Existence and blow-up rate of large solutions of p(x)-Laplacian equations with large perturbation and gradient terms

Qihu Zhang (1) , Dumitru Motreanu (2)

1 Zhengzhou University of Light Industry
2 LAMPS - LAboratoire de Modélisation Pluridisciplinaire et Simulations

In this paper, we investigate boundary blow-up solutions of the problem ⎧⎨⎩−Δp(x)u+f(x,u)=ρ(x,u)+K(x)|∇u|m(x) in Ω,u(x)→+∞ as d(x,∂Ω)→0, where Δp(x)u=div(|∇u|p(x)−2∇u) is called p(x)-Laplacian. Our results extend the previous work of J. García-Melián, A. Suárez [23] from the case where p(⋅)≡2, without gradient term, to the case where p(⋅) is a function, with gradient term. It also extends the previous work of Y. Liang, Q.H. Zhang and C.S. Zhao [38] from the radial case in the problem to the non-radial case. The existence of boundary blow-up solutions is established and the singularity of boundary blow-up solution is also studied for several cases including when ρ(x,u(x))f(x,u(x))→1 as x→∂Ω, which means that ρ(x,u) is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of d(x,∂Ω). Hence, the results of this paper are new even in the canonical case p(⋅)≡2. In particular, we do not have the comparison principle, because we don't make the monotone assumption of nonlinear term.

Domaines: Mathématiques [math]

Qihu Zhang, Dumitru Motreanu. Existence and blow-up rate of large solutions of p(x)-Laplacian equations with large perturbation and gradient terms. Advances in Differential Equations, 2016, 21 (7/8), pp.699-734. ⟨10.57262/ade/1462298655⟩. ⟨hal-01368479⟩