Minimax Theory and its Applications 05 (2020), No. 1, 007--018
Copyright Heldermann Verlag 2020
Infinitely Many Solutions for Semilinear Δγ-Differential Equations in RN without the Ambrosetti-Rabinowitz Condition
Duong Trong Luyen
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
and: Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
duongtrongluyen@tdtu.edu.vn
Le Thi Hong Hanh
Dept. of Mathematics, Hoa Lu University, Ninh Nhat, Ninh Binh City, Vietnam
honghanhtn2212@gmail.com
[Abstract-pdf]
We study the existence of infinitely many nontrivial solutions of the semilinear $\Delta_{\gamma}$-differential equations in $\mathbb{R}^N$ $$ - \Delta_{\gamma}u+ b(x)u=f(x,u)\quad \mbox{ in } \mathbb{R}^N, $$ where $\Delta_{\gamma}$ is the subelliptic operator of the type $$ \Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \quad \partial_{x_j}: =\frac{\partial }{\partial x_{j}},\quad \gamma: = (\gamma_1, \gamma_2, ..., \gamma_N), $$ and the potential $b(x)$ and nonlinearity $f(x, u)$ are not assumed to be continuous, moreover $f$ may not satisfy the Ambrosetti-Rabinowitz (AR) condition. Under some growth conditions on $b$ and $f$, we show that there are infinitely many solutions to the problem.
Keywords: Delta-sub-gamma-Laplace problems, Cerami condition, variational method, weak solutions, Mountain Pass Theorem.
MSC: 35J70, 35J20; 35J10.
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