Blow-up Rates Of Large Solutions Research Articles

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{−Δp(x)u+f(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 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)|∇u(x)|m(x) is a small perturbation, to the case in which ±K(x)|∇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,∂Ω) 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 Furthermore, the comparison principle is no longer applicable in our context, since f(x,⋅) is not assumed to be monotone in this paper.

Blow-up rates of large solutions for infinity Laplace equations

In this paper, by constructing suitable comparison functions, we mainly give the boundary behavior of solutions to boundary blow-up elliptic problems ▵∞u=b(x)f(u),x∈Ω,u|∂Ω=+∞, where Ω is a bounded domain with smooth boundary in RN, the operator △∞ is the ∞-Laplacian, b∈Cα(Ω¯) which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary and the nonlinear term f is a Γ-varying function at infinity, whose variation at infinity is not regular.

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

In this paper, we investigate boundary blow-up solutions of the problem \begin{equation*} \quad \left\{ \begin{array}{l} -\Delta _{p(x)}u+f(x,u)=\rho (x,u)+K(x)|\nabla u|^{m(x)}\mbox{ in }\Omega , \\[2mm] u(x)\rightarrow +\infty \mbox{ as }d(x,\partial \Omega )\rightarrow 0, \end{array} \right. \end{equation*} where $\Delta _{p(x)}u=\mathrm{div}\,(|\nabla u|^{p(x)-2}\nabla u)$ is called $p(x)$-Laplacian. Our results extend the previous work of J. García-Melián, A. Suárez [23] from the case where $p(\cdot )\equiv 2 $, without gradient term, to the case where $p(\cdot )$ 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 $\frac{\rho (x,u(x))}{f(x,u(x))}\rightarrow 1$ as $x\rightarrow \partial \Omega $, which means that $\rho (x,u)$ is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of $d(x,\partial \Omega )$. Hence, the results of this paper are new even in the canonical case $p(\cdot )\equiv 2$. In particular, we do not have the comparison principle, because we don't make the monotone assumption of nonlinear term.

Blow-up rates of large solutions for a ϕ-Laplacian problem with gradient term

We study the behaviour of solutions of a boundary blow-up elliptic problem on a bounded domain Ω with smooth boundary in ℝN. The data of the problem consist of an increasing function f : ℝ+ → ℝ+ and two real regularly varying functions ϕ and g.

Existence, Uniqueness and Blow-up Rate of Large Solutions of Quasi-linear Elliptic Equations with Higher Order and Large Perturbation

Existence, Uniqueness and Blow-up Rate of Large Solutions of Quasi-linear Elliptic Equations with Higher Order and Large Perturbation

Blow-up Rates of Large Solutions for Quasilinear Elliptic Equations

In this paper, we mainly study the boundary behavior of solutions to boundary blow-up quasilinear elliptic problem { div(|∇u|m−2∇u) = b(x)f(u), x ∈ Ω, u|∂Ω = +∞, where Ω is a bounded domain with smooth boundary in R , m > 1, b ∈ C(Ω) which is positive in Ω and maybe vanishing on the boundary and rapidly varying near the boundary.

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

We use Karamata regular variation theory to study the exact asymptotic behavior of large solutions near the boundary to a class of quasilinear elliptic equations with convection terms

Blow-up rate of the unique solution for a class of one-dimensional [formula omitted]-Laplacian equations

For more general nonlinear term f, this paper shows the exact blow-up rate of the unique solution l(t) to the singular boundary value problem (|u′(t)|p−2u′(t))′=b(t)f(u(t)),u(t)>0,t>0,1<p≤2,u(0)=∞,u(∞)=0, where b∈C1(0,∞), which is positive and non-decreasing on (0,∞), may be vanishing on the boundary. Our results extend some previous findings of [L. Wei, J. Zhu, The existence and blow-up rate of large solutions of one-dimensional p-Laplacian equations, Nonlinear Anal.: Real World Appl. 13 (2012) 665–676] in a certain sense.

Blow-up rate and uniqueness of singular radial solutions for a class of quasi-linear elliptic equations

We establish the uniqueness and the blow-up rate of the large positive solution of the quasi-linear elliptic problem − Δ p u = λ u p − 1 − b ( x ) h ( u ) in B R ( x 0 ) with boundary condition u = + ∞ on ∂ B R ( x 0 ) , where B R ( x 0 ) is a ball centered at x 0 ∈ R N with radius R, N ⩾ 3 , 2 ⩽ p < ∞ , λ > 0 are constants and the weight function b is a positive radially symmetrical function. We only require h ( u ) to be a locally Lipschitz function with h ( u ) / u p − 1 increasing on ( 0 , ∞ ) and h ( u ) ∼ u q − 1 for large u with q > p − 1 . Our results extend the previous work [Z. Xie, Uniqueness and blow-up rate of large solutions for elliptic equation − Δ u = λ u − b ( x ) h ( u ) , J. Differential Equations 247 (2009) 344–363] from case p = 2 to case 2 ⩽ p < ∞ .

