Nontrivial solutions to boundary value problems for semilinear $\Delta _\gamma $-differential equations
Applications of Mathematics, Tome 66 (2021) no. 4, pp. 461-478
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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.
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.
DOI :
10.21136/AM.2021.0363-19
Classification :
35D30, 35J20, 35J25, 35J70
Keywords: $\Delta _\gamma $-Laplace problem; Cerami condition; variational method; weak solution; Mountain Pass Theorem
Keywords: $\Delta _\gamma $-Laplace problem; Cerami condition; variational method; weak solution; Mountain Pass Theorem
@article{10_21136_AM_2021_0363_19,
author = {Luyen, Duong Trong},
title = {Nontrivial solutions to boundary value problems for semilinear $\Delta _\gamma $-differential equations},
journal = {Applications of Mathematics},
pages = {461--478},
year = {2021},
volume = {66},
number = {4},
doi = {10.21136/AM.2021.0363-19},
mrnumber = {4283300},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.2021.0363-19/}
}
TY - JOUR AU - Luyen, Duong Trong TI - Nontrivial solutions to boundary value problems for semilinear $\Delta _\gamma $-differential equations JO - Applications of Mathematics PY - 2021 SP - 461 EP - 478 VL - 66 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.21136/AM.2021.0363-19/ DO - 10.21136/AM.2021.0363-19 LA - en ID - 10_21136_AM_2021_0363_19 ER -
%0 Journal Article %A Luyen, Duong Trong %T Nontrivial solutions to boundary value problems for semilinear $\Delta _\gamma $-differential equations %J Applications of Mathematics %D 2021 %P 461-478 %V 66 %N 4 %U http://geodesic.mathdoc.fr/articles/10.21136/AM.2021.0363-19/ %R 10.21136/AM.2021.0363-19 %G en %F 10_21136_AM_2021_0363_19
Luyen, Duong Trong. Nontrivial solutions to boundary value problems for semilinear $\Delta _\gamma $-differential equations. Applications of Mathematics, Tome 66 (2021) no. 4, pp. 461-478. doi: 10.21136/AM.2021.0363-19
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