Related papers: Sign-changing solutions for the one-dimensional no…
We establish the existence of finitely many sign-changing solutions to the Lane-Emden system $$-\Delta u=|v|^{q-2}v,\quad -\Delta v=|u|^{p-2}u \quad \text{ in }\mathbb{R}^N, \ \ N\geq 4,$$ where the exponents $p$ and $q$ lie on the critical…
We study Smoluchowski-Poisson equation in two space dimensions provided with Dirichlet boundary condition for the Poisson part. For this equation several profiles of blowup solution have been noticed. Here we show the residual vanishing.
The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage…
Two families of sign-changing solutions of higher-order semilinear parabolic equations are studied.
In this paper, we are concerned with the following Schr\"{o}dinger-Poisson system with critical nonlinearity and critical nonlocal term due to the Hardy-Littlewood-Sobolev inequality \begin{equation}\begin{cases} -\Delta u+u+\lambda\phi…
Several new methods of numerical integration of Cauchy problems with blow-up solutions for nonlinear ordinary differential equations of the first- and second-order are described. Solutions of such problems have singularities whose positions…
We study the blow-up problem of one-dimensional nonlinear heat equations. Our result shows that for a certain class of initial conditions, the solutions blow up in finite time and we characterize the asymptotic dynamics of these solutions.…
In this paper we study the asymptotic behavior of solutions for a non-local non-autonomous scalar quasilinear parabolic problem in one space dimension. Our aim is to give a fairly complete description of the the forwards asymptotic behavior…
We study the existence of sign changing solutions to the following problem $$ (P) \quad \quad \quad \left\{ \begin{array}{ll} \Delta u+|u|^{p-1}u=0 \quad & {\rm in} \quad \Omega_\epsilon; u=0 \quad & {\rm on} \quad\partial \Omega_\epsilon,…
By a perturbative argument, we construct solutions for a plasma-type problem with two opposite-signed sharp peaks at levels $1$ and $-\gamma$, respectively, where $0<\gamma<1$. We establish some physically relevant qualitative properties…
We analyze the blowup behaviour of solutions to the focusing nonlinear Klein--Gordon equation in spatial dimensions $d\geq 2$. We obtain upper bounds on the blowup rate, both globally in space and in light cones. The results are sharp in…
We present a blow-up rate estimate for a solution to the parabolic Gross-Pitaevskii and related systems on entire space with Sobolev subcritical nonlinearity. We extend the results of [Y. Giga, S. Matsui and S. Sasayama, Indiana Univ. Math.…
We study a competitive nonlinear Schr\"odinger system in $\mathbb{R}^N$ whose nonlinear potential is localized in small regions that shrink to isolated points. Within a variational framework based on a fully sign-changing Nehari constraint…
We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the solution. In this paper, we are interested in…
Let $\Omega$ be an open, possibly unbounded, set in Euclidean space $\R^m$ with boundary $\partial\Omega,$ let $A$ be a measurable subset of $\Omega$ with measure $|A|$, and let $\gamma \in (0,1)$. We investigate whether the solution…
In this work, we study the existence of sign-changing solutions for the Schr\"odinger-Bopp-Podolsky system.
We study the existence of positive and sign-changing multipeak solutions for the stationary Nonlinear Schroedinger Equation. Here no symmetry on $V$ is assumed. It is known that this equation has positive multipeak solutions with all peaks…
We prove the existence of nonradial solutions for the H\'enon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent $\alpha$. For sign-changing solutions, the case $\alpha=0$ -- Lane-Emden equation…
In this work, we establish the existence of solutions that change sign at low energy for a non-local weighted Kirchhoff problem in the set $\mathbb{R}^{N}, N>2$. The non-linearity of the equation is assumed to have exponential growth in…
This work establishes conditions for the existence and uniqueness of local mild solutions to the Boussinesq equations with fractional dissipations in Sobolev-Gevrey spaces. We prove that a unique mild solution exists in an appropriate…