The existence and blow-up rate of large solutions of one-dimensional [formula omitted]-Laplacian equations

In this paper, we consider the existence, uniqueness and blow-up rate of positive solutions of the singular boundary value problem { ( | u ′ ( t ) | p − 2 u ′ ( t ) ) ′ = f ( t ) h ( u ( t ) ) , t > 0 , u ( 0 ) = + ∞ , u ( + ∞ ) = 0 , where f is a nondecreasing and continuous function such that f ( 0 ) ≥ 0 , and h is a non-negative function satisfying the Keller–Osserman condition. Our results extend some previous findings of [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (2008) 3180–3203] in a certain sense.

Blow-up rates of large solutions for semilinear elliptic equations

In this paper we analyze the blow-up rates of large solutions tothe semilinear elliptic problem $\Delta u =b(x)f(u), x\in \Omega, u|_{\partial \Omega} = +\infty,$ where $\Omega$ is a bounded domain with smooth boundaryin $R^N$, $f$ is rapidly varying or normalised regularly varying with index $p$ ($p>1$) at infinity, and $b \inC^\alpha (\bar{\Omega})$ which is non-negative in $\Omega$ and positivenear the boundary and may be vanishing on the boundary.

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 { Δ u = b ( x ) f ( u ) , x ∈ Ω , u ( x ) = + ∞ , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R N . The weight function b ( x ) is a non-negative continuous function in the domain, which vanishes on the boundary of the underlying domain Ω at different rates according to the point of the boundary. f ( u ) is locally Lipschitz continuous satisfying the Keller–Osserman condition and f ( u ) / u is increasing on ( 0 , ∞ ) . It is worth emphasizing that we obtain the main results for a large class of nonlinear terms f , which is regularly varying at infinity with index p ∈ R (that is for all ξ > 0 , lim u → ∞ f ( ξ u ) / f ( u ) = ξ p ), instead of the restriction: f ( u ) ∼ H u p for sufficiently large u and some positive constants H > 0 , p > 1 as in the series of papers of J. L. Gómez.

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 { Δ u ± | ∇ u | q = b ( x ) f ( u ) , x ∈ Ω , u ( x ) = + ∞ , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R N . The weight function b ( x ) is a non-negative continuous function in the domain, which may be vanishing on the boundary or be singular on the boundary. f ( u ) ∈ C 2 [ 0 , + ∞ ) is increasing on ( 0 , ∞ ) satisfying the Keller–Osserman condition, and regularly varying at infinity with index ρ > 1 .

Blow-up rates of large solutions for elliptic equations

In this paper, we mainly study the boundary behavior of solutions to boundary blow-up elliptic problems for more general nonlinearities f (which may be rapidly varying at infinity) Δ u = b ( x ) f ( u ) , x ∈ Ω , u | ∂ Ω = + ∞ , where Ω is a bounded domain with smooth boundary in R N , and b ∈ C α ( Ω ¯ ) which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary. Further, when f ( s ) = s p ± f 1 ( s ) for s sufficiently large, where p > 1 and f 1 is normalized regularly varying at infinity with index p 1 ∈ ( 0 , p ) , we show the influence of the geometry of Ω on the boundary behavior for solutions to the problem. We also give the existence and uniqueness of solutions.

Uniqueness and blow-up rate of large solutions for elliptic equation [formula omitted

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem { − Δ u = λ u − b ( x ) h ( u ) in Ω , u = + ∞ on ∂ Ω , where Ω is a smooth bounded domain in R N . The weight function b ( x ) is a non-negative continuous function in the domain. h ( u ) is locally Lipschitz continuous and h ( u ) / u is increasing on ( 0 , ∞ ) and h ( u ) ∼ H u p for sufficiently large u with H > 0 and p > 1 . Naturally, the blow-up rate of the problem equals its blow-up rate for the very special, but important, case when h ( u ) = H u p . We distinguish two cases: (I) Ω is a ball domain and b is a radially symmetric function on the domain in Theorem 1.1; (II) Ω is a smooth bounded domain and b satisfies some local condition on each boundary normal section assumed in Theorem 1.2. The blow-up rate is explicitly determined by functions b and h. In case (I), the singular boundary value problem has a unique solution u satisfying lim d ( x ) → 0 u ( x ) K H − β ( b ∗ ( ‖ x − x 0 ‖ ) ) − β = 1 , where d ( x ) = dist ( x , ∂ Ω ) , b ∗ ( r ) and K are defined by b ∗ ( r ) = ∫ r R ∫ s R b ( t ) d t d s , K = [ β ( ( β + 1 ) C 0 − 1 ) ] 1 p − 1 , β : = 1 p − 1 . In case (II), the blow-up rates of the solutions to the boundary value problem are established and the uniqueness is proved.

