| Title:
|
Nontrivial solutions to boundary value problems for semilinear $\Delta _\gamma $-differential equations (English) |
| Author:
|
Luyen, Duong Trong |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
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1572-9109 (online) |
| Volume:
|
66 |
| Issue:
|
4 |
| Year:
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2021 |
| Pages:
|
461-478 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this article, we study the existence of nontrivial weak solutions for the following boundary value problem: $$ -\Delta _\gamma u=f(x,u) \ \text {in} \ \Omega , \quad u=0 \ \text {on} \ \partial \Omega , $$ where $\Omega $ is a bounded domain with smooth boundary in $\mathbb {R}^N$, $\Omega \cap \{x_j=0\}\ne \emptyset $ for some $j$, $\Delta _{\gamma }$ is a subelliptic linear operator of the type $$ \Delta _\gamma : =\sum _{j=1}^{N}\partial _{x_j} (\gamma _j^2 \partial _{x_j} ), \quad \partial _{x_j}:=\frac {\partial }{\partial x_{j}}, \quad N\ge 2, $$ where $\gamma (x) = (\gamma _1(x), \gamma _2(x),\dots ,\gamma _N(x))$ satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity $f(x,\xi )$ is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition. (English) |
| Keyword:
|
$\Delta _\gamma $-Laplace problem |
| Keyword:
|
Cerami condition |
| Keyword:
|
variational method |
| Keyword:
|
weak solution |
| Keyword:
|
Mountain Pass Theorem |
| MSC:
|
35D30 |
| MSC:
|
35J20 |
| MSC:
|
35J25 |
| MSC:
|
35J70 |
| idZBL:
|
07396164 |
| idMR:
|
MR4283300 |
| DOI:
|
10.21136/AM.2021.0363-19 |
| . |
| Date available:
|
2021-07-09T08:10:23Z |
| Last updated:
|
2023-09-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148968 |
| . |
| Reference:
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