Let $\varOmega$ be a bounded domain in $\mathbb R^{2}$ with smooth boundary, $a > 0, \lambda>0$ and $0 < \delta < 3$. We consider the following critical problem with singular and discontinuous nonlinearity: \begin{eqnarray*} \begin{array}{rl} -\Delta u & = \lambda ( {\chi_{\{u < a\}}}{u^{-{\delta}}} + h(u) e^{u^2})~~\text{in} ~~\Omega, \\ u & > 0 ~\text{ in }~ \Omega,\\ u & = 0 ~\text{ on }~ \partial \Omega, \end{array} \end{eqnarray*} where $\chi$ is the characteristic function and $h(u)$ is a smooth nonlinearity that is a "perturbation" of $e^{u^2}$ as $u \to \infty$ (for precise definitions, see hypotheses (H1)-(H5) in Section 1). With these assumptions we study the existence of multiple positive solutions to the above problem.

The boundary-value problem for the generalized Navier-Stokes equations with anisotropic diffusion is considered in this work. For this problem, we prove the existence of weak solutions in the sense that solutions and test functions are considered in the same admissible function space. We also prove the existence of very weak solutions, i.e., solutions for which the test functions have more regularity. By exploiting several examples we show, in the case of dimension $3$, that these existence results improve its isotropic versions in almost all directions or for particular choices of all the diffusion coefficients.

We prove that, for $\alpha > 2$, $\frac{N}{N-2+c} < p <\infty$, the operator $Lu=(1+|x|^\alpha)\Delta u+c|x|^{\alpha-1}\tfrac{x}{|x|}\cdot\nabla$ generates an analytic semigroup in $L^p$, which is contractive if and only if $p \ge \frac{N+\alpha-2}{N-2+c}$. Moreover, for $\alpha <\frac{N}{p'}+c$, we provide an explicit description of the domain. Spectral properties of the operator $L$ and kernel estimates are also obtained.

In this work, we shall present some new variational principles for evolutionary equations by the virtue of the Non-convex self-dual (Nc-SD) Lagrangians. It is established that how lifting Nc-SD Lagrangians to path spaces allows one to associate to an evolution boundary value problem several potential functions, which can often be used with relative ease compared to other methods such as the use of Euler-Lagrange functionals. These Lagrangians, indeed provide new representations and formulations for the superposition of semi-convex functions and symmetric operators. They yield new variational resolutions for large class of hamiltonian partial differential equations with a variety of linear and nonlinear boundary conditions including many of the standard ones. They can be adapted to easily deal with both nonlinear and homogeneous boundary value problems and, in most cases, solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler-Lagrange function.

In this paper, we study an initial-boundary value problem of the Korteweg-de Vries equation posed on a bounded interval $(0,L)$ with nonhomogeneous boundary conditions, which is known to be locally well-posed in the Sobolev space $H^s(0,L)$ with $s\gt-3/4$. Taking the advantage of the hidden dissipative mechanism and the sharp trace regularities of its solutions, we show that the problem is locally well-posed in the space $H^s(0,L)$ with $s\gt-1$.

In this paper, we prove the existence of positive weak solutions for the homogeneous Dirichlet problem associated to the equation \begin{equation*} -\Delta u=f(x,u)+\lambda h(x,u),\quad \text{in } \Omega, \end{equation*} where $ \lambda\ge 0, $ $ f(x,u) $ can be singular as $ u \rightarrow0^+ $ and $ h(x,u) $ can diverge as $ u \rightarrow\infty. $ We assume that $ 0\le f(x,u)\le\frac{\psi_0(x)}{u^p} $ with $ \psi_0\in L^m(\Omega),\,\, m\ge 1, $ and $ 0\le h(x,u)\le\psi_\infty(x)u^q $ with $ \psi_\infty\in L^M(\Omega),\,\, M\gt\frac{N}{2}. $ We do not have any monotonicity assumption on $ f(x,\cdot)$, and $ h(x,\cdot)$. Moreover, we do not assume the existence of any super or sub solution.