The existence of nontrivial solutions to nonlinear elliptic equation of image–image-Laplacian type on image Original Research Article

In this paper, we study the following nonlinear elliptic equation of pp–qq-Laplacian type on RNRN: View the MathML source{−Δpu+a(x)|u|p−2u−Δqu+b(x)|u|q−2u=f(x,u)+g(x),x∈RNu∈W≡W1,p(RN)⋂W1,q(RN)(⋆) Turn MathJax on where 10a(x)≡m>0, b(x)≡n>0b(x)≡n>0 for some constants mm and nn, then the problem (⋆⋆) has at least one nontrivial weak solution (see Theorem 1.12), generalizing a similar result for pp-Laplacian type equation in [J.F. Yang, X.P. Zhu, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded Domains(I)Positive mass case, Acta Math. Sci. 7 (1987) 341–359]. Also, we prove that under essentially the same assumptions on f(x,t)f(x,t) as that in Theorem 1.12, there exists a constant C>0C>0, such that if ‖g‖∗