On the stationary solutions of generalized reaction diffusion equations with \(p\)\& \(q\)-Laplacian
The authors deal with the study of the existence and non-existence results for solutions to the following stationary nonlinear equation \[ -\text{div}([|\nabla_x u|^{p-2}+ |\nabla_x u|^{q-2}]\,\nabla_x u)+ q(x)|u|^{p-2} u+ w(x)|u|^{q-2}_x= \lambda f(x)|u|^{q-2} u\tag{1} \] considered in a bounded domain \(\Omega\subset\mathbb{R}^d\), \(d\geq 1\), subject to the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). The authors study problem (1) with respect to the real parameter \(\lambda\). They introduce a new type variational principles corresponding to this family of equations.
scientific article; zbMATH DE number 2177439