Existence Of Boundary Blow-up Solutions Research Articles

Sharp conditions for the existence of boundary blow-up solutions to the Monge–Ampère equation

In this paper we give sharp conditions on K(x) and f(u) for the existence of strictly convex solutions to the boundary blow-up Monge–Ampere problem $$\begin{aligned} M[u](x)=K(x)f(u) \quad \hbox {for } x \in \Omega ,\; u(x)\rightarrow +\,\infty \quad \hbox {as } \mathrm{dist}(x,\partial \Omega )\rightarrow 0. \end{aligned}$$ Here $$M[u]=\det \, (u_{x_{i}x_{j}})$$ is the Monge–Ampere operator, and $$\Omega $$ is a smooth, bounded, strictly convex domain in $$ \mathbb {R}^N \, (N\ge 2)$$ . Further results are obtained for the special case that $$\Omega $$ is a ball. Our approach is largely based on the construction of suitable sub- and super-solutions.

Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term

AbstractMotivated by the work [9], in this paper we investigate the infinite boundary value problem associated with the semilinear PDE{Lu=f(u)+h(x)}on bounded smooth domains{\Omega\subseteq\mathbb{R}^{n}}, whereLis a non-divergence structure uniformly elliptic operator with singular lower-order terms. In the equation,fis a continuous non-decreasing function that satisfies the Keller–Osserman condition, whilehis a continuous function in Ω that may change sign, and which may be unbounded on Ω. Our purpose is two-fold. First we study some sufficient conditions onfandhthat would ensure existence of boundary blow-up solutions of the above equation, in which we allow the lower-order coefficients to be singular on the boundary. The second objective is to provide sufficient conditions onfandhfor the uniqueness of boundary blow-up solutions. However, to obtain uniqueness, we need the lower-order coefficients ofLto be bounded in Ω, but we still allowhto be unbounded on Ω.

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.

On the Existence of Boundary Blow-up Solutions for a General Class of Quasilinear Elliptic Systems

Abstract In this paper, we investigate the following quasilinear elliptic system (P) with explosive boundary conditions: Δpu = f1(x, u, v) in Ω; u|∂Ω = +∞, u >0 in Ω, Δqv = f2(x, u, v) in Ω; v|∂Ω = +∞, v >0 in Ω, where Ω is a smooth bounded domain of ℝN, N ≥ 2, 1 < p, q < +∞ and f1, f2 are two Carathéodory functions in Ω × (ℝ+* × ∗ ℝ+*). Under rather general conditions on f1 and f2 and assuming the existence of a sub and supersolutions pair, we prove the existence of a large solution to (P) by a fixed point approach. Then, we apply this result considering particular systems arising in Biology.

ON THE BOUNDARY BLOW-UP SOLUTIONS OF $p(x)$-LAPLACIAN EQUATIONS WITH GRADIENT TERMS

In this paper we investigate boundary blow-up solutions of the problem $$ \left\{ \begin{array}{l} -\triangle _{p(x)}u+f(x,u)=\rho (x,u)+K(\left\vert x\right\vert )\left\vert \nabla u\right\vert ^{\delta (\left\vert x\right\vert )}\text{ in }\Omega \text{,} \\[12pt] \text{ }u(x)\rightarrow +\infty \text{ as }d(x,\text{ }\partial \Omega )\rightarrow 0\text{,} \end{array} \right. $$ where $-\triangle _{p(x)}u=-\mbox{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)$ is called $p(x)$-Laplacian. The existence of boundary blow-up solutions is proved and the singularity of boundary blow-up solution is also given for several cases including the case of $\rho (x,u)$ being a large perturbation (namely, $\frac{\rho (x,u(x))}{f(x,u(x))}\rightarrow 1$ as $x\rightarrow \partial \Omega $). In particular, we do not have the comparison principle.

Existence and Asymptotic Behavior of Boundary Blow-Up Solutions for Weightedp(x)-Laplacian Equations with Exponential Nonlinearities

This paper investigates the followingp(x)-Laplacian equations with exponential nonlinearities:−Δp(x)u+ρ(x)ef(x,u)=0inΩ,u(x)→+∞asd(x,∂Ω)→0, where−Δp(x)u=−div(|∇u|p(x)−2∇u)is calledp(x)-Laplacian,ρ(x)∈C(Ω). The asymptotic behavior of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given.

Existence of boundary blow-up solutions for a quasilinear elliptic equation

In this paper, the existence of solution for a class of quasilinear elliptic problem div(|∇ u| p−2 ∇ u)=a(x)f(u), u≥0 in Ω=B (the unit ball), with the boundary blow-up condition u| ∂Ω=+∞ is established, where a(x)∈C(Ω) blows up on ∂Ω,p>1 and f is assumed to satisfy (f 1) and (f 2). The results are obtained by using sub-supersolution methods.

Boundary Blow-Up Solutions to p(x)-Laplacian Equations with Exponential Nonlinearities

This paper investigates the -Laplacian equations with exponential nonlinearities in , as , where is called - Laplacian. The singularity of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given.

Existence of boundary blow-up solutions for a class of quasilinear elliptic systems with critical case

We consider the quasilinear elliptic system div ( | ∇ u | p - 2 ∇ u ) = u m 1 v n 1 , div ( | ∇ v | q - 2 ∇ v ) = u m 2 v n 2 in Ω , where m 1 > p - 1 , n 2 > q - 1 , m 2 , n 1 > 0 , and Ω ⊂ R N is a smooth bounded domain, subject to three different types of Dirichlet boundary conditions: u = λ , v = μ or u = v = + ∞ or u = + ∞ , v = μ on ∂ Ω , where λ , μ > 0 . Under several hypotheses on the parameters m 1 , n 1 , m 2 , n 2 which is a critical case, we show that the existence of positive solutions.

Existence of boundary blow-up solutions for a class of quasiliner elliptic systems for the subcritical case

In this paper, we study positivesolution of the following system of quasilinear elliptic equations div$(|\nabla u|^{p-2}\nabla u)=u^{m_1}v^{n_1},$ in $\Omega$div$(|\nabla v|^{q-2}\nabla v)=u^{m_2}v^{n_2},$ in $\Omega,$ $\qquad\qquad\qquad\qquad$ (0.1) where$m_1>p-1,n_2>q-1, m_2,n_1>0$, and $\Omega\subset R^N$ is a smoothbounded domain, subject to three different types of Dirichletboundary conditions: $u=\lambda, v=\mu$ or $u=v=+\infty$ or$u=+\infty, v=\mu$ on $\partial\Omega$, where $\lambda, \mu>0$.Under several hypotheses on the parameters $m_1,n_1,m_2,n_2$, weshow that the existence of positive solutions. We further provide the asymptoticbehaviors of the solutions near $\partial\Omega$. Some more generalrelated problems are also studied.

Boundary blow-up solutions of elliptic equations involving regional fractional Laplacian

In this paper, we study existence of boundary blow-up solutions for elliptic equations involving regional fractional Laplacian. We also discuss the optimality of our results.