For one spatial variable, a new kind of nonlinear wave equation for Emden-Fowler type is considered with boundary value null and initial values. Under certain conditions on the initial data and the exponent p, we exhibit that the visco-elastic term leads our problem to be dissipative and the global solutions still non-exist in L2 at given finite time.
Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term
研究了奇异椭圆型方程$-Δu= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$（$N\geq 3$），其中$p$为正权，$0< a <1$为正权.在假设$f$是次线性增长的非减函数，而$g$是减函数且在原点附近无界的条件下，我们建立了在无穷远处消失的基态解的存在性.我们的论证基本上依赖于最大值原理。
Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence
Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam（经济数学学院，经济与法律大学）；Vietnam National University, Ho Chi Minh City, Vietnam（越南国立大学）
Combined effects in nonlinear problems arising in the study of anisotropic continuous media
本文研究了具有Dirichlet边界条件和变势函数的Lane-Emden-Fowler方程。在本文中开发的分析相结合的单调性方法与变分参数。备注（2020年4月21日）：我们的结果后来也被D.- P. Covei，具有凸和凹非线性和可变势之间竞争的拟线性问题，Internat。J. Math.24：1（2013），art.1350005），arXiv：1104.4626v1.
Nonradial stable solutions near the Joseph--Lundgren threshold
University of Science and Technology of China（中国科学技术大学）；Beijing Technology and Business University（北京工商大学）；Chinese University of Hong Kong（香港中文大学）；University of Macau（澳门大学）
We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation $$ -Δu+ u=u^{2^*-1}+λu^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ is an integer, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $2<q<2^*$ and $λ>0$ is a parameter. It is known that as $λ\to 0$, after a rescaling the ground state solutions of the equation converge to a particular solution of the critical Emden-Fowler equation $-Δu=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension $N=…
We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)Δu+ λu= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $λ\to 0$ and $λ\to +\infty$, where $N=3$ or $N= 4$, $2<q\le p\le 2^*$, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $a>0$, $b\ge 0$ are constants and $λ>0$ is a parameter. In particular, we prove that in the case $2<q<p=2^*$, as $λ\to 0$, after a suitable rescaling the ground state solutions of the problem converge to the unique positive solution of the equation $-Δu+u=u^{q-1}$ a…
In this paper, we apply blow-up analysis and Liouville type theorems to study pointwise a priori estimates for some quasilinear equations with p-Laplace operator. We first obtain pointwise interior estimates for the gradient of p-harmonic function, i.e., the solution of $Δ_{p}u=0,\ x\inΩ$, which extends the well-established results of the interior estimates of the gradient of harmonic function. We then get singularity and decay estimates of the sign changing solution of Lane-Emden-Fowler type p-Laplace equation $-Δ_{p}u=|u|^{λ-1}u, \ x\inΩ$, which are then generalized for the equation with gen…
We classify the local asymptotic behavior of positive singular solutions to a class of subcritical sixth order equations on the punctured ball. Initially, using a version of the integral moving spheres technique, we prove that solutions are asymptotically radially symmetric solutions with respect to the origin. We divide our approach into some cases concerning the growth of nonlinearity. In general, we use an Emden--Fowler change of variables to translate our problem to a cylinder. In the lower critical regime, this is not enough, thus, we need to introduce a new notion of change of variables.…
By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main theorems, we prove the existence of at least one solution to the following singular Yamabe-type problem $$ \left\lbrace \begin{array}{ll} -Δ_g w + α(σ)w = μK(σ) w^\frac{d+2}{d-2} +λ\left( w^{r-1} + f(w)\right), \quad σ\in\mathcal{M} &\\ &\\ w\in H^2_α(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ where, as usual, $Δ_g$ denote…
We are concerned with the half-space Dirichlet problem \[\left\{\begin{array}{ll} -Δv+v=|v|^{p-1}v & \textrm{in}\ \mathbb{R}^N_+, v=c\ \textrm{on}\ \partial\mathbb{R}^N_+, &\lim_{x_N\to \infty}v(x',x_N)=0\ \textrm{uniformly in}\ x'\in\mathbb{R}^{N-1}, \end{array}\right. \] where $\mathbb{R}^N_+=\{x\in \mathbb{R}^N \ : \ x_N>0\}$ for some $N\geq 2$, and $p>1$, $c>0$ are constants. It was shown recently by Fernandez and Weth [Math. Ann. (2021)] that there exists an explicit number $c_p\in (1,\sqrt{e})$, depending only on $p$, such that for $0<c<c_p$ there are infinitely many bounded positive sol…
