• Peihao Zhao

Articles written in Proceedings – Mathematical Sciences

• Remarks on Hausdorff Measure and Stability for the 𝑝-Obstacle Problem $(1 &lt; p &lt; 2)$

In this paper, we consider the obstacle problem for the inhomogeneous 𝑝-Laplace equation

$$\mathrm{div}(\nabla u|^{p-2}\nabla u)=f\cdot p\chi\{u&gt;0\},\quad 1 &lt; p &lt; 2,$$

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 (J. Fourier Anal. Appl. 4(4--5) (1998) 383--402) for $p=2$, and Lee and Shahgholian (J. Differ. Equ. 195 (2003) 14--24) for $2 &lt; p &lt; \infty$.

• Porosity of Free Boundaries in the Obstacle Problem for Quasilinear Elliptic Equations

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 et al. (J. Diff. Equ. 164 (2000) 110–117) for 𝑝-Laplacian equations, and by Zheng and Zhang (J. Shaanxi Normal Univ. 40(2) (2012) 11–13, 18) for 𝑝-Laplacian type equations.

• 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.

• Positive solutions with single and multi-peak for semilinear elliptic equations with nonlinear boundary condition in the half-space

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 ${\mathcal Q}$.

• Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
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• Editorial Note on Continuous Article Publication

Posted on July 25, 2019