首页|Multiplicity of solutions for the semilinear subelliptic Dirichlet problem

Multiplicity of solutions for the semilinear subelliptic Dirichlet problem

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In this paper,we study the semilinear subelliptic equation {-ΔXu=f(x,u)+g(x,u)in Ω,u=0 on ∂Ω,where ΔX=-∑mi=1X*iXi is the self-adjoint Hörmander operator associated with the vector fields X=(X1,X2,.…,Xm)satisfying the Hörmander's condition,f(x,u)∈ C(-Ω) ×R),g(x,u)is a Carathéodory function on Ω × R,and Ω is an open bounded domain in Rn with smooth boundary.Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values,we obtain two kinds of existence results for multiple weak solutions to the problem above.Furthermore,we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations.Compared with the classical elliptic cases,both approaches here have their own strengths in the degenerate cases.This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases.

degenerate elliptic equationsHörmander operatorsperturbation methodMorse index

Hua Chen、Hong-Ge Chen、Jin-Ning Li、Xin Liao

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School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China

Wuhan Institute of Physics and Mathematics,Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences,Wuhan 430071,China

国家自然科学基金国家重点研发计划国家自然科学基金Innovation Program of Wuhan-Shuguang Project中国博士后科学基金China National Postdoctoral Program for Innovative Talents

121310172022YFA10056021220160720230102010202862023T160655BX20230270

2024

10.1007/s11425-023-2242-6

中国科学:数学(英文版)
中国科学院

中国科学:数学(英文版)

CSTPCD
影响因子：0.36
ISSN：1674-7283
年,卷(期)：2024.67(3)
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