Uniqueness Of Boundary Blow-up Solutions Research Articles

Boundary blow-up solutions to a class of degenerate elliptic equations

Let \(\Omega \) be a bounded domain in \(\mathbb {R}^N=\mathbb {R}^{N_1} \times \mathbb {R}^{N_2}\) with \(N_1, N_2 \ge 1\), and \(N(s) = N_1 + (1+s)N_2\) be the homogeneous dimension of \(\mathbb {R}^N\) for \(s \ge 0\). In this paper, we prove the existence and uniqueness of boundary blow-up solutions to the following semilinear degenerate elliptic equation $$\begin{aligned} {\left\{ \begin{array}{ll} G_s u = {|x|^{2s}} u^p_+ \; &{}\text { in } \Omega ,\\ u(z)\rightarrow +\infty \; &{}\text { as } {d}(z) \rightarrow 0, \end{array}\right. } \end{aligned}$$ where \(u_+ = \max \{u,0\}\), \(1<p<{{N(s)+2s} \over {N(s)-2}}\), and d(z) denotes the Grushin distance from z to the boundary of \(\Omega \). Here \(G_s\) is the Grushin operator of the form $$\begin{aligned} G_s u= \Delta _x u + |x|^{2s}\Delta _y u, \; s\ge 0. \end{aligned}$$ It is worth noticing that our results do not require any assumption on the smoothness of the domain \(\Omega \), and when \(s=0\), we cover the previous results for the Laplace operator \(\Delta \).

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.

Elliptic systems with boundary blow-up: existence, uniqueness and applications to removability of singularities

In this paper we consider the elliptic system $\Delta u = u^p -v^q$, $\Delta v= -u^r +v^s$ in $\Omega$, where the exponents verify $p,s>1$, $q,r>0$ and $ps>qr$, and $\Omega$ is a smooth bounded domain of $R^N$. First, we show existence and uniqueness of boundary blow-up solutions, that is, solutions $(u,v)$ verifying $u=v=+\infty$ on $\partial \Omega$. Then, we use them to analyze the removability of singularities of positive solutions of the system in the particular case $qr\leq 1$, where comparison is available.

Asymptotic behavior and uniqueness of boundary blow-up solutions to elliptic equations

In this paper, under some structural assumptions of weight function b(x) and nonlinear term f (u), we establish the asymptotic behavior and uniqueness of boundary blow-up solutions to semilinear elliptic equations { ∆u = b(x) f (u), x ∈ Ω, u(x) = ∞, x ∈ ∂Ω, where Ω ⊂ RN is a bounded smooth domain. Our analysis is based on the Karamata regular variation theory and the Lopez-Gomez localization method.

Uniqueness of boundary blowup solutions to [formula omitted]-curvature equation

We consider the boundary blowup problem for the k-curvature equation, i.e., Hk[u]=f(u)g(|Du|) in an n-dimensional domain Ω, with the boundary condition u(x)→∞ as dist(x,∂Ω)→0. We prove the uniqueness result under some hypotheses.

Uniqueness of boundary blow-up solutions on unbounded domain of [formula omitted

In this paper, we consider the existence of positive solutions of the degenerate logistic type elliptic equation − Δ u = a ( x ) u − b ( x ) f ( u ) , x ∈ R N ∖ D u | ∂ D = ∞ , where N ≥ 2 , D ⊂ R N is a bounded smooth domain and a ( x ) , b ( x ) are continuous functions on R N with b ( x ) ≥ 0 , b ( x ) ≢ 0 , especially b ( x ) = 0 on ∂ D and f ( u ) ∈ C ( 0 , ∞ ) . We show that under rather general conditions on a ( x ) and b ( x ) for large | x | and f ( u ) behaves like u q where q > 1 . Without the behavior of b ( x ) near the boundary ∂ D , it will be shown that there exists a unique positive solution. Our results improve the corresponding ones in [W. Dong, Pang, Uniqueness of boundary blow-up solutions on exterior domain of R N , J. Math. Anal. Appl. 330 (2007), 654–664] and [Y. Du, L. Ma, Logistic type equations on R N by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., (2) 64 (2001), 107–124].

On Uniqueness of Boundary Blow-Up Solutions of a Class of Nonlinear Elliptic Equations

We study boundary blow-up solutions of semilinear elliptic equations Lu = u + p with p > 1, or Lu = e au with a > 0, where L is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.

Uniqueness of boundary blow-up solutions on exterior domain of [formula omitted

In this paper, we consider the existence and uniqueness of positive solutions of the degenerate logistic type elliptic equation − Δ u = a ( x ) u − b ( x ) | u | q − 1 u , x ∈ R N ∖ D , u | ∂ D = ∞ , where N ⩾ 2 , D ⊂ R N is a bounded domain with smooth boundary and a ( x ) , b ( x ) are continuous functions on R N with b ( x ) ⩾ 0 , b ( x ) ≢ 0 . We show that under rather general conditions on a ( x ) and b ( x ) for large | x | , there exists a unique positive solution. Our results improve the corresponding ones in [W. Dong, Y. Du, Unbounded principal eigenfunctions and the logistic equation on R N , Bull. Austral. Math. Soc. 67 (2003) 413–427] and [Y. Du, L. Ma, Logistic type equations on R N by a squeezing method involving boundary blow-up solutions, J. London Math. Soc. (2) 64 (2001) 107–124].

Remarks on uniqueness of boundary blow-up solutions

We show that there is at most one nonnegative boundary blow-up solution for the one-dimensional boundary blow-up problem ( | u ′ | p − 2 u ′ ) ′ = f ( u ) in ( 0 , 1 ) , u ( 0 ) = ∞ , u ( 1 ) = ∞ where p > 1 provided f ∈ C 1 ( 0 , ∞ ) ∩ C 0 [ 0 , ∞ ) with f ( s ) > 0 and f ′ ( s ) ≥ 0 for s ∈ ( 0 , ∞ ) . We see that the same result still holds for some equations with special nonlinearities satisfying f ( s ) > 0 and f ′ ( s ) ≥ 0 for s ∈ ( 0 , ∞ ) in higher dimensions, but we conjecture that the same result should be true for equations with general nonlinearities satisfying f ( s ) > 0 and f ′ ( s ) ≥ 0 for s ∈ ( 0 , ∞ ) in higher dimensions.