Positive entire solutions for singular image-Laplacian equations on image with image Original Research Article

Suppose that constants p≥N≥2p≥N≥2, β≥0β≥0 and that View the MathML sourcef:RN×R+×RN→R is a continuous function with View the MathML sourceR+:=(0,∞). This paper discusses the existence of the positive entire solutions of the singular pp-Laplacian equation View the MathML sourcediv(|∇u|p−2∇u)=f(x,u,∇u)u−β,x∈RN Turn MathJax on and gives some sufficient conditions for the equation to have infinitely many positive entire solutions u(x)u(x) satisfying View the MathML sourceC1α(|x|)≤u(x)≤C2α(|x|)for |x|≥1, Turn MathJax on where C1C1, C1>0C1>0 are constants depending only on uu, α(|x|)=max{1,log|x|}α(|x|)=max{1,log|x|} for p=Np=N and α(|x|)=|x|(p−N)/(p−1)α(|x|)=|x|(p−N)/(p−1) for p>Np>N. The super–subsolution method is used to prove the existence of such solutions.