Multiplicity of positive solutions for a \(p\)-\(q\)-Laplacian type equation with critical nonlinearities
Summary: 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 \(\lambda>0\), there exist at least \(k\) positive solutions of the following \(p\)-\(q\)-Laplacian equation, \(-\Delta_p u-\Delta_q u=f(x)|u|^{p^*-2}u+\lambda g(x)|u|^{r-2}u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(\Omega \subset \mathbb R^N\) is a bounded smooth domain, \(N>p\), \(1<q<N(p - 1)/(N - 1)<p \leq\max \{p,p^* - q/(p - 1) \} < r < p^*\), \(p^* = Np/(N - p)\) is the critical Sobolev exponent, and \(\Delta_s u = \operatorname{div}(|\nabla u|^{s-2}\nabla u)\) is the \(s\)-Laplacian of \(u\).
scientific article; zbMATH DE number 7023153
scientific article