Applications of Mathematics, Vol. 66, No. 4, pp. 461-478, 2021
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.
References: [1] A. Ambrosetti, A. Malchiodi: Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge Studies in Advanced Mathematics 104. Cambridge University Press, Cambridge (2007). DOI 10.1017/CBO9780511618260 | MR 2292344 | Zbl 1125.47052
[2] A. Ambrosetti, P. H. Rabinowitz: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349-381. DOI 10.1016/0022-1236(73)90051-7 | MR 0370183 | Zbl 0273.49063
[3] C. T. Anh, B. K. My: Existence of solutions to $\Delta_\lambda$-Laplace equations without the Ambrosetti-Rabinowitz condition. Complex Var. Elliptic Equ. 61 (2016), 137-150. DOI 10.1080/17476933.2015.1068762 | MR 3428858 | Zbl 1336.35164
[4] H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011). DOI 10.1007/978-0-387-70914-7 | MR 2759829 | Zbl 1220.46002
[5] G. Cerami: An existence criterion for the critical points on unbounded manifolds. Rend., Sci. Mat. Fis. Chim. Geol. 112 (1978), 332-336. (In Italian.) MR 0581298 | Zbl 0436.58006
[6] G. Cerami: Sull'esistenza di autovalori per un problema al contorno non lineare. Ann. Mat. Pura Appl., IV. Ser. 124 (1980), 161-179. (In Italian.) DOI 10.1007/BF01795391 | MR 0591554 | Zbl 0441.35054
[7] B. Franchi, E. Lanconelli: An embedding theorem for Sobolev spaces related to non-smooth vector fields and Harnack inequality. Commun. Partial Differ. Equations 9 (1984), 1237-1264. DOI 10.1080/03605308408820362 | MR 0764663 | Zbl 0589.46023
[8] N. Garofalo, E. Lanconelli: Existence and nonexistence results for semilinear equations on the Heisenberg group. Indiana Univ. Math. J. 41 (1992), 71-98. DOI 10.1512/iumj.1992.41.41005 | MR 1160903 | Zbl 0793.35037
[9] V. V. Grushin: On a class of hypoelliptic operators. Math. USSR, Sb. 12 (1970), 458-476; translation from Mat. Sb. (N.S.) 83 (1970), 456-473. DOI 10.1070/SM1970v012n03ABEH000931 | MR 0279436 | Zbl 0252.35057
[10] D. S. Jerison: The Dirichlet problem for the Kohn Laplacian on the Heisenberg group II. J. Funt. Anal. 43 (1981), 224-257. DOI 10.1016/0022-1236(81)90031-8 | MR 0633978 | Zbl 0493.58022
[11] D. S. Jerison, J. M. Lee: The Yamabe problem on CR manifolds. J. Diff. Geom. 25 (1987), 167-197. DOI 10.4310/jdg/1214440849 | MR 0880182 | Zbl 0661.32026
[12] A. E. Kogoj, E. Lanconelli: On semilinear $\Delta_\lambda$-Laplace equation. Nonlinear Anal., Theory Methods Appl., Ser. A 75 (2012), 4637-4649. DOI 10.1016/j.na.2011.10.007 | MR 2927124 | Zbl 1260.35020
[13] A. E. Kogoj, E. Lanconelli: Linear and semilinear problems involving $\Delta_\lambda$-Laplacians. Electron. J. Differ. Equ. 2018 (2018), Article ID 25, 167-178. MR 3883635 | Zbl 1400.35130
[14] N. Lam, G. Lu: $N$-Laplacian equations in $\mathbb{R}^N$ with subcritical and critical growth without the Ambrosetti-Rabinowitz condition. Adv. Nonlinear Stud. 13 (2013), 289-308. DOI 10.1515/ans-2013-0203 | MR 3076792 | Zbl 1283.35049
[15] N. Lam, G. Lu: Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition. J. Geom. Anal. 24 (2014), 118-143. DOI 10.1007/s12220-012-9330-4 | MR 3145918 | Zbl 1305.35069
[16] G. Li, C. Wang: The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition. Ann. Acad. Sci. Fenn., Math. 36 (2011), 461-480. DOI 10.5186/aasfm.2011.3627 | MR 2865507 | Zbl 1234.35095
[17] S. Li, S. Wu, H.-S. Zhou: Solutions to semilinear elliptic problems with combined nonlinearities. J. Differ. Equations 185 (2002), 200-224. DOI 10.1006/jdeq.2001.4167 | MR 1935276 | Zbl 1032.35072
[18] S. Liu: On superlinear problems without the Ambrosetti and Rabinowitz condition. Nonlinear Anal., Theory Methods Appl., Ser. A 73 (2010), 788-795. DOI 10.1016/j.na.2010.04.016 | MR 2653749 | Zbl 1192.35074
