On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states

Consider the Hénon equation with the homogeneous Neumann boundary condition

$?\mathrm{\Delta }u+u=|x{|}^{\alpha }{u}^p,u>0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{ on }?\mathrm{\Omega },$

where $\mathrm{\Omega }?B(0,1)?{\mathbb{R}}^N,N\ge 2$ and $?\mathrm{\Omega }\cap ?B(0,1)\ne ?$. We are concerned on the asymptotic behavior of ground state solutions as the parameter $\alpha \to \infty$. As $\alpha \to \infty$, the non-autonomous term $|x{|}^{\alpha }$ is getting singular near $|x|=1$. The singular behavior of $|x{|}^{\alpha }$ for large $\alpha >0$ forces the solution to blow up. Depending subtly on the $(N?1)?$dimensional measure $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$ and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$. In particular, the critical exponent $2_{?}=\frac{2(N?1)}{N?2}$ for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any $p\in (1,2^{?}?1)$ and a smooth domain Ω.     

On the positive radial solutions of a class of singular semilinear elliptic equations

In this paper, we consider the following elliptic equation(0.1)$\mathit{div}\left(A\right(|x|\left)\mathrm{?}u\right)+B(|x|)u^p=0\text{in}{\mathbb{R}}^n,$ where $p>1$, $n?3$, $A(|x|)>0$ is differentiable in ${\mathbb{R}}^n?\{0\}$ and $B(|x|)$ is a given nonnegative Hölder continuous function in ${\mathbb{R}}^n?\{0\}$. The asymptotic behavior at infinity and structure of separation property of positive radial solutions with different initial data for (0.1) are discussed. Moreover, the existence and separation property of infinitely many positive solutions for Hardy equation and an equation related to Caffarelli–Kohn–Nirenberg inequality are obtained respectively, as special cases.     

A Hardy type inequality for W02,1(Ω) functions

We consider functions $u\in W_0^{2,1}(\Omega )$, where $\Omega ?{\mathbb{R}}^N$ is a smooth bounded domain. We prove that $\frac{u(x)}{d(x)}\in W_0^{1,1}(\Omega )$ with$6\mathrm{?}\left(\frac{u(x)}{d(x)}\right)6_{L^1(\Omega )}?C6u{6}_{W^{2,1}(\Omega )},$ where d is a smooth positive function which coincides with $\mathrm{dist}(x,?\Omega )$ near ?Ω and C is a constant depending only on d and Ω.     

On a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.     

Quasilinear elliptic problems with critical exponents and Hardy terms

Let $\Omega \ni 0$ be an open bounded domain, $\Omega \subset R^N(N>p^2)$. We are concerned with the multiplicity of positive solutions of $-{\mathrm{\Delta }}_{p}u-\mu \frac{|u{|}^{p-2}u}{|x{|}^p}=\lambda |u{|}^{p-2}u+Q(x)|u{|}^{p^{*}-2}u,u\in W_0^{1,p}(\Omega )\text{,}$where $-{\mathrm{\Delta }}_{p}u=-\mathrm{div}(|\nabla u{|}^{p-2}\nabla u),1<p<N,p^{*}=\frac{\mathit{Np}}{N-p},0<\mu <{\left(\frac{N-p}{p}\right)}^p,\lambda >0$and $Q(x)$ is a nonnegative function on $\overline{\Omega }$. By investigating the effect of the coefficient of the critical nonlinearity, we, by means of variational method, prove the existence of multiple positive solutions.     

Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball

In this article we obtain positive singular solutions of

(1)$?\mathrm{\Delta }u=|\mathrm{?}u{|}^p\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a small $C^2$ perturbation of the unit ball in ${\mathbb{R}}^N$. For $\frac{N}{N?1}<p<2$ we prove that if Ω is a sufficiently small $C^2$ perturbation of the unit ball there exists a singular positive weak solution u of (1). In the case of $p>2$ we prove a similar result but now the positive weak solution u is contained in $C^{0,\frac{p?2}{p?1}}(\stackrel{￣}{\mathrm{\Omega }})$ and yet is not in $C^{0,\frac{p?2}{p?1}+\epsilon }(\stackrel{￣}{\mathrm{\Omega }})$ for any $\epsilon >0$.