions is needed in the case of convex domains. An explicit estimate for the constants appearing in the relevant estimates is exhibited in terms of the Lipschitz characteristic of the domains, when their boundary is endowed with Hölder continuous curvatures.

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arXiv2023

Symmetry breaking and instability for semilinear elliptic equations in spherical sectors and cones

We consider semilinear elliptic equations with mixed boundary conditions in spherical sectors inside a cone. The aim of the paper is to show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones. This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone. In particular it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index. A similar breaking of symmetry result is obtained for the positive solutions of the critical Neumann problem in the whole unbounded cone. In this case it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones.

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