Multiplicity of Positive Solutions for a p-q-Laplacian Type Equation with Critical Nonlinearities - Tsing-San Hsu; Huei-Li Lin (Abstract and Applied Analysis)
We study the effect of the coefficient f(x) of the critical nonlinearity on the number of positive solutions for a p-q-Laplacian equation. Under suitable assumptions for f(x) and g(x), we should prove that for sufficiently small λ>0, there exist at least k positive solutions of the following p-q-Laplacian equation, -Δpu-Δqu=fxu|p*-2u+λgxu|r-2u  in  Ω, u=0   on   ∂Ω, where Ω⊂RN is a bounded smooth domain, N>p, 1<q<N(p-1)/(N-1)<p≤max⁡{p,p^*-q/(p-1)}<r<p^*, p^*=Np/(N-p) is the critical Sobolev exponent, and Δsu=div(|∇u|s-2∇u is the s-Laplacian of u.