Ústav matematiky
Browse
Recent Submissions
Now showing 1 - 5 of 78
- ItemVanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point(De Gruyter, 2024-02-05) Diblík, Josef; Růžičková, MiroslavaA singular nonlinear differential equation z(sigma) dw/dz = aw + zwf(z , w), where sigma > 1, is considered in a neighbourhood of the point z = 0 z=0 located either in the complex plane C if sigma is a natural number, in a Riemann surface of a rational function if sigma is a rational number, or in the Riemann surface of logarithmic function if sigma is an irrational number. It is assumed that w = w ( z ) w=w\left(z) , a is an element of C { 0 } a, and that the function f f is analytic in a neighbourhood of the origin in C x C . Considering sigma to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions w = w (z ) w=w(z) in a domain that is part of a neighbourhood of the point z = 0 z=0 in C or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property lim z -> 0 w (z) = 0 is proved and an asymptotic behaviour of w (z) s established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.
- ItemMultiple normalized solutions for the planar Schrödinger–Poisson system with critical exponential growth(Springer Nature, 2024-02-16) Chen, Sitong; Radulescu, Vicentiu; Tang, XianhuaThe paper deals with the existence of normalized solutions for the following Schr & ouml;dinger-Poisson system with -constraint: { -Delta u+lambda u+mu(log||& lowast;u2)u=(e(u2-)1-u2)u,x is an element of R-2, integral R(2)u(2)dx=c, where mu>0,lambda is an element of R , will arise as a Lagrange multiplier and the nonlinearity enjoys critical exponential growth of Trudinger-Moser type. By specifying explicit conditions on the energy level c, we detect a geometry of local minimum and a minimax structure for the corresponding energy functional, and prove the existence of two solutions, one being a local minimizer and one of mountain-pass type. In particular, to catch a second solution of mountain-pass type, some sharp estimates of energy levels are proposed, suggesting a new threshold of compactness in the -constraint. Our study extends and complements the results of Cingolani-Jeanjean (SIAM J Math Anal 51(4): 3533-3568, 2019) dealing with the power nonlinearity a|u|p-2uin the case ofa>0andp>4, in the case of and , which seems to be the first contribution in the context of normalized solutions. Our model presents some new difficulties due to the intricate interplay between a logarithmic convolution potential and a nonlinear term of critical exponential type and requires a novel analysis and the implementation of new ideas, especially in the compactness argument. We believe that our approach will open the door to the study of other -constrained problems with critical exponential growth, and the new underlying ideas are of future development and applicability.
- ItemGroundstates of the planar Schrodinger-Poisson system with potential well and lack of symmetry(Cambridge University Press, 2023-06-06) Liu, Zhisu; Radulescu, Vicentiu; Zhang, JianjunThe Schrodinger-Poisson system describes standing waves for the nonlinear Schrodinger equation interacting with the electrostatic field. In this paper, we are concerned with the existence of positive ground states to the planar Schrodinger-Poisson system with a nonlinearity having either a subcritical or a critical exponential growth in the sense of Trudinger-Moser. A feature of this paper is that neither the finite steep potential nor the reaction satisfies any symmetry or periodicity hypotheses. The analysis developed in this paper seems to be the first attempt in the study of planar Schrodinger-Poisson systems with lack of symmetry.
- ItemNon-autonomous double phase eigenvalue problems with indefinite weight and lack of compactness(London Mathematical Society, 2024-02-08) Tianxiang, Gou; Radulescu, VicentiuIn this paper, we consider eigenvalues to the following double phase problem with unbalanced growth and indefinite weight,-Delta pau-Delta qu=lambda m(x)|u|q-2uinRN,$$\begin{equation*} \hspace*{3pc}-\Delta _pa u-\Delta _q u =\lambda m(x)|u|{q-2}u \quad \mbox{in} \,\, \mathbb {R}<^>N, \end{equation*}$$where N > 2$N \geqslant 2$, 1{0, 1}(\mathbb {R}N, [0, +\infty))$, a not equivalent to 0$a \not\equiv 0$ and m:RN -> R$m: \mathbb {R}N \rightarrow \mathbb {R}$ is an indefinite sign weight which may admit non-trivial positive and negative parts. Here, Delta q$\Delta _q$ is the q$q$-Laplacian operator and Delta pa$\Delta _pa$ is the weighted p$p$-Laplace operator defined by Delta pau:=div(a(x)| backward difference u|p-2 backward difference u)$\Delta _pa u:=\textnormal {div}(a(x)|\nabla u|{p-2} \nabla u)$. The problem can be degenerate, in the sense that the infimum of a$a$ in RN$\mathbb {R}N$ may be zero. Our main results distinguish between the cases p
- ItemModelling precipitation extremes in the Czech Republic: Update of intensity-duration-frequency curves(Estonian Academy Publishers, 2016-12-20) Fusek, Michal; Hellebrand, Radek; Michálek, JaroslavPrecipitation records from six stations of the Czech Hydrometeorological Institute were subject to statistical analysis with the objectives of updating the Intensity-Duration-Frequency (IDF) curves, by applying extreme value distributions, and comparing the updated curves against those produced by an empirical procedure in 1958. Another objective was to investigate differences between both sets of curves, which could be explained by such factors as different measuring instruments, measuring stations altitudes, and data analysis methods. It has been shown that the differences between the two sets of IDF curves are significantly influenced by the chosen method of data analysis.