Related papers: Morse indices of multiple blow-up solutions to the…
We consider the Gelfand problem on a planar domain. Under some conditions on the potential, we provide the first examples of multiplicity for blowing-up solutions at a given point in the domain. The argument is based on a refined…
We construct families of blowing-up solutions to elliptic systems on smooth bounded domains in the Euclidean space, which are variants of the critical Lane-Emden system and analogous to the Brezis-Nirenberg problem. We find a function which…
In this paper, we investigate carefully the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension two containing a nonlinearity with Trudinger-Moser growth. A quantification result had been obtained by the first…
In this paper we consider the Dirichlet problem \begin{equation} \label{iniz} \begin{cases} -\Delta u = \rho^2 (e^{u} - e^{-u}) & \text{ in } \Omega\\ u=0 & \text{ on } \partial \Omega, \end{cases} \end{equation} where $\rho$ is a small…
This paper is concerned with finite blow-up solutions of a one dimensional complex-valued semilinear heat equation. We provide locations and the number of blow-up points from the viewpoint of zeros of the solution.
We derive a second order estimate for the first m eigenvalues and eigenfunctions of the linearized Gel'fand problem associated to solutions which blow-up at m points. This allows us to determine, in some suitable situations, some…
Using the method of blow-up analysis, we obtain two sharp Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary, as well as the existence of the corresponding extremals. This generalizes early results of Chang-Yang…
This paper is concerned with blow-up solutions of the 4-dimensional energy critical heat equation $u_t=\Delta u + u^3$. Our main result is to show that the existence of type II blowup solutions, and…
Blowing-up solutions of Klein-Gordon equations with gauge variant semilinear terms are considered in Friedmann-Lema\'itre-Robertson-Walker spacetimes. Effects of spatial expansion or contraction on the solutions are studied through the…
We are concerned with blow-up solutions of the 5-dimensional energy critical heat equation $u_t=\Delta u + | u |^{\frac{4}{3}}u$. Our main result is to show that the existence of type 2 solutions blows up at 2 points with 2 different…
We consider the Gelfand problem with rapidly growing nonlinearities in the two-dimensional bounded strictly convex domains. In this paper, we prove the uniformly boundedness of finite Morse index solutions. As a result, we show that there…
Prescribing conformally the scalar curvature of a Riemannian manifold as a given function consists in solving an elliptic PDE involving the critical Sobolev exponent. One way of attacking this problem consist in using subcritical…
We analyze the blowup behaviour of solutions to the focusing nonlinear Klein--Gordon equation in spatial dimensions $d\geq 2$. We obtain upper bounds on the blowup rate, both globally in space and in light cones. The results are sharp in…
This paper establishes a blow-up criterion of strong solutions to the two-dimensional compressible magnetohydrodynamic (MHD) flows. The criterion depends on the density, but is independent of the velocity and the magnetic field. More…
The aim of this paper is to use a selection process and a careful study of the interaction of bubbling solutions to show a classification result for the blow-up values of the elliptic sinh-Gordon equation $$\Delta u+h_1e^u-h_2e^{-u}=0 \quad…
For a solution of the Allen-Cahn equation in $\mathbb{R}^2$, under the natural linear growth energy bound, we show that the blowing down limit is unique. Furthermore, if the solution has finite Morse index, the blowing down limit satisfies…
We investigate the (linearized) Morse index of solutions to Hamiltonan systems, with a focus on convex Hamiltonians functions and sign-changing radial solutions. For strongly coupled systems, we describe the profile of the radial solutions…
We consider, on a connected metric graph $\mathcal{G}$, a family of nonlinear Schr\"odinger equations $$ -u'' + W_n(x) u + \lambda_n u = \rho_n(x)|u|^{p-2}u, \quad n \in \mathbb{N}. \qquad (*) $$ We assume that $p > 2$, $(W_n)$, $(\rho_n)…
Let $\Omega$ be a flat torus and $G$ be the green's function of $-\Delta$ on $\Omega$. One intriguing mystery of $G$ is how the number of its critical points is related to blowup solutions of certain PDEs. In this article we prove that for…
We prove that finite Morse index solutions to the Allen-Cahn equation in $\R^2$ have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index…