Applications of Mathematics, first online, pp. 1-18


Nontrivial solutions to boundary value problems for semilinear $\Delta_\gamma$-differential equations

Duong Trong Luyen

Received December 15, 2019.   Published online February 8, 2021.

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 $\mathbb{R}^N$, $\Omega\cap\{x_j=0\}\ne\emptyset$ for some $j$, $\Delta_{\gamma}$ is a subelliptic linear operator of the type $$ \Delta_\gamma: =\sum_{j=1}^N\partial_{x_j} (\gamma_j^2 \partial_{x_j} ), \quad\partial_{x_j}:=\frac{\partial}{\partial x_j}, \quad N\ge2, $$ 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.
Keywords:  $\Delta_\gamma$-Laplace problem; Cerami condition; variational method; weak solution; Mountain Pass Theorem
Classification MSC:  35J70, 35J20, 35J25, 35D30
DOI:  10.21136/AM.2021.0363-19

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Affiliations:   Duong Trong Luyen, Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, 19 Nguyen Huu Tho street, Tan Phong ward, District 7, Ho Chi Minh City, Vietnam; Faculty of Mathematics and Statistics, Ton Duc Thang University, 19 Nguyen Huu Tho street, Tan Phong ward, District 7, Ho Chi Minh City, Vietnam, e-mail: duongtrongluyen@tdtu.edu.vn


 
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