Least energy solutions with sign information for parametric double phase problems
The authors study a parametric double phase Dirichlet problem of the form \begin{align*} \begin{cases} -\Delta_p^q u-\Delta_qu=\lambda|u|^{q-2}u +f(x,u)& \text{in }\Omega,\\ u\big|_{\partial\Omega}=0, \ 1<q<p, \ \lambda\in\mathbb{R}, & \end{cases} \end{align*} where \(a\in L^\infty(\Omega)\) with \(a(z)>0\) for a.\,a.\,\(x\in\Omega\), \(\Delta_p^a\) denotes the weighted \(p\)-Laplace differential operator defined by \begin{align*} \Delta_p^au=\text{div}\left(a(z)|\nabla u|^{p-2}\nabla u\right), \end{align*} while the superlinear perturbation in the reaction satisfies a weak Nehari-type monotonicity. The authors show that the problem has at least three nontrivial solutions all with sign information.
parametric double phase Dirichlet problem
scientific article; zbMATH DE number 7466710
scientific article