Positive solutions of critical quasilinear elliptic equations in $R^N$

Binding, Paul A., Drábek, Pavel, and Huang, Yin Xi. "Positive solutions of critical quasilinear elliptic equations in $R ^N$." Mathematica Bohemica 124.2-3 (1999): 149-166. <http://eudml.org/doc/248458>.

@article{Binding1999,
abstract = {We consider the existence of positive solutions of -pu=g(x)|u|p-2u+h(x)|u|q-2u+f(x)|u|p*-2u(1) in $\mathbb \{R\}^N$, where $\lambda , \alpha \in \mathbb \{R\}$, $1<p<N$, $p^*=Np/(N-p)$, the critical Sobolev exponent, and $1<q<p^*$, $q\ne p$. Let $\lambda _1^+>0$ be the principal eigenvalue of -pu=g(x)|u|p-2u    in ,        g(x)|u|p>0, (2) with $u_1^+>0$ the associated eigenfunction. We prove that, if $\int _\{\mathbb \{R\}^N\}f|u_1^+|^\{p^*\}<0$, $\int _\{\mathbb \{R\}^N\}h|u_1^+|^q>0$ if $1<q<p$ and $\int _\{\mathbb \{R\}^N\}h|u_1^+|^q<0$ if $p<q<p^*$, then there exist $\lambda ^*>\lambda _1^+$ and $\alpha ^*>0$, such that for $\lambda \in [\lambda _1^+, \lambda ^*)$ and $\alpha \in [0, \alpha ^*)$, (1) has at least one positive solution.},
author = {Binding, Paul A., Drábek, Pavel, Huang, Yin Xi},
journal = {Mathematica Bohemica},
keywords = {positive solutions; critical exponent; the $p$-Laplacian; -Laplacian; positive solutions; critical exponent},
language = {eng},
number = {2-3},
pages = {149-166},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Positive solutions of critical quasilinear elliptic equations in $R ^N$},
url = {http://eudml.org/doc/248458},
volume = {124},
year = {1999},
}

TY - JOUR
AU - Binding, Paul A.
AU - Drábek, Pavel
AU - Huang, Yin Xi
TI - Positive solutions of critical quasilinear elliptic equations in $R ^N$
JO - Mathematica Bohemica
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 124
IS - 2-3
SP - 149
EP - 166
AB - We consider the existence of positive solutions of -pu=g(x)|u|p-2u+h(x)|u|q-2u+f(x)|u|p*-2u(1) in $\mathbb {R}^N$, where $\lambda , \alpha \in \mathbb {R}$, $1<p<N$, $p^*=Np/(N-p)$, the critical Sobolev exponent, and $1<q<p^*$, $q\ne p$. Let $\lambda _1^+>0$ be the principal eigenvalue of -pu=g(x)|u|p-2u    in ,        g(x)|u|p>0, (2) with $u_1^+>0$ the associated eigenfunction. We prove that, if $\int _{\mathbb {R}^N}f|u_1^+|^{p^*}<0$, $\int _{\mathbb {R}^N}h|u_1^+|^q>0$ if $1<q<p$ and $\int _{\mathbb {R}^N}h|u_1^+|^q<0$ if $p<q<p^*$, then there exist $\lambda ^*>\lambda _1^+$ and $\alpha ^*>0$, such that for $\lambda \in [\lambda _1^+, \lambda ^*)$ and $\alpha \in [0, \alpha ^*)$, (1) has at least one positive solution.
LA - eng
KW - positive solutions; critical exponent; the $p$-Laplacian; -Laplacian; positive solutions; critical exponent
UR - http://eudml.org/doc/248458
ER -