We consider the problem $$ \begin{alignat}{2} &u_t = \Delta\log u\quad && x\in\mathbb{R}^2, 0<t<T,\\ &u(x,0)=u_0(x)\quad && x\in\mathbb{R}^2, \end{alignat} $$ with nonnegative and integrable data $u_0(x)$, which appears in Riemannian geometry. We construct a class of maximal solutions of this problem and prove that they generate a semigroup of contractions in $L^1(\mathbb R^2)$, characterized by the property of area loss $$ \dfrac{d}{dt}\int u(x,t)\,dx= -4\pi, $$ which holds until $u(x,t)$ vanishes identically at the time $T=\int u_0(x)\, dx/(4\pi)$. Several constructions of the maximal solutions are proposed and alternative characterizations proved. Sharp uniqueness criteria are obtained.

A bifurcation problem or a nonlinear eigenvalue problem for degenerate quasilinear elliptic equations with Dirichlet boundary condition is studied. Uniform estimates of partial derivatives of solutions are calculated. By virtue of these estimatates, we can apply the Leray-Schauder degree theory to our problem and obtain the bifurcation of nontrivial weak solutions.

An analysis of the nonlocal parabolic equation $$ u_t-\Delta u=\frac{\delta f(u)}{(\int_\Omega f(u))^p},\quad x\in\Omega, t>0, \tag"(P)" $$ and its associated steady state equation $$ -\Delta u=\frac{\delta f(u)}{(\int_\Omega f(u))^p}, \quad x\in \Omega, \tag"(S)" $$ with Dirichlet boundary conditions is given, assuming $\Omega\subset\Bbb R^n$ is a smooth bounded domain, $\delta>0$, $p\ge0$, and $f$ is positive and locally Lipschitz continuous. Existence-nonexistence results are proven for (S) when $f(u)=e^u$ or $e^{-u}$ and $\Omega$ is a ball or star-shaped domain. For $f(u)\ge c>0$, $n=1$, and $p\ge1$, we prove that (P) has a global bounded solution for any nonnegative initial data $u_0(x)$ and any $\delta>0$. For $f(u)=e^u$, $n=2$, $\Omega=B_1(0)$, $u_0(x)$ radially symmetric, nonnegative, if $p>1$, (P) has a unique, globally bounded solution for any $\delta>0$. If $p=1$, $0<\delta<8\pi$, (P) again has a bounded global solution. For $f(u)=e^u$, $n=1$ or 2, if $p<1$ and $\delta>\delta^*$ where $\delta^*$ is critical value for existence-nonexistence for (S), then the solution $u$ of (P) blows up in finite time.

We improve some previous results for the principal eigenvalue of the $p$-Laplacian defined on $\mathbb{R}^N,$ study regularity of the corresponding eigenfunctions and give an existence result of the type below the first eigenvalue (or between the first eigenvalues) for a certain perturbed problem. Based on our approach for the equation we deduce existence, uniqueness and simplicity of positive principal eigenvalues for the $p$-Laplacian system $$ \begin{align} &{-\Delta}_{p} u = \lambda a(x) |u|^{p-2}u + \lambda b(x) |u|^{\alpha - 1} u |v|^{\beta +1}, \quad x \in \mathbb{R}^N, \\ &{-\Delta}_{q} v = \lambda b(x) |u|^{\alpha + 1} |v|^{\beta -1}v + \lambda d(x) |v|^{q - 2}v, \quad x \in \mathbb{R}^N, \\ & 0 <u, 0<v, \text{ in } \quad \mathbb{R}^N, \lim_{|x| \rightarrow +\infty} u(x) = \lim_{|x| \rightarrow +\infty} v(x) = 0. \end{align} $$ We also establish the regularity of the corresponding eigenfunctions.

We prove $W^{1,p}$-interior estimates for viscosity solutions of fully nonlinear, uniformly elliptic equations. The estimates complement $C^{1,\alpha}$-regularity results of L. Caffarelli. We apply the estimates to obtain existence and uniqueness of viscosity and strong solutions of Dirichlet boundary value problems.