Boundary Blow-up Solutions Research Articles

The Classical Boundary Blow-Up Solutions for a Class of Gaussian Curvature Equations

The Classical Boundary Blow-Up Solutions for a Class of Gaussian Curvature Equations

Boundary blow-up solutions for the Monge-Ampère equation with an invariant gradient type term

We prove existence and non-existence of the boundary blow-up solutions for the Monge–Ampère equation of the form detD2u+g(u)ϕ(∇u,D2u)=b(x)f(u) in a smooth, bounded, strictly convex domain Ω⊂Rn(n≥2), where b∈C∞(Ω) is positive in Ω, the pair of functions f and g satisfies conditions of Keller–Osserman type on [0,∞), and ϕ:Rn×Rn2→R is a quadratic term which is invariant for both the gradient ∇u and the Hessian D2u in the sense discussed by Kazdan and Kramer (1978).

Boundary blow-up solutions to singular k-Hessian equations with gradient terms: Sufficient and necessary conditions and asymptotic behavior

Let Ω be a smooth, bounded and strictly (k−1)-convex domain in RN(N≥2). Assume that b and f are smooth positive functions and b may be singular near ∂Ω. We provide sufficient and necessary conditions on b, f and q for the existence of k-convex solutions to the boundary blow-up k-Hessian problemSk(D2u)=b(x)f(u)|∇u|q for x∈Ω,u(x)→+∞ as dist(x,∂Ω)→0. The asymptotic behavior of such solutions is also investigated. Moreover, when b has strong singularity near ∂Ω, we study the existence and nonexistence of boundary blow-up solutions.

The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation

Abstract The primary objective of this article is to analyze the existence of infinitely many radial p p - k k -convex solutions to the boundary blow-up p p - k k -Hessian problem σ k ( λ ( D i ( ∣ D u ∣ p − 2 D j u ) ) ) = H ( ∣ x ∣ ) f ( u ) in Ω , u = + ∞ on ∂ Ω . {\sigma }_{k}\left(\lambda \left({D}_{i}\left({| Du| }^{p-2}{D}_{j}u)))=H\left(| x| )f\left(u)\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{0.33em}u=+\infty \hspace{0.33em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega . Here, k ∈ { 1 , 2 , … , N } k\in \left\{1,2,\ldots ,N\right\} , σ k ( λ ) {\sigma }_{k}\left(\lambda ) is the k k -Hessian operator, and Ω \Omega is a ball in R N ( N ≥ 2 ) {{\mathbb{R}}}^{N}\hspace{0.33em}\left(N\ge 2) . Our methods are mainly based on the sub- and super-solutions method.

Sufficient and Necessary Conditions on the Existence and Estimates of Boundary Blow-Up Solutions for Singular p-Laplacian Equations

Sufficient and Necessary Conditions on the Existence and Estimates of Boundary Blow-Up Solutions for Singular p-Laplacian Equations

Qualitative properties of blow-up solutions to some semilinear elliptic systems in non-convex domains

In this paper we study qualitative properties of boundary blow-up solutions to some semilinear elliptic cooperative systems in bounded non-convex domains. In particular, by a careful adaptation of the celebrated moving plane procedure of Alexandrov–Serrin, we deduce symmetry and monotonicity results for blow-up solutions for this class of systems.

BOUNDARY BLOW-UP SOLUTIONS TO EQUATIONS INVOLVING THE INFINITY LAPLACIAN

AbstractIn this paper, we study the boundary blow-up problem related to the infinity Laplacian $$ \begin{align*}\begin{cases} \Delta_{\infty}^h u=u^q &\mathrm{in}\; \Omega, \\ u=\infty &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$ where $\Delta _{\infty }^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator associated with the infinity Laplacian arising from the stochastic game named Tug-of-War. When $q>h>1$ , we establish the existence of the boundary blow-up viscosity solution. Moreover, when the domain satisfies some regular condition, we establish the asymptotic estimate of the blow-up solution near the boundary. As an application of the asymptotic estimate and the comparison principle, we obtain the uniqueness result of the large solution. We also give the nonexistence of the large solution for the case $q \leq h.$

