Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Topics in Functional Analysis and Applications

In this paper we consider the problem

 $$\left\{\begin{array}{ll}-\Delta u=u^{p}\quad {\rm in}\, \Omega_R,\\ u=0 \quad \quad \quad {\rm on}\, \partial\Omega_R,\quad\quad\quad (0.1)\end{array}\right.$$
 where p > 1 and Ω
 R
 is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as $${R \longrightarrow \infty}$$
 . The main goal of the paper is to prove, for large R, the existence of a positive solution to (0.1) which is close to the positive radial solution in the corresponding diffeomorphic annulus. The proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions.

Asymptotically radial solutions in expanding annular domains

Given 
$$n \ge 2$$

 and 
$$1<p<n$$

, we consider the critical p-Laplacian equation 
$$\Delta _p u + u^{p^*-1}=0$$

, which corresponds to critical points of the Sobolev inequality. Exploiting the moving planes method, it has been recently shown that positive solutions in the whole space are classified. Since the moving plane method strongly relies on the symmetries of the equation and the domain, in this paper we provide a new approach to this Liouville-type problem that allows us to give a complete classification of solutions in an anisotropic setting. More precisely, we characterize solutions to the critical p-Laplacian equation induced by a smooth norm inside any convex cone. In addition, using optimal transport, we prove a general class of (weighted) anisotropic Sobolev inequalities inside arbitrary convex cones.

Symmetry results for critical anisotropic $p$-Laplacian equations in convex cones

We study the existence of ground state solutions for a class of discrete nonlinear Schrodinger equations with a sign-changing potential V that converges at infinity and a nonlinear term being asymptotically linear at infinity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite and the other is that, due to the convergency of V at infinity, the classical methods such as periodic translation technique and compact inclusion method cannot be employed directly to deal with the lack of compactness of the Cerami sequence. New techniques are developed in this work to overcome these two major difficulties. This enables us to establish the existence of a ground state solution and derive a necessary and sufficient condition for a special case. To the best of our knowledge, this is the first attempt in the literature on the existence of a ground state solution for the strongly indefinite problem under no periodicity condition on the bounded potential and the nonlinear term being asymptotically linear at infinity. Moreover, our conditions can also be used to significantly improve the well-known results of the corresponding continuous nonlinear Schrodinger equation.

Ground State Solutions of Discrete Asymptotically Linear Schrödinger Equations with Bounded and Non-periodic Potentials

We consider the semilinear Lane-Emden problem 1uDjuj p 1 u in; uD 0 on@; (Ep) where p > 1 andis a smooth bounded domain of R 2 . The aim of the paper is to analyze the asymptotic behavior of sign-changing solutions of (Ep) as p! 1. Among other results we show, under some symmetry assumptions on , that the positive and negative parts of a family of symmetric solutions concentrate at the same point as p ! 1, and the limit profile looks like a tower of two bubbles given by superposition of a regular and a singular solution of the Liouville problem in R 2 .

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=17&iss=8&rank=4

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Lane Emden problems with large exponents and singular Liouville equations

Let Γ denote a smooth simple curve in ℝ N , N ≥ 2, possibly with boundary. Let Ω R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx | x ∈ Γ∖∂Γ}. Consider the supe...

Alternating Sign Multibump Solutions of Nonlinear Elliptic Equations in Expanding Tubular Domains

We consider families $u_p$ of solutions to the problem 


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where $p>1$ and $\Omega $ is a smooth bounded domain of $\mathbb {R}^2$. Under the condition 


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we give a complete description of the asymptotic behavior of $u_p$ as $p\rightarrow +\infty $.

Asymptotic profile of positive solutions of Lane–Emden problems in dimension two

We consider a partially overdetermined problem in a sector-like domain Ω in a cone Σ in RN, N≥2, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that Ω is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces Γ with boundary which satisfy a 'gluing' condition with respect to the cone Σ. We prove that if either the cone is convex or the surface is a radial graph then Γ must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed.

Filomena Pacella

Papers

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Lane Emden problems with large exponents and singular Liouville equations

Alternating Sign Multibump Solutions of Nonlinear Elliptic Equations in Expanding Tubular Domains

Asymptotic profile of positive solutions of Lane–Emden problems in dimension two

Overdetermined problems and constant mean curvature surfaces in cones