Semilinear elliptic problems with nonlinearities depending on the derivative
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 3, pp. 413-426
We deal with the boundary value problem $$ \alignat2 -\Delta u(x) = \lambda _{1}u(x)+g(\nabla u(x))+h(x), \quad x\in \Omega \ u(x) = 0, x\in \partial \Omega \endalignat $$ where $\Omega \subset \Bbb R^N$ is an smooth bounded domain, $\lambda _{1}$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega $, $h\in L^{\max \{2,N/2\}}(\Omega )$ and $g:\Bbb R^N\longrightarrow \Bbb R$ 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.
We deal with the boundary value problem $$ \alignat2 -\Delta u(x) = \lambda _{1}u(x)+g(\nabla u(x))+h(x), \quad x\in \Omega \ u(x) = 0, x\in \partial \Omega \endalignat $$ where $\Omega \subset \Bbb R^N$ is an smooth bounded domain, $\lambda _{1}$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega $, $h\in L^{\max \{2,N/2\}}(\Omega )$ and $g:\Bbb R^N\longrightarrow \Bbb R$ 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.
Classification :
35B32, 35B34, 35J25, 35J60, 35J65, 47J15
Keywords: nonlinear boundary value problems; elliptic partial differential equations; bifurcation; resonace
Keywords: nonlinear boundary value problems; elliptic partial differential equations; bifurcation; resonace
@article{CMUC_2003_44_3_a2,
author = {Arcoya, David and del Toro, Naira},
title = {Semilinear elliptic problems with nonlinearities depending on the derivative},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {413--426},
year = {2003},
volume = {44},
number = {3},
mrnumber = {2025810},
zbl = {1105.35038},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_2003_44_3_a2/}
}
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 SP - 413 EP - 426 VL - 44 IS - 3 UR - http://geodesic.mathdoc.fr/item/CMUC_2003_44_3_a2/ LA - en ID - CMUC_2003_44_3_a2 ER -
%0 Journal Article %A Arcoya, David %A del Toro, Naira %T Semilinear elliptic problems with nonlinearities depending on the derivative %J Commentationes Mathematicae Universitatis Carolinae %D 2003 %P 413-426 %V 44 %N 3 %U http://geodesic.mathdoc.fr/item/CMUC_2003_44_3_a2/ %G en %F CMUC_2003_44_3_a2
Arcoya, David; del Toro, Naira. Semilinear elliptic problems with nonlinearities depending on the derivative. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 3, pp. 413-426. http://geodesic.mathdoc.fr/item/CMUC_2003_44_3_a2/