Large solutions of a class of degenerate equations associated with infinity Laplacian

Abstract In this article, we investigate the boundary blow-up problem Δ ∞ h u = f ( x , u ) , in Ω , u = ∞ , on ∂ Ω , \left\{\begin{array}{ll}{\Delta }_{\infty }^{h}u=f\left(x,u),& {\rm{in}}\hspace{0.33em}\Omega ,\\ u=\infty ,& {\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ ∞ h u = ∣ D u ∣ h − 3 ⟨ D 2 u D u , D u ⟩ {\Delta }_{\infty }^{h}u=| Du\hspace{-0.25em}{| }^{h-3}\langle {D}^{2}uDu,Du\rangle is the highly degenerate operator related to infinity Laplacian which comes from the absolutely minimizing Lipschitz extension and has a close relationship with the random game named tug-of-war. When the function f f satisfies the Keller-Osserman-type condition, we establish the existence of the boundary blow-up viscosity solution. Moreover, for the separable case f ( x , u ) = b ( x ) g ( u ) f\left(x,u)=b\left(x)g\left(u) , we establish the asymptotic estimate of the blow-up solution near the boundary under some regular conditions of the domain. Based on the asymptotic estimate and comparison principle, we obtain the uniqueness of the large viscosity solution. During this procedure, we also study the non-existence of the large solution. For the separable case, we show that the Keller-Osserman-type condition is sufficient and necessary for the existence of the boundary blow-up viscosity solution.

Boundary blow-up solutions to the k-Hessian equation with the logarithmic nonlinearity and singular weights

Boundary blow-up solutions to the k-Hessian equation with the logarithmic nonlinearity and singular weights

Exact principal blowup rate near the boundary of boundary blowup solutions to k-curvature equation

We consider boundary blowup problems for k-curvature equations of the form $$H_k[u] = f(u)g(|Du|)$$ in a bounded smooth domain $$\Omega \subset \mathbb {R}^n$$ , where f(s) behaves like $$s^p$$ as $$s \rightarrow \infty $$ and g(t) behaves like $$t^{-q}$$ as $$t \rightarrow \infty $$ . We obtain the exact principal blowup rate of a solution u near the boundary $$\partial \Omega $$ under some conditions.

Second order boundary estimate of boundary blowup solutions to k-Hessian equation

We consider boundary blowup problem for k-Hessian equation of the form Fk[u]=μdα(x)up+φ(x,u) in a uniformly (k−1)-convex domain Ω⊂Rn. We obtain the asymptotic behavior of a solution u near the boundary ∂Ω up to the second order.

Precise blowup rate near the boundary of boundary blowup solutions to k-Hessian equation

We consider boundary blowup problem for k-Hessian equation of the form $$F_k[u] = f(x)g(u)$$ in a uniformly $$(k-1)$$ -convex domain $$\Omega \subset {\mathbb {R}}^n$$ , where f(x) behaves like $$\mathop {\mathrm {dist}} \nolimits (x,\partial \Omega )^{\alpha }$$ as $$\text {dist}(x,\partial \Omega ) \rightarrow 0$$ and g(u) behaves like $$u^p$$ as $$u \rightarrow \infty$$ . We obtain the precise blowup rate of a solution u near the boundary $$\partial \Omega$$ .

Boundary blow-up solutions to the [formula omitted]-Hessian equation with singular weights

In this paper, we discuss the k-convex solution to the following boundary blow-up k-Hessian equation with singular weights: SkD2u=H(x)up,inΩ,u=∞,on∂Ω, where k∈{1,2,…,n}, SkD2u is the k-Hessian operator, Ω is a smooth, bounded and strictly convex domain in Rn(n≥2),p>0.

Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior

AbstractConsider the boundary blow-up Monge-Ampère problem$$\begin{array}{} \displaystyle M[u]=K(x)f(u) \mbox{ for } x \in {\it\Omega},\; u(x)\rightarrow +\infty \mbox{ as } {\rm dist}(x,\partial {\it\Omega})\rightarrow 0. \end{array}$$HereM[u] = det (uxixj) is the Monge-Ampère operator, andΩis a smooth, bounded, strictly convex domain in ℝN(N≥ 2). UnderK(x) satisfying appropriate conditions, we first prove that the boundary blow-up Monge-Ampère problem has a strictly convex solution if and only iffsatisfies Keller-Osserman type condition. Then the asymptotic behavior of strictly convex solutions to the boundary blow-up Monge-Ampère problem is considered under weaker condition with respect to previous references. Finally, iffdoes not satisfy Keller-Osserman type condition, we show the existence of strictly convex solutions under different conditions onK(x). The proof combines standard techniques based upon the sub-supersolution method with non-standard arguments, such as the Karamata regular variation theory.

The existence and asymptotic behavior of boundary blow-up solutions to the k-Hessian equation

In this paper we consider the existence and asymptotic behavior of k-convex solution to the boundary blow-up k-Hessian problemSk(D2u)=H(x)f(u) for x∈Ω,u(x)→+∞ as dist(x,∂Ω)→0, where k∈{1,2,⋯,N}, Sk(D2u) is the k-Hessian operator, Ω is a smooth, bounded, strictly convex domain in RN(N≥2), H∈C∞(Ω) is positive in Ω, but is not necessarily bounded on ∂Ω, and f is a smooth positive function that satisfies the so-called Keller-Osserman condition. Further results are obtained for the special case that Ω is a ball. Our approach to show the existence and asymptotic behavior, exploits the method of sub- and super-solutions and Karamata regular variation theory.

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 \(\Delta \).

On the moving plane method for boundary blow-up solutions to semilinear elliptic equations

Abstract We consider weak solutions to - Δ ⁢ u = f ⁢ ( u ) {-\Delta u=f(u)} on Ω 1 ∖ Ω 0 {\Omega_{1}\setminus\Omega_{0}} , with u = c ≥ 0 {u=c\geq 0} in ∂ ⁡ Ω 1 {\partial\Omega_{1}} and u = + ∞ {u=+\infty} on ∂ ⁡ Ω 0 {\partial\Omega_{0}} , and we prove monotonicity properties of the solutions via the moving plane method. We also prove the radial symmetry of the solutions in the case of annular domains.

Boundary blow-up solutions to the k-Hessian equation with a weakly superlinear nonlinearity

By constructing new sub- and super-solutions, we are concerned with determining values of β, for which there exist k-convex solutions to the boundary blow-up k-Hessian problemSk(D2u(x))=H(x)[u(x)]k[ln⁡u(x)]β>0 for x∈Ω,u(x)→+∞ as dist(x,∂Ω)→0. Here k∈{1,2,⋯,N}, Sk(D2u) is the k-Hessian operator, β>0 and Ω is a smooth, bounded, strictly convex domain in RN(N≥2). We suppose that the nonlinearity behaves like uklnβ⁡u as u→∞, which is more complex and difficult to deal with than the nonlinearity grows like up with p>k or faster at infinity. Further, several new results of nonexistence, global estimates and estimates near the boundary for the solutions are also given.

Boundary Blow-up Solutions to Nonlocal Elliptic Systems of Cooperative Type

In this paper, we consider the boundary blow-up solutions to the nonlocal elliptic systems of cooperative type. By introducing the boundary measure, the boundary blow-up problem becomes a Cauchy problem. Then using the super-subsolution method, we obtain the existence and nonexistence of positive solutions. Moreover, we study the stability of the minimal solution to the Cauchy problem.

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.