Peihao Zhao
Articles written in Proceedings – Mathematical Sciences
Volume 122 Issue 1 February 2012 pp 129-137
Remarks on Hausdorff Measure and Stability for the 𝑝-Obstacle Problem $(1 < p < 2)$
In this paper, we consider the obstacle problem for the inhomogeneous 𝑝-Laplace equation
$$\mathrm{div}(\nabla u|^{p-2}\nabla u)=f\cdot p\chi\{u>0\},\quad 1 < p < 2,$$
where 𝑓 is a positive, Lipschitz function. We prove that the free boundary has finite $(N-1)$-Hausdorff measure and stability property, which completes previous works by Caffarelli (
Volume 123 Issue 3 August 2013 pp 373-382
Porosity of Free Boundaries in the Obstacle Problem for Quasilinear Elliptic Equations
Jun Zheng Zhihua Zhang Peihao Zhao
In this paper, we establish growth rate of solutions near free boundaries in the identical zero obstacle problem for quasilinear elliptic equations. As a result, we obtain porosity of free boundaries, which is naturally an extension of the previous works by Karp
Volume 125 Issue 3 August 2015 pp 307-321
Soliton solutions for a quasilinear Schrödinger equation via Morse theory
In this paper, Morse theory is used to show the existence of nontrivial weak solutions to a class of quasilinear Schrödinger equation of the form
$$-\Delta u - \frac{p}{2^{p-1}} u \Delta_p (u^2) = f(x, u)$$
in a bounded smooth domain $\Omega \subset \mathbb{R}^N$ with Dirichlet boundary condition.
Volume 128 Issue 2 April 2018 Article ID 0022 Research Article
We consider the existence of single and multi-peak solutions of thefollowing nonlinear elliptic Neumann problem \begin{align*} &\left\{ \begin{array}{1} -\Delta u + \lambda^{2} u = Q(x)|u|^{p-2}u & {\rm in}\quad {\mathbb R}^N_+,\\ \frac{\partial u}{\partial n} = f(x,u) & {\rm on} \quad \partial {\mathbb R}^N_+, \end{array} \right. \end{align*} where $\lambda$ is a large number, $p \in (2, \frac{2N}{N−2})$ for $N \geq 3, f (x, u)$ is subcritical about $u$ and ${\mathcal Q}$ is positive and has some non-degenerate critical points in ${\mathbb R}^{N}_{+}$. For $\lambda$ large, we can get solutions which have peaks near the non-degenerate critical points of ${\mathcal Q}$.
Volume 133, 2023
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