Nontrivial solutions to boundary value problems for semilinear Δ γ -differential equations

Duong Trong Luyen

Applications of Mathematics (2021)

  • Volume: 66, Issue: 4, page 461-478
  • ISSN: 0862-7940

Abstract

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In this article, we study the existence of nontrivial weak solutions for the following boundary value problem: - Δ γ u = f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain with smooth boundary in ℝ N , Ω ∩ { x j = 0 } ≠ ∅ for some j , Δ γ is a subelliptic linear operator of the type Δ γ : = ∑ j = 1 N ∂ x j ( γ j 2 ∂ x j ) , ∂ x j : = ∂ ∂ x j , N ≥ 2 , where γ ( x ) = ( γ 1 ( x ) , γ 2 ( x ) , ⋯ , γ N ( x ) ) satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity f ( x , ξ ) is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.

How to cite

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Luyen, Duong Trong. "Nontrivial solutions to boundary value problems for semilinear $\Delta _\gamma $-differential equations." Applications of Mathematics 66.4 (2021): 461-478. <http://eudml.org/doc/297746>.

@article{Luyen2021,
abstract = {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 \lbrace x_j=0\rbrace \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.},
author = {Luyen, Duong Trong},
journal = {Applications of Mathematics},
keywords = {$\Delta _\gamma $-Laplace problem; Cerami condition; variational method; weak solution; Mountain Pass Theorem},
language = {eng},
number = {4},
pages = {461-478},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nontrivial solutions to boundary value problems for semilinear $\Delta _\gamma $-differential equations},
url = {http://eudml.org/doc/297746},
volume = {66},
year = {2021},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 4
SP - 461
EP - 478
AB - 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 \lbrace x_j=0\rbrace \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.
LA - eng
KW - $\Delta _\gamma $-Laplace problem; Cerami condition; variational method; weak solution; Mountain Pass Theorem
UR - http://eudml.org/doc/297746
ER -

References

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