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We show the existence of a weak solution of a semilinear elliptic Dirichlet problem on an arbitrary open set. We make no assumptions about the open set, very mild regularity assumptions on the semilinearity, plus a coerciveness assumption…

Analysis of PDEs · Mathematics 2016-07-19 Reinhard Stahn

In this paper, we first prove the existence of solutions to Dirichlet problems involving the fractional $g$-Laplacian operator and lower order terms by appealing to sub- and supersolution methods. Moreover, we also state the existence of…

Analysis of PDEs · Mathematics 2023-05-04 Pablo Ochoa , Analía Silva , Maria José Suarez Marziani

We prove a two-term asymptotic expansion of eigenvalue sums of the Laplacian on a bounded domain with Neumann, or more generally, Robin boundary conditions. We formulate and prove the asymptotics in terms of semi-classical analysis. In this…

Spectral Theory · Mathematics 2012-08-14 Rupert L. Frank , Leander Geisinger

We obtain multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals. There is no assumption to the…

Analysis of PDEs · Mathematics 2014-09-25 Cyril J. Batkam , Fabrice Colin , Tomasz Kaczynski

In this paper, we investigate the existence and uniqueness of solutions for the following model problem, involving singularities and inhomogeneous Robin boundary conditions \begin{equation*} \left\{ \begin{array}{ll}…

Analysis of PDEs · Mathematics 2024-10-29 Mohamed El Hichami , Youssef El Hadfi

In this paper, we obtain nonexistence results of positive solutions, and also the existence of an unbounded sequence of solutions that changing sign for some critical problems involving conformally invariant operators on the standard unit…

Differential Geometry · Mathematics 2021-02-24 Emerson Abreu , Ezequiel Barbosa , Joel Cruz Ramirez

Let $1<p<+\infty$ and let $\Omega\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the…

Analysis of PDEs · Mathematics 2020-02-28 Alberto Boscaggin , Francesca Colasuonno , Benedetta Noris

This paper is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an…

Analysis of PDEs · Mathematics 2018-08-21 Soeren Fournais , Loïc Le Treust , Nicolas Raymond , Jean Van Schaftingen

This paper explores the complexity associated with solving the inverse Sturm-Liouville problem with Robin boundary conditions: given a sequence of eigenvalues and a sequence of norming constants, how many limits does a universal algorithm…

Numerical Analysis · Mathematics 2023-12-27 Jonathan Ben-Artzi , Marco Marletta , Frank Rösler

We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in \cite{DFMST}. We apply it to generalized $p$-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative…

Analysis of PDEs · Mathematics 2020-08-19 Humberto Ramos Quoirin

In this paper, we will prove the existence of infinitely many positive solutions to the following supercritical problem by using the Liapunov-Schmidt reduction method and asymptotic analysis: {ll}\Delta u + u^{p}+f(x)=0, u>0 {in} R^{n},…

Analysis of PDEs · Mathematics 2010-01-13 Baishun Lai , Zhihao Ge

In this paper, we consider the large-time behavior of solutions of a reaction diffusion system arising from a nuclear reactor model with the Robin boundary conditions, which consists of two real-valued unknown functions. In particular, we…

Analysis of PDEs · Mathematics 2018-11-12 Kosuke Kita , Mitsuharu Ôtani

The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is to classify the solutions which have a singularity on the…

Analysis of PDEs · Mathematics 2015-11-03 Nicola Abatangelo

We introduce tow assumptions weaker than the classical Ambrosetti-Rabinowitz and the subcritical polynomial growth conditions to obtain the Palais-Smale Condition. Therefore, we improve the Ambrosetti- Rabinowitz existence theorems. Also,…

Analysis of PDEs · Mathematics 2013-10-30 Abdellaziz Harrabi

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic…

Spectral Theory · Mathematics 2007-05-23 Michael Levitin , Leonid Parnovski

We establish the existence of multiple solutions for a nonlinear problem of critical type. The problem considered is fractional in nature, since it is obtained by the superposition of $(s,p)$-fractional Laplacians of different orders. The…

Analysis of PDEs · Mathematics 2026-03-12 Serena Dipierro , Kanishka Perera , Caterina Sportelli , Enrico Valdinoci

We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools…

Analysis of PDEs · Mathematics 2018-05-04 Anup Biswas

In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the…

Analysis of PDEs · Mathematics 2021-05-19 Said El Manouni , Greta Marino , Patrick Winkert

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on…

Analysis of PDEs · Mathematics 2018-07-20 Dario D. Monticelli , Fabio Punzo , Marco Squassina

We provide results on the existence, non-existence, multiplicity and localization of positive radial solutions for semi linear elliptic systems with Dirichlet or Robin boundary conditions on an annulus. Our approach is topological and…

Functional Analysis · Mathematics 2018-04-06 F. Cianciaruso , P. Pietramala