Nonstandard Dirichlet problems with competing (p,q)-Laplacian, convection, and convolution

Dumitru Motreanu, Viorica Venera Motreanu


The paper focuses on a nonstandard Dirichlet problem driven by the operator $-\Delta_p +\mu\Delta_q$, which is a competing $(p,q)$-Laplacian with lack of ellipticity if $\mu>0$, and exhibiting a reaction term in the form of a convection (i.e., it depends on the solution and its gradient) composed with the convolution of the solution with an integrable function. We prove the existence of a generalized solution through a combination of fixed-point approach and approximation. In the case $\mu\leq 0$, we obtain the existence of a weak solution to the respective elliptic problem.


competing (p,q)-Laplacian; Dirichlet problem; convection; convolution; generalized solution; weak solution

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