Results 51 to 60 of about 526 (85)

Boundary regularity, Pohozaev identities, and nonexistence results

, 2017
In this expository paper we survey some recent results on Dirichlet problems of the form $Lu=f(x,u)$ in $\Omega$, $u\equiv0$ in $\mathbb R^n\backslash\Omega$. We first discuss in detail the boundary regularity of solutions, stating the main known results
Ros-Oton, Xavier
core   +1 more source

Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

Advances in Nonlinear Analysis, 2018
We study the semilinear elliptic ...
Ghergu Marius, Kim Sunghan, Shahgholian Henrik   +2 more
doaj   +1 more source

Elliptic equations involving general subcritical source nonlinearity and measures [PDF]

, 2014
In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta)^\alpha u=g(u)+\sigma\nu \quad{\rm in}\quad \Omega,$$ subject to the condition (E2) $$u=\varrho\mu\quad {\rm on}\quad \partial\Omega\ \ {\rm if}\ \alpha=1 ...
Chen, Huyuan, Felmer, Patricio, Véron, Laurent   +2 more
core   +2 more sources

Uniqueness and nondegeneracy of ground states for −Δu+u=(Iα⋆u2)u-\Delta u+u=\left({{\rm{I}}}_{\alpha }\star {u}^{2})u in R3{{\mathbb{R}}}^{3} when α\alpha is close to 2

Advances in Nonlinear Analysis
In this article, we study the following Choquard equation: −Δu+u=(Iα⋆u2)u,x∈R3,-\Delta u+u=\left({{\rm{I}}}_{\alpha }\star {u}^{2})u,\hspace{1.0em}x\in {{\mathbb{R}}}^{3}, where Iα{{\rm{I}}}_{\alpha } is the Riesz potential and α\alpha is sufficiently ...
Luo Huxiao, Zhang Dingliang, Xu Yating
doaj   +1 more source

Extremal values of the (fractional) Weinstein functional on the hyperbolic space

, 2016
We study Weinstein functionals, first defined in [33], mainly on the hyperbolic space ℍ n
Mukherjee, M.
core   +1 more source

Semilinear fractional elliptic equations with gradient nonlinearity involving measures [PDF]

, 2013
We study the existence of solutions to the fractional elliptic equation (E1) $(-\Delta)^\alpha u+\epsilon g(|\nabla u|)=\nu $ in a bounded regular domain $\Omega$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $\Omega^c$, where $\epsilon=1$ or
Chen, Huyuan, Veron, Laurent
core   +2 more sources

Large versus bounded solutions to sublinear elliptic problems

, 2018
Let $L $ be a second order elliptic operator with smooth coefficients defined on a domain $\Omega \subset \mathbb{R}^d$ (possibly unbounded), $d\geq 3$.
Damek, Ewa, Ghardallou, Zeineb
core   +1 more source

Existence of a positive solution for nonlinear Schrödinger equations with general nonlinearity

Advances in Nonlinear Analysis, 2014
We study the following nonlinear Schrödinger equations: -Δu+V(x)u=f(u)inℝN.$ - \Delta u + V(x) u = f(u) \quad \text{in } {\mathbb {R}^N}. $ The purpose of this paper is to establish the existence of a positive solution under general conditions which are ...
Sato Yohei, Shibata Masataka
doaj   +1 more source

An upper bound for the least energy of a sign-changing solution to a zero mass problem

Advanced Nonlinear Studies
We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation −Δu=f(u),u∈D1,2(RN), $-{\Delta}u=f\left(u\right), u\in {D}^{1,2}\left({\mathrm{R}}^{N}\right),$ where N ≥ 5 and the nonlinearity f is
Clapp Mónica, Maia Liliane, Pellacci Benedetta   +2 more
doaj   +1 more source

Extremals for Fractional Moser–Trudinger Inequalities in Dimension 1 via Harmonic Extensions and Commutator Estimates

Advanced Nonlinear Studies, 2020
We prove the existence of extremals for fractional Moser–Trudinger inequalities in an interval and on the whole real line. In both cases we use blow-up analysis for the corresponding Euler–Lagrange equation, which requires new sharp estimates obtained ...
Mancini Gabriele, Martinazzi Luca
doaj   +1 more source