Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
Advances in Nonlinear AnalysisIt is well-known that it is a quite interesting topic to study the asymptotic expansions of entire large solutions of nonlinear elliptic equations near infinity. But very little is done.
Wan Haitao
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A classification result for the quasi-linear Liouville equation [PDF]
arXiv, 2016 Entire solutions of the $n-$Laplace Liouville equation in $\mathbb{R}^n$ with finite mass are completely classified.
arxiv
Some remarks on the structure of finite Morse index solutions to the Allen-Cahn equation in $\mathbb{R}^2$ [PDF]
arXiv, 2015 For a solution of the Allen-Cahn equation in $\mathbb{R}^2$, under the natural linear growth energy bound, we show that the blowing down limit is unique. Furthermore, if the solution has finite Morse index, the blowing down limit satisfies the multiplicity one property.
arxiv
Uniform Lipschitz regularity of flat segregated interfaces in a singularly perturbed problem [PDF]
arXiv, 2015 For the singularly perturbed system \[\Delta u_{i,\beta}=\beta u_{i,\beta}\sum_{j\neq i}u_{j,\beta}^2, \quad 1\leq i\leq N,\] we prove that flat interfaces are uniformly Lipschitz. As a byproduct of the proof we also obtain the optimal lower bound near the flat interfaces, \[\sum_iu_{i,\beta}\geq c\beta^{-1/4}.\]
arxiv
Front blocking in the presence of gradient drift [PDF]
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arxiv
Entire radial and nonradial solutions for systems with critical growth [PDF]
arXiv, 2016 In this paper we establish existence of radial and nonradial solutions to the system $$ \begin{array}{ll} -\Delta u_1 = F_1(u_1,u_2) &\text{in }\mathbb{R}^N,\newline -\Delta u_2 = F_2(u_1,u_2) &\text{in }\mathbb{R}^N,\newline u_1\geq 0,\ u_2\geq 0 &\text{in }\mathbb{R}^N,\newline u_1,u_2\in D^{1,2}(\mathbb{R}^N), \end{array} $$ where ...
arxiv
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arxiv
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arXiv, 2017 For differential inequalities of the form $$ \sum_{|\alpha| = m} (- 1)^m \partial^\alpha a_\alpha (x, u) \ge b (x) |u|^\lambda \quad \mbox{in } {\mathbb R}^n, \: n \ge 1, $$ where $a_\alpha$ and $b$ are some functions, we obtain conditions guaranteeing that any solution is identically equal to zero.
arxiv
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arXiv, 2010 In this paper we deal with the cubic Schr\"odinger system $ -\Delta u_i = \sum_{j=1}^n \beta_{ij}u_j^2 u_i$, $u_1,\dots,u_n \geq 0$ in $\mathbb{R}^N (N\leq 3)$, where $\beta=(\beta_{i,j})_{ij}$ is a symmetric matrix with real coefficients and $\beta_{ii}\geq 0$ for every $i=1,\ldots,n$.
arxiv
Generalized transition waves and their properties [PDF]
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arxiv