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 (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 $2 < p < \infty$.

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.

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.

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 ${\mathcal Q}$.