We study the nonlinear Schrödinger equation with general nonlinearity of competing type. This equation has traveling waves solution with non-vanishing condition at infinity in one dimension. We give a sharp condition for the stability and instability of these solutions. This justifies the previous prediction posed in physical literature.

We prove the existence of multipeak positive solutions for an elliptic Neumann problem in exterior domains involving nearly critical Sobolev exponent.

An improved Poincaré inequality is shown for $p>2$: There exists a constant $c>0$ such that for all $u\in W_0^{1,p}(\Omega)$, \begin{equation*} \tag*{(P)} \int_\Omega |\nabla u|^p \,{\rm d}x - \lambda_1 \int_\Omega |u|^p \,{\rm d}x \geq c \Big( | u^\parallel |^{p-2} \int_\Omega |\nabla\varphi_1|^{p-2} |\nabla u^\top|^2 \,{\rm d}x + \int_\Omega |\nabla u^\top|^p \,{\rm d}x \Big) . \end{equation*} Here, a function $u\in L^2(\Omega)$ is decomposed as an orthogonal sum \[ u = u^\parallel\cdot \varphi_1 + u^\top \;\mbox{ where }\; u^\parallel {\stackrel{{\mathrm {def}}}{=}} \|\varphi_1\|_{ L^2(\Omega) }^{-2} \langle u, \varphi_1 \rangle \;\mbox{ and }\; \langle u^\top, \varphi_1 \rangle = 0 , \] $\lambda_1$ denotes the first eigenvalue of the positive Dirichlet $p$-Laplacian $-\Delta_p$, $\Delta_p u\equiv {\mathop{\mathrm {div}}} ( |\nabla u|^{p-2} \nabla u )$, $\lambda_1$ is simple, and $\varphi_1$ stands for the corresponding eigenfunction. Inequality (P) is applied to show the existence of a weak solution to the following degenerate quasi\-linear boundary value problem at resonance, where $f\in L^2(\Omega)$ with $\langle f,\varphi_1 \rangle = 0$: \[ - \Delta_p u = \lambda_1 |u|^{p-2} u + f(x) \;\mbox{ in } \Omega ;\qquad u = 0 \;\mbox{ on } \partial\Omega . \]

We consider blow-up problems of the Cauchy-Neumann problem for semilinear heat equations with large diffusion. We prove that, in cylindrical domains, the solutions blow up only at the edge of the domain for almost all initial data. Furthermore, we give an estimate of the blow-up time of the solutions.