Search results for "35b33"
showing 6 items of 6 documents
Uniqueness of positive radial solutions to singular critical growth quasilinear elliptic equations
2015
In this paper, we prove that there exists at most one positive radial weak solution to the following quasilinear elliptic equation with singular critical growth \[ \begin{cases} -\Delta_{p}u-{\displaystyle \frac{\mu}{|x|^{p}}|u|^{p-2}u}{\displaystyle =\frac{|u|^{\frac{(N-s)p}{N-p}-2}u}{|x|^{s}}}+\lambda|u|^{p-2}u & \text{in }B,\\ u=0 & \text{on }\partial B, \end{cases} \] where $B$ is an open finite ball in $\mathbb{R}^{N}$ centered at the origin, $1<p<N$, $-\infty<\mu<((N-p)/p)^{p}$, $0\le s<p$ and $\lambda\in\mathbb{R}$. A related limiting problem is also considered.
Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry
2014
Let $1p^{2}+p,a(0)>0$ and $\Omega$ satisfies some geometry conditions if $0\in\partial\Omega$, say, all the principle curvatures of $\partial\Omega$ at $0$ are negative, then the above problem has infinitely many solutions.
On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain
2021
Abstract We study the wave inequality with a Hardy potential ∂ t t u − Δ u + λ | x | 2 u ≥ | u | p in ( 0 , ∞ ) × Ω , $$\begin{array}{} \displaystyle \partial_{tt}u-{\it\Delta} u+\frac{\lambda}{|x|^2}u\geq |u|^p\quad \mbox{in } (0,\infty)\times {\it\Omega}, \end{array}$$ where Ω is the exterior of the unit ball in ℝ N , N ≥ 2, p > 1, and λ ≥ − N − 2 2 2 $\begin{array}{} \displaystyle \left(\frac{N-2}{2}\right)^2 \end{array}$ , under the inhomogeneous boundary condition α ∂ u ∂ ν ( t , x ) + β u ( t , x ) ≥ w ( x ) on ( 0 , ∞ ) × ∂ Ω , $$\begin{array}{} \displaystyle \alpha \frac{\partial u}{\partial \nu}(t,x)+\beta u(t,x)\geq w(x)\quad\mbox{on } (0,\infty)\times \partial{\it\Omega}, \e…
Asymptotic Behaviors of Solutions to quasilinear elliptic Equations with critical Sobolev growth and Hardy potential
2015
Abstract Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations − Δ p u − μ | x | p | u | p − 2 u = Q ( x ) | u | N p N − p − 2 u , x ∈ R N , where 1 p N , 0 ≤ μ ( ( N − p ) / p ) p and Q ∈ L ∞ ( R N ) .
Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces
2011
Let $(X,d)$ be a complete, pathwise connected metric measure space with locally Ahlfors $Q$-regular measure $\mu$, where $Q>1$. Suppose that $(X,d,\mu)$ supports a (local) $(1,2)$-Poincar\'e inequality and a suitable curvature lower bound. For the Poisson equation $\Delta u=f$ on $(X,d,\mu)$, Moser-Trudinger and Sobolev inequalities are established for the gradient of $u$. The local H\"older continuity with optimal exponent of solutions is obtained.
Gradient estimates for solutions to quasilinear elliptic equations with critical sobolev growth and hardy potential
2015
This note is a continuation of the work \cite{CaoXiangYan2014}. We study the following quasilinear elliptic equations \[ -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\, x\in\mathbb{R}^{N}, \] where $1<p<N,0\leq\mu<\left((N-p)/p\right)^{p}$ and $Q\in L^{\infty}(\R^{N})$. Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.