Blow-up rate of the unique solution for a class of one-dimensional problems on the half-line

For more general nonlinear term g, the paper shows the exact blow-up rate of the unique solution ψ ( t ) to the singular boundary value problem u ″ ( t ) = b ( t ) g ( u ( t ) ) , u ( t ) > 0 , t > 0 , u ( 0 ) = ∞ , u ( ∞ ) = 0 , where b ∈ C 1 ( 0 , ∞ ) , which is positive and non-decreasing on ( 0 , ∞ ) (may vanish at zero). Our results are obtained in a more general setting to those in [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (12) (2008) 3180–3203], where g ( u ) ≅ u p ( p > 1 ) for sufficiently large u.

Blow-up rates of radially symmetric large solutions

This paper adapts a technical device going back to [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385–439] to ascertain the blow-up rate of the (unique) radially symmetric large solution given through the main theorem of [J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst., Supplement dedicated to the 6th AIMS Conference, Poitiers, France, 2007, pp. 677–686]. The requested underlying estimates are based upon the main theorem of [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (2008) 3180–3203]. Precisely, we show that if Ω is a ball, or an annulus, f ∈ C [ 0 , ∞ ) is positive and non-decreasing, V ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) satisfies V ( 0 ) = 0 , V ′ ( u ) > 0 , V ″ ( u ) ⩾ 0 , for every u > 0 , and V ( u ) ∼ H u p − 1 as u ↑ ∞ , for some H > 0 and p > 1 , then, for each λ ⩾ 0 , − Δ u = λ u − f ( dist ( x , ∂ Ω ) ) V ( u ) u possesses a unique positive large solution in Ω, L, which must be radially symmetric, by uniqueness, and we can estimate the exact blow-up rate of L ( x ) at ∂ Ω in terms of p, H and f (see Theorem 1.1).

Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line

This paper shows the existence and the uniqueness of the positive solution ℓ ( t ) of the singular boundary value problem { u ″ ( t ) = f ( t ) h ( u ( t ) ) , t > 0 , u ( 0 ) = ∞ , u ( ∞ ) = 0 , where f is a continuous non-decreasing function such that f ( 0 ) ⩾ 0 , and h is a non-negative function satisfying the Keller–Osserman condition. Moreover, it also ascertains the exact blow-up rate of ℓ ( t ) at t = 0 in the special case when there exist H > 0 and p > 1 such that h ( u ) ∼ H u p for sufficiently large u. Naturally, the blow-up rate of the problem in such a case equals its blow-up rate for the very special, but important, case when h ( u ) = H u p for all u ⩾ 0 . So, our results are substantial improvements of some previous findings of [J. López-Gómez, Uniqueness of large solutions for a class of radially symmetric elliptic equations, in: S. Cano-Casanova, J. López-Gómez, C. Mora-Corral (Eds.), Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, World Scientific, 2005, pp. 75–110] and [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385–439].

The exact boundary blow-up rate of large solutions for semilinear elliptic problems

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem − △ u = λ u − a ( x ) u p , u | ∂ Ω = + ∞ , where Ω is a bounded smooth domain in R N . The weight function a ( x ) in front of the nonlinearity can vanish on the boundary of the domain Ω at different rates according to the point x 0 of the boundary. The decay rate of the weight function a ( x ) may not be approximated by a power function of distance near the boundary ∂ Ω . We combine the localization method of [J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] with some previous radially symmetric results of [T. Ouyang, Z. Xie, The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation, Nonlinear Anal. 64 (9) (2006) 2129–2142] to prove that any large solution u ( x ) must satisfy lim x → x 0 u ( x ) K ( b x 0 ∗ ( dist ( x , ∂ Ω ) ) ) − β = 1 for each x 0 ∈ ∂ Ω , where b x 0 ∗ ( r ) = ∫ 0 r ∫ 0 s b x 0 ( t ) d t d s , K = [ β ( ( β + 1 ) C 0 − 1 ) ] 1 p − 1 , β = 1 p − 1 , C 0 = lim r → 0 ( ∫ 0 r b x 0 ( t ) d t ) 2 b x 0 ∗ ( r ) b x 0 ( r ) and b x 0 ( r ) is the boundary normal section of a ( x ) at x 0 ∈ ∂ Ω , i.e., b x 0 ( r ) = a ( x 0 − r n x 0 ) , r > 0 , r ∼ 0 , and n x 0 stands for the outward unit normal vector at x 0 ∈ ∂ Ω .

On the blow-up rate of large solutions for a porous media logistic equation on radial domain

In this paper we establish the exact blow-up rate of the large solutions of a porous media logistic equation. We consider the carrying capacity function with a general decay rate at the boundary instead of the usual cases when it can be approximated by a distant function. Obtaining the accurate blow-up rate allows us to establish the uniqueness result. Our result covers all previous results on the ball domain and can be further adapted in a more general domain.