arXiv subjects

Yavdat Il'yasov

Publications and source records attributed to Yavdat Il'yasov.

21 records · Page 2Linked to original sources

arXiv2015

Maximal existence domains of positive solutions for two-parametric systems of elliptic equations

The paper is devoted to the study of two-parametric families of Dirichlet problems for systems of equations with $p, q$-Laplacians and indefinite nonlinearities. Continuous and monotone curves $Γ_f$ and $Γ_e$ on the parametric plane $λ\times μ$, which are the lower and upper bounds for a maximal domain of existence of weak positive solutions are introduced. The curve $Γ_f$ is obtained by developing our previous work \cite{BobkovIlyasov} and it determines a maximal domain of the applicability of the Nehari manifold and fibering methods. The curve $Γ_e$ is derived explicitly via minimax variational principle of the extended functional method.

math.AP↗

arXiv2014

Computation of maximal turning points by a variational approach

We develop a new paradigm for finding bifurcations of solutions of nonlinear problems, which is based on the detection of extreme values of new type of variational functional associated with the considering problem. The variational functional is obtained constructively by the extended functional method which can be applied to a wide class of parametric problems including nonlinear partial differential equations and problems in non-variational form. The main feature of the approach is that it allows to consider the problem of finding bifurcations from another geometric point of view, namely using the extended variational functional. Sufficient and necessary conditions for the existence of a maximal turning point by the approach are proved. Based on these, an algorithm of the quasi-direction of steepest ascent is introduced. Simulation experiments are used to illustrate the behavior of the method and to discuss its advantages and disadvantages in comparison with the alternatives.

math.AP↗

arXiv2009

Hopf maximum principle violation for elliptic equations with non-Lipschitz nonlinearity

We consider elliptic equations with non-Lipschitz nonlinearity $$ -Δu = λ|u|^{β-1}u-|u|^{α-1}u$$ in a smooth bounded domain $Ω\subset \mathbb{R}^n$, $n\geq 3$, with Dirichlet boundary conditions; here $0<α<β<1$. We prove the existence of a weak nonnegative solution which does not satisfy the Hopf boundary maximum principle, provided that $λ$ is large enough and $n>2(1+α) (1+β)/(1-α)(1-β)$.

math.AP↗