Abstract
Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.
Highlights
IntroductionSuch a problem without any available ellipticity but with a variational structure that prevents to have convection was studied for the first time in [1]
The object of the paper is to study the following quasilinear problem with homogeneous Dirichlet boundary condition − Δpu + Δqu = f (x, u, ∇u) in Ω, (1)u = 0 on ∂Ω on a bounded domain Ω ⊂ N with the boundary ∂Ω
Note that in −Δp + Δq there is competition between −Δp and −Δq taking their difference and destroying the ellipticity in contrast to what happens in the case of −Δp − Δq
Summary
Such a problem without any available ellipticity but with a variational structure that prevents to have convection was studied for the first time in [1]. A minimal condition of ellipticity for an operator in divergence form −div(a(|∇u|)) (see, e.g., [2]) is to have, among other things, a(t) > 0 for all t > 0, which is not satisfied in the case of a(t) = t p−2 − tq−2 Another important aspect of problem (1) is the presence of the convection term f (x, u, ∇u). Such methods cannot be directly implemented in the case of problem (1) taking into account the lack of needed ellipticity We overcome this difficulty by resolving finite dimensional approximated problems and passing to the limit in an appropriate sense. The rest of the paper consists of sections regarding mathematical background and hypotheses, approximate solutions and existence of generalized solutions to problem (1)
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