Jérôme Vétois

MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS ON COMPACT RIEMANNIAN MANIFOLDS

Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 4, and h be a Holder continuous function on M. We prove multiplicity of changing sign solutions for equations like Δg u + hu = |u|2* - 2 u, where Δg is the Laplace–Beltrami operator and 2* = 2n/(n - 2) is critical from the Sobolev viewpoint.

Sign-Changing Blow-Up for Scalar Curvature Type Equations

Given (M, g) a compact Riemannian manifold of dimension n ≥ 3, we are interested in the existence of blowing-up sign-changing families (u ϵ)ϵ>0 ∈ C 2, θ(M), θ ∈ (0, 1), of solutions to where Δ g : = −div g (∇) and h ∈ C 0, θ(M) is a potential. Assuming the existence of a nondegenerate solution to the limiting equation (which is a generic assumption), we prove that such families exist in two main cases: in small dimension n ∈ {3, 4, 5, 6} for any potential h or in dimension 3 ≤ n ≤ 9 when . These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of Druet [11] and Khuri et al. [19].

Fundamental Solutions for Anisotropic Elliptic Equations: Existence and A Priori Estimates

We study anisotropic equations such as (with Dirac mass δ0 at 0) in a domain Ω ⊂ ℝ n (n ≥ 2) with 0 ∈ Ω and u|∂Ω = 0. Suppose that p i ∈ (1, ∞) for all i with their harmonic mean p satisfying either Case 1: p < n and or Case 2: p = n and Ω is bounded. We establish the existence of a suitable notion of fundamental solution (or Greens function), together with sharp pointwise upper bound estimates near zero via an anisotropic Moser-type iteration scheme. As critical tools, we derive generalized anisotropic Sobolev inequalities and estimates in weak Lebesgue spaces.

Strong Maximum Principles for Anisotropic Elliptic and Parabolic Equations

Abstract We investigate vanishing properties of nonnegative solutions of anisotropic elliptic and parabolic equations. We describe the optimal vanishing sets, and we establish strong maximum principles.

Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four

Abstract We extend Chen, Wei and Yan’s constructions of families of solutions with unbounded energies [5] to the case of cubic nonlinear Schrödinger equations in the optimal dimension four.

Blow-up solutions for linear perturbations of the Yamabe equation

For a smooth, compact Riemannian manifold (M,g) of dimension \(N \geq 3\) we are interested in the critical equation

A General Theorem for the Construction of Blowing-up Solutions to Some Elliptic Nonlinear Equations via Lyapunov– Schmidt’s Finite-dimensional Reduction

\Delta_g{u} + \left(\frac{N-2}{4(N-1)}S_{g}+\varepsilon\mathrm{h}\right)u\;=\;u^{\frac{N+2}{N-2}}\qquad \mathrm{in}\;M,\quad u>0\quad \mathrm{in}\;M

Sharp Sobolev Asymptotics for Critical Anisotropic Equations

where \(\Delta_{g}\) is the Laplace–Beltrami operator, Sg is the scalar curvature of \((M,g),h\in C^{0,\propto}(M)\) and e is a small parameter.