Multiplicity of solutions for a quasilinear elliptic equation with (p,q) $(p,q)$-Laplacian and critical exponent on RN $\mathbb{R}^{N}$

Boundary Value Problems

Chen Huang¹  Gao Jia²  Tiansi Zhang² 

[1] Business School, University of Shanghai for Science and Technology;College of Science, University of Shanghai for Science and Technology;

关键词: ( p , q ) $(p,q)$ -Laplacian;    Critical exponents;    Variational methods;    Genus;   

DOI  :  10.1186/s13661-018-1068-x

来源: DOAJ

【 摘 要 】

Abstract The multiplicity of solutions for a (p,q) $(p,q)$-Laplacian equation involving critical exponent −Δpu−Δqu=λV(x)|u|k−2u+K(x)|u|p∗−2u,x∈RN $$ -\Delta_{p}u-\Delta_{q}u=\lambda V(x) \vert u \vert ^{k-2}u+K(x) \vert u \vert ^{p^{\ast }-2}u,\quad x\in \mathbb{R}^{N} $$ is considered. By variational methods and the concentration–compactness principle, we prove that the problem possesses infinitely many weak solutions with negative energy for λ∈(0,λ∗) $\lambda\in(0,\lambda^{\ast})$. Moreover, the existence of infinitely many solutions with positive energy is also given for all λ>0 $\lambda>0$ under suitable conditions.

【 授权许可】

Unknown