[19] Z. Liu, Z.-Q. Wang: On the Ambrosetti-Rabinowitz superlinear condition. Adv. Nonlinear Stud. 4 (2004), 563-574. DOI 10.1515/ans-2004-0411 | MR 2100913 | Zbl 1113.35048
[20] D. T. Luyen: Two nontrivial solutions of boundary-value problems for semilinear $\Delta_{\gamma}$ differential equations. Math. Notes 101 (2017), 815-823. DOI 10.1134/S0001434617050078 | MR 3669606 | Zbl 1375.35200
[21] D. T. Luyen: Existence of nontrivial solution for fourth-order semilinear $\Delta_{\gamma}$-Laplace equation in $\mathbb{R}^N$. Electron. J. Qual. Theory Differ. Equ. 2019 (2019), Article ID 78, 12 pages. DOI 10.14232/ejqtde.2019.1.78 | MR 4028910 | Zbl 07174921
[22] D. T. Luyen, N. M. Tri: Existence of solutions to boundary-value problems for semilinear $\Delta_{\gamma}$ differential equations. Math. Notes 97 (2015), 73-84. DOI 10.1134/S0001434615010101 | MR 3394492 | Zbl 1325.35051
[23] D. T. Luyen, 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), 141-161. DOI 10.4064/ap3831-3-2016 | MR 3539074 | Zbl 1356.35057
[24] D. T. Luyen, N. M. Tri: Large-time behavior of solutions to degenerate damped hyperbolic equations. Sib. Math. J. 57 (2016), 632-649; translation from Sib. Mat. Zh. 57 (2016), 809-829. DOI 10.1134/S0037446616040078 | MR 3601331 | Zbl 1364.35045
[25] D. T. Luyen, N. M. Tri: Existence of infinitely many solutions for semilinear degenerate Schrödinger equations. J. Math. Anal. Appl. 461 (2018), 1271-1286. DOI 10.1016/j.jmaa.2018.01.016 | MR 3765489 | Zbl 1392.35146
[26] D. T. Luyen, N. M. Tri: Infinitely many solutions for a class of perturbed degenerate elliptic equations involving the Grushin operator. Complex Var. Elliptic Equ. 65 (2020), 2135-2150. DOI 10.1080/17476933.2020.1730824 | MR 4170200
[27] O. H. Miyagaki, M. A. S. Souto: Superlinear problems without Ambrosetti and Rabinowitz growth condition. J. Differ. Equations 245 (2008), 3628-3638. DOI 10.1016/j.jde.2008.02.035 | MR 2462696 | Zbl 1158.35400
[28] P. H. Rabinowitz: Minimax Methods in Critical Point theory with Applications to Differential Equations. Regional Conference Series in Mathematics 65. American Mathematical Society, Providence (1986). DOI 10.1090/cbms/065 | MR 0845785 | Zbl 0609.58002
[29] M. Schechter, W. Zou: Superlinear problems. Pac. J. Math. 214 (2004), 145-160. DOI 10.2140/pjm.2004.214.145 | MR 2039130 | Zbl 1134.35346
[30] N. T. C. Thuy, N. M. Tri: Some existence and nonexistence results for boundary value problems for semilinear elliptic degenerate operators. Russ. J. Math. Phys. 9 (2002), 365-370. MR 1965388 | Zbl 1104.35306
[31] P. T. Thuy, N. M. Tri: Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations. NoDEA, Nonlinear Differ. Equ. Appl. 19 (2012), 279-298. DOI 10.1007/s00030-011-0128-z | MR 2926298 | Zbl 1247.35028
[32] P. T. Thuy, N. M. Tri: Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators. NoDEA, Nonlinear Differ. Equ. Appl. 20 (2013), 1213-1224. DOI 10.1007/s00030-012-0205-y | MR 3057173 | Zbl 1268.35018
[33] N. M. Tri: Critical Sobolev exponent for degenerate elliptic operators. Acta Math. Vietnam. 23 (1998), 83-94. MR 1628086 | Zbl 0910.35060
[34] N. M. Tri: On Grushin's equation. Math. Notes 63 (1998), 84-93; translation from Mat. Zametki 63 (1998), 95-105. DOI 10.1007/BF02316146 | MR 1631852 | Zbl 0913.35049
[35] N. M. Tri: Semilinear Degenerate Elliptic Differential Equations: Local and Global Theories. Lambert Academic Publishing, Saarbrücken (2010).
[36] N. M. Tri: Recent Progress in the Theory of Semilinear Equations Involving Degenerate Elliptic Differential Operators. Publ. House Sci. Technology, Hanoi (2014).
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
