arXiv subjects

Bartosz Bieganowski

Publications and source records attributed to Bartosz Bieganowski.

40 records · Page 3Linked to original sources

arXiv2018

The semirelativistic Choquard equation with a local nonlinear term

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in $\mathbb{R}^N$ \begin{equation*} \sqrt{\strut -Δ+ m^2} u - mu + V(x)u = \left( \int_{\mathbb{R}^N} \frac{|u(y)|^p}{|x-y|^{N-α}} \, dy \right) |u|^{p-2}u - Γ(x) |u|^{q-2}u, \end{equation*} where $m > 0$ and the potential $V$ is decomposed as the sum of a $\mathbb{Z}^N$-periodic term and of a bounded term that decays at infinity. The result is proved by variational methods applied to an auxiliary problem in the half-space $\mathbb{R}_{+}^{N+1}$.

math.AP↗

arXiv2018

Boundedness of solutions to the critical fully parabolic quasilinear one-dimensional Keller-Segel system

In this paper we consider a one-dimensional fully parabolic quasilinear Keller-Segel system with critical nonlinear diffusion. We show uniform-in-time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well-known Lyapunov functional and a recently introduced new Lyapunov-like functional in [3].

math.AP↗

arXiv2016

Solutions of the fractional Schrödinger equation with sign-changing nonlinearity

We look for a solutions to a nonlinear, fractional Schrödinger equation $$(-Δ)^{α/ 2}u + V(x)u = f(x,u)-Γ(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where potential $V$ is coercive or $V=V_{per} + V_{loc}$ is a sum of periodic in $x$ potential $V_{per}$ and localized potential $V_{loc}$, $Γ\in L^{\infty}(\mathbb{R}^N)$ is periodic in $x$, $Γ(x)\geq 0$ for a.e. $x\in\mathbb{R}^N$ and $2<q<2^*_α$. If $f$ has the subcritical growth, but higher than $Γ(x)|u|^{q-2}u$, then we find a ground state solution being a minimizer on the Nehari manifold. Moreover we show that if $f$ is odd in $u$ and $V$ is periodic, this equation admits infinitely many solutions, which are pairwise geometrically distinct. Finally, we obtain the existence result in the case of coercive potential $V$.

math.AP↗

arXiv2016

Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities

We look for ground state solutions to the following nonlinear Schrödinger equation $$-Δu + V(x)u = f(x,u)-Γ(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic potential $V_{per}$ and a localized potential $V_{loc}$, $Γ\in L^{\infty}(\mathbb{R}^N)$ is periodic and $Γ(x)\geq 0$ for a.e. $x\in\mathbb{R}^N$ and $2\leq q<2^*$. We assume that $\infσ(-Δ+V)>0$, where $σ(-Δ+V)$ stands for the spectrum of $-Δ+V$ and $f$ has the subcritical growth but higher than $Γ(x)|u|^{q-2}u$, however the nonlinearity $f(x,u)-Γ(x)|u|^{q-2}u$ may change sign. Although a Nehari-type monotonicity condition for the nonlinearity is not satisfied we investigate the existence of ground state solutions being minimizers on the Nehari manifold.

math.AP↗