| Title:
|
Positive solutions of critical quasilinear elliptic equations in $R \sp N$ (English) |
| Author:
|
Binding, Paul A. |
| Author:
|
Drábek, Pavel |
| Author:
|
Huang, Yin Xi |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
124 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
149-166 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the existence of positive solutions of
-\Delta_pu=\lambda g(x)|u|^{p-2}u+\alpha h(x)|u|^{q-2}u+f(x)|u|^{p^*-2}u\eqno(1)
in $\Bbb R^N$, where $\lambda, \alpha\in\Bbb R$, $1<p<N$, $p^*=Np/(N-p)$, the critical Sobolev exponent, and $1<q<p^*$, $q\ne p$. Let $\lambda_1^+>0$ be the principal eigenvalue of
-\Delta_pu=\lambda g(x)|u|^{p-2}u \quad\text{in} \Rn, \qquad\int_{\Rn} g(x)|u|^p>0, \eqno(2)
with $u_1^+>0$ the associated eigenfunction. We prove that, if $\int_{\Bbb R^N}f|u_1^+|^{p^*}<0$, $\int_{\Bbb R^N}h|u_1^+|^q>0$ if $1<q<p$ and $\int_{\Bbb R^N}h|u_1^+|^q<0$ 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. (English) |
| Keyword:
|
positive solutions |
| Keyword:
|
critical exponent |
| Keyword:
|
the $p$-Laplacian |
| MSC:
|
35B33 |
| MSC:
|
35J70 |
| MSC:
|
35P30 |
| MSC:
|
47J30 |
| MSC:
|
58E05 |
| idZBL:
|
Zbl 0937.35075 |
| idMR:
|
MR1780688 |
| DOI:
|
10.21136/MB.1999.126255 |
| . |
| Date available:
|
2009-09-24T21:36:30Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126255 |
| . |
| Reference:
|
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| Reference:
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