Three nontrivial solutions of boundary value problems for semilinear $\Delta_{\gamma}-$Laplace equation
Resumo
In this paper, we study the existence of multiple solutions for the boundary value problem
\begin{equation}
\Delta_{\gamma} u+f(x,u)=0 \quad \mbox{ in } \Omega, \quad \quad
u=0 \quad \mbox{ on } \partial \Omega, \notag
\end{equation}
where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N \ (N \ge 2)$
and $\Delta_{\gamma}$ is the subelliptic operator of the type $$
\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \ \partial_{x_j}
=\frac{\partial }{\partial x_{j}}, \gamma = (\gamma_1, \gamma_2, ..., \gamma_N),
$$
the nonlinearity $f(x , \xi)$ is subcritical growth and may be not satisfy the Ambrosetti-Rabinowitz (AR) condition. We establish the existence of three nontrivial solutions by using Morse theory.
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Referências
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
C. T. Anh and B. K. My, Existence of solutions to ∆λ−Laplace equations without the Ambrosetti - Rabinowitz condition, Complex Var. Elliptic Equ. 61(2016), no. 1, 137-150. https://doi.org/10.1080/17476933.2015.1068762
C. T. Anh and B. K. My, Liouville - type theorems for elliptic inequalities involving the ∆λ−Laplace operator, Complex Var. Elliptic Equ. 61(2016), no. 7, 1002-1013. https://doi.org/10.1080/17476933.2015.1131685
K. C. Chang; Infinite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, 6. Birkhauser Boston, Inc., Boston, MA, 1993. x+312 pp. https://doi.org/10.1007/978-1-4612-0385-8_1
N. Garofalo and E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41(1992), no. 1, 71-98. https://doi.org/10.1512/iumj.1992.41.41005
V. V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.) 83 (1970), no. 125, 456-473 [in Russian].
L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on Rn, Proc. Roy. Soc. Edinburgh Sect. A 129(1999), no. 4, 787-809. https://doi.org/10.1017/S0308210500013147
D. S. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25(1987),no. 2, 167-197. https://doi.org/10.4310/jdg/1214440849
M. Y. Jiang, Critical groups and multiple solutions of the p−Laplacian equations, Nonlinear Anal., 59 (2004), 1221-1241. https://doi.org/10.1016/S0362-546X(04)00324-4
D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, II, J. Funct. Anal. 43(1981), no. 2, 224-257. https://doi.org/10.1016/0022-1236(81)90031-8
A. E. Kogoj and E. Lanconelli, On semilinear ∆λ−Laplace equation, Nonlinear Analysis. 75 (2012), no. 12, 4637-4649. https://doi.org/10.1016/j.na.2011.10.007
A. E. Kogoj and E. Lanconelli, Linear and semilinear problems involving ∆λ−Laplacians, Electron. J. Differential Equations 2018, no. 25, 12 pp.
D. T. Luyen, D. T. Huong and L. T. H. Hanh, Existence of infinitely many solutions for ∆γ−Laplace problems Math. Notes 103 (2018), no. 5, 724-736. https://doi.org/10.1134/S000143461805005X
D. T. Luyen, Two nontrivial solutions of boundary value problems for semilinear ∆γ differential equations, Math. Notes 101 (2017), no. 5, 815-823. https://doi.org/10.1134/S0001434617050078
D. T. Luyen and N. M. Tri, Existence of solutions to boundary value problems for semilinear ∆γ differential equations, Math. Notes 97 (2015), no. 1, 73-84. https://doi.org/10.1134/S0001434615010101
D. T. Luyen and N. M. Tri, Large-time behavior of solutions to damped hyperbolic equation involving strongly degenerate elliptic differential operators, Siberian Math. J. 57 (2016), no. 4, 632-649. https://doi.org/10.1134/S0037446616040078
D. T. Luyen and N. M. Tri, Global attractor of the Cauchy problem for a semilinear degenerate damped hyperbolic equation involving the Grushin operator, Ann. Pol. Math. 117 (2016), no. 2, 141-162. https://doi.org/10.4064/ap3831-3-2016
D. T. Luyen and N. M. Tri, Existence of infinitely many solutions for semilinear degenerate Schr¨odinger equations, J. Math. Anal. Appl. 461 (2018), no. 2, 1271-1286. https://doi.org/10.1016/j.jmaa.2018.01.016
D. T. Luyen and N. M. Tri, On the existence of multiple solutions to boundary value problems for semilinear elliptic degenerate operators. Complex Var. Elliptic Equ. 64 (2019), no. 6, 1050-1066. https://doi.org/10.1080/17476933.2018.1498086
S. I. Pohozaev, On the eigenfunctions of the equation ∆u +λf(u) = 0. (Russian) Dokl. Akad. Nauk SSSR, 165, (1965), 36-39.
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. viii+100 pp. https://doi.org/10.1090/cbms/065
B. Rahal and M. K. Hamdani, Infinitely many solutions for ∆α−Laplace equations with sign-changing potential, J. Fixed Point Theory Appl. 20 (2018), no. 4, 20:137. https://doi.org/10.1007/s11784-018-0617-3
N. T. C. Thuy and N. M. Tri, Some existence and nonexistence results for boundary value problems for semilinear elliptic degenerate operators, Russ. J. Math. Phys. 9(2002), no. 3, 365-370.
P. T. Thuy and N. M. Tri, Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations, NoDEA Nonlinear Differential Equations Appl. 19 (2012), no. 3, 279-298. https://doi.org/10.1007/s00030-011-0128-z
N. M. Tri, On the Grushin equation, Mat. Zametki 63(1998), no. 1 , 95-105. https://doi.org/10.4213/mzm1251
N. M. Tri, Critical Sobolev exponent for hypoelliptic operators, Acta Math. Vietnam. 23 (1998), no. 1, 83-94.
N. M. Tri, Semilinear Degenerate Elliptic Differential Equations, Local and global theories (Lambert Academic Publishing, 2010).
N. M. Tri, Recent Progress in the Theory of Semilinear Equations Involving Degenerate Elliptic Differential Operators (Publishing House for Science and Technology of the Vietnam Academy of Science and Technology, 2014).
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Funding data
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National Foundation for Science and Technology Development
Grant numbers 101.02-2017.21