Multiple and Nodal Solutions for Parametric Dirichlet Equations Driven by the Double Phase Differential Operator

Abstract

We consider a nonlinear parametric Dirichlet problem driven by the double phase differential operator. Using variational tools combined with critical groups, we show that for all small values of the parameter, the problem has at least three nontrivial bounded solutions which are ordered and we provide the sign information for all of them. Two solutions are of constant sign and the third one is nodal. Finally, we determine the asymptotic behavior of the nodal solution as the parameter converges to zero.We consider a nonlinear parametric Dirichlet problem driven by the double phase differential operator. Using variational tools combined with critical groups, we show that for all small values of the parameter, the problem has at least three nontrivial bounded solutions which are ordered and we provide the sign information for all of them. Two solutions are of constant sign and the third one is nodal. Finally, we determine the asymptotic behavior of the nodal solution as the parameter converges to zero.