A second-order estimate for blow-up solutions of elliptic equations

We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, in a bounded smooth domain Ω⊂R^N. b(x) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). The main results show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.     

Asymptotic behavior of large solutions to -Laplacian of Bieberbach–Rademacher type

By the Karamata regular variation theory and the method of lower and upper solutions, we establish the asymptotic behavior of boundary blow-up solutions of the quasilinear elliptic equation div(|u|^p−2u)=b(x)f(u) in a bounded ΩR^N subject to the singular boundary condition u(x)=∞, where the weight b(x) is non-negative and non-trivial in Ω, which may be vanishing on the boundary or go to unbounded, the nonlinear term f is a Γ-varying function at infinity, whose variation at infinity is not regular.     

Positive solutions for logistic type quasilinear elliptic equations on R

In this paper, we consider positive solutions of the logistic type p-Laplacian equation −Δ_pu=a(x)|u|^p−2u−b(x)|u|^q−1u, xR^N (N2). We show that under rather general conditions on a(x) and b(x) for large |x|, the behavior of the positive solutions for large |x| can be determined. This is then used to show that there is a unique positive solution. Our results improve the corresponding ones in J. London Math. Soc. (2) 64 (2001) 107–124 and J. Anal. Math., in press.     

Boundary blow-up rates of large solutions for elliptic equations with convection terms

By Karamata regular variation theory, a perturbation method and constructing comparison functions, we show the exact asymptotic behavior of large solutions to the semilinear elliptic equations with convection terms

     

Asymptotic behavior of large solution to elliptic equation of Bieberbach–Rademacher type with convection terms

We analyze the asymptotic behavior of solutions to nonlinear elliptic equation Δu±|u|^q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)=∞ as , where Ω is a smooth bounded domain in R^N, for some , and . Our approach employs Karamata regular variation theory combined with the method of lower and supper solution.     

Remarks on large solutions of a class of semilinear elliptic equations

In this paper, we show existence, uniqueness and exact asymptotic behavior of solutions near the boundary to a class of semilinear elliptic equations −Δu=λg(u)−b(x)f(u) in Ω, where λ is a real number, b(x)>0 in Ω and vanishes on ∂Ω. The special feature is to consider g(u) and f(u) to be regularly varying at infinity and b(x) is vanishing on the boundary with a more general rate function. The vanishing rate of b(x) determines the exact blow-up rate of the large solutions. And the exact blow-up rate allows us to obtain the uniqueness result.     

Singular solutions of perturbed logistic-type equations

We are concerned with the qualitative analysis of positive singular solutions with blow-up boundary for a class of logistic-type equations with slow diffusion and variable potential. We establish the exact blow-up rate of solutions near the boundary in terms of Karamata regular variation theory. This enables us to deduce the uniqueness of the singular solution.     

Asymptotic behavior of large solution for boundary blowup problems with non-linear gradient terms

Let Ω⊂R^N(N?3) be a bounded domain with smooth boundary. We show the asymptotic behavior of boundary blowup solutions to non-linear elliptic equation Δu±|∇u|^q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)=∞ as dist(x,∂Ω)→0,f is Γ-varying at ∞. Our analysis is based on the Karamata regular variation theory combined with the method of lower and supper solution.     

The exact blow-up rates of large solutions for semilinear elliptic equations

In this paper, combining the method of lower and upper solutions with the localization method, we establish the boundary blow-up rate of the large positive solutions to the singular boundary value problem

     

Symmetry of large solutions of nonlinear elliptic equations in a ball

Let g be a locally Lipschitz continuous real-valued function which satisfies the Keller-Osserman condition and is convex at infinity, then any large solution of −Δu+g(u)=0 in a ball is radially symmetric.     

Sharp estimates of bounded solutions to some semilinear second order dissipative equations

Let H, V be two real Hilbert spaces such that VH with continuous and dense imbedding, and let FC¹(V) be convex. By using differential inequalities, a close-to-optimal ultimate bound of the energy is obtained for solutions in to u^″+cu^′+bu+F(u)=f(t) whenever .     

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

In this paper we investigate boundary blow-up solutions of the problem

$\{\begin{array}{c}?{\mathrm{\Delta }}_{p(x)}u+f(x,u)=\pm K(x)|\mathrm{?}u{|}^{m(x)}\text{ in }\mathrm{\Omega },\hfill \\ u(x)\to +\infty \text{as }d(x,?\mathrm{\Omega })\to 0,\hfill \end{array}$

where ${\mathrm{\Delta }}_{p(x)}u=\mathrm{div}(|\mathrm{?}u{|}^{p(x)?2}\mathrm{?}u)$ is called the $p(x)$-Laplacian. Our results extend the previous work [25] of Y. Liang, Q.H. Zhang and C.S. Zhao from the radial case to the non-radial setting, and [46] due to Q.H. Zhang and D. Motreanu from the assumption that $K(x)|\mathrm{?}u(x){|}^{m(x)}$ is a small perturbation, to the case in which $\pm K(x)|\mathrm{?}u{|}^{m(x)}$ is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of $d(x,?\mathrm{\Omega })$ and in terms of the growth of the exponents. Furthermore, the comparison principle is no longer applicable in our context, since $f(x,?)$ is not assumed to be monotone in this paper.     

Boundary blow-up solutions to elliptic systems of competitive type

We consider the elliptic system Δu=u^pv^q, Δv=u^rv^s in Ω, where p,s>1, q,r>0, and Ω⊂R^N is a smooth bounded domain, subject to different types of Dirichlet boundary conditions: (F) u=λ, v=μ, (I) u=v=+∞ and (SF) u=+∞, v=μ on ∂Ω, where λ,μ>0. Under several hypotheses on the parameters p,q,r,s, we show existence and nonexistence of positive solutions, uniqueness and nonuniqueness. We further provide the exact asymptotic behaviour of the solutions and their normal derivatives near ∂Ω. Some more general related problems are also studied.     

Entire large solutions to semilinear elliptic systems

We consider the problem of existence of positive solutions to the elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on Rn (n?3) which satisfies . The parameters α and β are positive, and the nonnegative functions p and q are continuous and min{p(r),q(r)} does not have compact support. We show that if αβ?1, then such a solution exists if and only if the functions p and q satisfy