Abstract
This paper provides a positive solution for the $(p,q)$ -Laplace equation in $\mathbb{R}^{N}$ with a nonlinear term depending on the gradient. The solution is constructed as the limit of positive solutions in bounded domains. Strengthening the growth condition, it is shown that the solution is also bounded. The positivity of the solution is obtained through a new comparison principle. Finally, under a stronger growth condition, we show the existence of a vanishing at infinity solution.
Highlights
1 Introduction In this paper, we study the existence of a solution for the following quasi-linear elliptic equation: (P)
The right-hand side of the equation in (P) is in the form of convection term, meaning a nonlinearity that depends on the point x in RN, on the solution u, and on its gradient ∇u
Under a stronger version of the growth condition (F ), we show that any positive solution disappear at infinity
Summary
We study the existence of a (positive) solution for the following quasi-linear elliptic equation:. The existence of positive solutions for problems with p-Laplacian and convection term on a bounded domain has been studied in [ – ]. Assume that u ∈ C (D) is a positive subsolution of problem ( ) and that h : D × R × RN → R is continuous function such that h(x, t, ξ ) ≥ g(t) for all x ∈ RN , ξ ∈ RN , and t ∈ ( , u L∞(D)]. In the case where u and u satisfy the homogeneous Dirichlet boundary condition we can state the following: Theorem Let g : R → R be a continuous function such that t –qg(t) is nonincreasing for t > if μ > and t –pg(t) is nonincreasing for t > if μ =. We can conclude as in the proof of Theorem
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