# Semilinear elliptic problems with nonlinearities depending on the derivative

• Volume: 44, Issue: 3, page 413-426
• ISSN: 0010-2628

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## Abstract

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We deal with the boundary value problem $\begin{array}{ccccccc}\hfill -\Delta u\left(x\right)& ={\lambda }_{1}u\left(x\right)+g\left(\nabla u\left(x\right)\right)+h\left(x\right),\phantom{\rule{1.0em}{0ex}}\hfill & & x\in \Omega \phantom{\rule{4pt}{0ex}}u\left(x\right)\hfill & \hfill =0,& & \hfill x\in \partial \Omega \end{array}$ where $\Omega \subset {ℝ}^{N}$ is an smooth bounded domain, ${\lambda }_{1}$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega$, $h\in {L}^{max\left\{2,N/2\right\}}\left(\Omega \right)$ and $g:{ℝ}^{N}⟶ℝ$ is bounded and continuous. Bifurcation theory is used as the right framework to show the existence of solution provided that $g$ satisfies certain conditions on the origin and at infinity.

## How to cite

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Arcoya, David, and del Toro, Naira. "Semilinear elliptic problems with nonlinearities depending on the derivative." Commentationes Mathematicae Universitatis Carolinae 44.3 (2003): 413-426. <http://eudml.org/doc/249153>.

@article{Arcoya2003,
abstract = {We deal with the boundary value problem \begin\{@align\}\{3\}2 -\Delta u(x) & = \lambda \_\{1\}u(x)+g(\nabla u(x))+h(x), \quad && x\in \Omega \ u(x) & = 0, && x\in \partial \Omega \end\{@align\} where $\Omega \subset \mathbb \{R\}^N$ is an smooth bounded domain, $\lambda _\{1\}$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega$, $h\in L^\{\max \lbrace 2,N/2\rbrace \}(\Omega )$ and $g:\mathbb \{R\}^N\longrightarrow \mathbb \{R\}$ is bounded and continuous. Bifurcation theory is used as the right framework to show the existence of solution provided that $g$ satisfies certain conditions on the origin and at infinity.},
author = {Arcoya, David, del Toro, Naira},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonlinear boundary value problems; elliptic partial differential equations; bifurcation; resonace; nonlinear boundary value problem; elliptic partial differential equations; bifurcation; resonance},
language = {eng},
number = {3},
pages = {413-426},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Semilinear elliptic problems with nonlinearities depending on the derivative},
url = {http://eudml.org/doc/249153},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Arcoya, David
AU - del Toro, Naira
TI - Semilinear elliptic problems with nonlinearities depending on the derivative
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 3
SP - 413
EP - 426
AB - We deal with the boundary value problem \begin{@align}{3}2 -\Delta u(x) & = \lambda _{1}u(x)+g(\nabla u(x))+h(x), \quad && x\in \Omega \ u(x) & = 0, && x\in \partial \Omega \end{@align} where $\Omega \subset \mathbb {R}^N$ is an smooth bounded domain, $\lambda _{1}$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega$, $h\in L^{\max \lbrace 2,N/2\rbrace }(\Omega )$ and $g:\mathbb {R}^N\longrightarrow \mathbb {R}$ is bounded and continuous. Bifurcation theory is used as the right framework to show the existence of solution provided that $g$ satisfies certain conditions on the origin and at infinity.
LA - eng
KW - nonlinear boundary value problems; elliptic partial differential equations; bifurcation; resonace; nonlinear boundary value problem; elliptic partial differential equations; bifurcation; resonance
UR - http://eudml.org/doc/249153
ER -

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