arXiv subjects

Andrzej Szulkin

Publications and source records attributed to Andrzej Szulkin.

22 records · Page 2Linked to original sources

arXiv2014

Bifurcation from infinity for an asymptotically linear Schrödinger equation

We consider an asymptotically linear Schrödinger equation $-Δu + V(x)u = λu + f(x,u), \ x\in R^N$, and show that if $λ_0$ is an isolated eigenvalue for the linearization at infinity, then under some additional conditions there exists a sequence $(u_n,λ_n)$ of solutions such that $\|u_n\|\to\infty$ and $λ_n\toλ_0$. Our results extend some recent work by Stuart. We use degree theory if the multiplicity of $λ_0$ is odd and Morse theory (or more specifically, Gromoll-Meyer theory) if it is not.

math.AP↗

arXiv2013

A supercritical elliptic problem in a cylindrical shell

We consider the problem \[ -Δu=|u|^{p-2}u in Ω, u=0 on \partialΩ, \] where $Ω:=\{(y,z)\in\mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1}: 0<a<|y|<b<\infty\}$, $0\leq m\leq N-1$ and $N\geq2$. Let $2_{N,m}^{\ast}:=2(N-m)/(N-m-2)$ if $m<N-2$ and $2_{N,m}^{\ast}:=\infty$ if $m=N-2$ or $N-1$. We show that $2_{N,m}^{\ast}$ is the true critical exponent for this problem, and that there exist nontrivial solutions if $2<p<2_{N,m}^{\ast}$ but there are no such solutions if $p\geq2_{N,m}^{\ast}$.

math.AP↗

arXiv2012

Multiple solutions to nonlinear Schrödinger equations with singular electromagnetic potential

We consider the semilinear electromagnetic Schrödinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(Ω,\mathbb{C}), where $Ω=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq m\leq N$, $N\geq3$, $2^{\ast}:= 2N/(N-2)$ is the critical Sobolev exponent, $V$ is a Hardy term and $A$ is a singular magnetic potential of a particular form which includes the Aharonov-Bohm potentials. Under some symmetry assumptions on $A$ we obtain multiplicity of solutions satisfying certain symmetry properties.

math.AP↗

arXiv2012

A concentration phenomenon for semilinear elliptic equations

For a domain $Ω\subset\dR^N$ we consider the equation $ -Δu + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $Ω$ and negative outside, and such that the sets $\{Q_n>0\}$ shrink to a point $x_0\inΩ$ as $n\to\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\{Q_n>0\}$ shrink to two points and $u_n$ are ground state solutions, then they concentrate at one of these points.

math.AP↗