Nonstandard competing anisotropic ( p , q ) $(p,q)$ -Laplacians with convolution

Abstract

Abstract A competing anisotropic ( p , q ) $(p,q)$ -Laplacian − ∑ i = 1 N ∂ ∂ x i ( | ∂ u ∂ x i | p i − 2 − μ | ∂ u ∂ x i | q i − 2 ) ∂ u ∂ x i = f ( x , ϕ ⋆ u , ∇ ( ϕ ⋆ u ) ) $$ -\overset{N}{\underset{i=1}{\sum}}\frac{\partial }{\partial x_{i}} \biggl( \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}-2}- \mu \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{q_{i}-2} \biggr) \frac{\partial u}{\partial x_{i}} =f \bigl(x, \phi \star u,\nabla (\phi \star u)\bigr) $$ as a nonstandard Dirichlet problem with convolutions on a bounded smooth domain in R N $\mathbb{R}^{N}$ , N ≥ 3 $N\geq 3$ is considered. Assume f : Ω × R × R N → R $f:\Omega \times \mathbb{R}\times \mathbb{R}^{N}\to \mathbb{R}$ is a Carathéodory function and ϕ ∈ L 1 ( R N ) $\phi \in L^{1}(\mathbb{R}^{N})$ . If μ > 0 $\mu >0$ , the existence of a generalized solution is proved. By the Galerkin basis for the space, a sequence that converges strongly to the solution is constructed. If μ ≤ 0 $\mu \leq 0$ , it is proved that any generalized solution is a weak solution.

Keywords

• Anisotropic operator
• Competing ( p , q ) $(p, q)$ -Laplacian
• Finite-dimensional approximation
• Weak solution
• Generalized solution