The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems

Drábek, Pavel. "The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems." Mathematica Bohemica 120.2 (1995): 169-195. <http://eudml.org/doc/247813>.

@article{Drábek1995,
abstract = {We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem \begin\{align\}-\operatorname\{div\}(a(x,u)||^\{p-2\}\nabla u) = &\lambda b(x,u)|u|^\{p-2\}u \quad \text\{ in \} \Omega , u = &0 \hspace\{56.9055pt\}\text\{ on \} \partial \Omega , \end\{align\} where $\Omega $ is a bounded domain, $p>1$ is a real number and $a(x,u)$, $b(x,u)$ satisfy appropriate growth conditions. Moreover, the coefficient $a(x,u)$ contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in $L^\infty (\Omega )$. The main tool is the investigation of the associated homogeneous eigenvalue problem and an application of the Schauder fixed point theorem.},
author = {Drábek, Pavel},
journal = {Mathematica Bohemica},
keywords = {boundedness of eigenfunction; weighted Sobolev space; Schauder fixed point theorem; degenerated quasilinear partial differential equations; weak solutions; eigenvalue problems; boundedness of the solution; boundedness of eigenfunction; weighted Sobolev space; Schauder fixed point theorem},
language = {eng},
number = {2},
pages = {169-195},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems},
url = {http://eudml.org/doc/247813},
volume = {120},
year = {1995},
}

TY - JOUR
AU - Drábek, Pavel
TI - The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems
JO - Mathematica Bohemica
PY - 1995
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 120
IS - 2
SP - 169
EP - 195
AB - We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem \begin{align}-\operatorname{div}(a(x,u)||^{p-2}\nabla u) = &\lambda b(x,u)|u|^{p-2}u \quad \text{ in } \Omega , u = &0 \hspace{56.9055pt}\text{ on } \partial \Omega , \end{align} where $\Omega $ is a bounded domain, $p>1$ is a real number and $a(x,u)$, $b(x,u)$ satisfy appropriate growth conditions. Moreover, the coefficient $a(x,u)$ contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in $L^\infty (\Omega )$. The main tool is the investigation of the associated homogeneous eigenvalue problem and an application of the Schauder fixed point theorem.
LA - eng
KW - boundedness of eigenfunction; weighted Sobolev space; Schauder fixed point theorem; degenerated quasilinear partial differential equations; weak solutions; eigenvalue problems; boundedness of the solution; boundedness of eigenfunction; weighted Sobolev space; Schauder fixed point theorem
UR - http://eudml.org/doc/247813
ER -

References

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Citations in EuDML Documents

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Abdelali Sabri, Ahmed Jamea, Hamad Talibi Alaoui, Weak solution for nonlinear degenerate elliptic problem with Dirichlet-type boundary condition in weighted Sobolev spaces

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