681 research outputs found
Large solutions of elliptic systems of second order and applications to the biharmonic equation
In this work we study the nonnegative solutions of the elliptic system \Delta
u=|x|^{a}v^{\delta}, \Delta v=|x|^{b}u^{\mu} in the superlinear case \mu
\delta>1, which blow up near the boundary of a domain of R^{N}, or at one
isolated point. In the radial case we give the precise behavior of the large
solutions near the boundary in any dimension N. We also show the existence of
infinitely many solutions blowing up at 0. Furthermore, we show that there
exists a global positive solution in R^{N}\{0}, large at 0, and we describe its
behavior. We apply the results to the sign changing solutions of the biharmonic
equation \Delta^2 u=|x|^{b}|u|^{\mu}. Our results are based on a new dynamical
approach of the radial system by means of a quadratic system of order 4,
combined with nonradial upper estimates
Nonmonotone local minimax methods for finding multiple saddle points
In this paper, by designing a normalized nonmonotone search strategy with the
Barzilai--Borwein-type step-size, a novel local minimax method (LMM), which is
a globally convergent iterative method, is proposed and analyzed to find
multiple (unstable) saddle points of nonconvex functionals in Hilbert spaces.
Compared to traditional LMMs with monotone search strategies, this approach,
which does not require strict decrease of the objective functional value at
each iterative step, is observed to converge faster with less computations.
Firstly, based on a normalized iterative scheme coupled with a local peak
selection that pulls the iterative point back onto the solution submanifold, by
generalizing the Zhang--Hager (ZH) search strategy in the optimization theory
to the LMM framework, a kind of normalized ZH-type nonmonotone step-size search
strategy is introduced, and then a novel nonmonotone LMM is constructed. Its
feasibility and global convergence results are rigorously carried out under the
relaxation of the monotonicity for the functional at the iterative sequences.
Secondly, in order to speed up the convergence of the nonmonotone LMM, a
globally convergent Barzilai--Borwein-type LMM (GBBLMM) is presented by
explicitly constructing the Barzilai--Borwein-type step-size as a trial
step-size of the normalized ZH-type nonmonotone step-size search strategy in
each iteration. Finally, the GBBLMM algorithm is implemented to find multiple
unstable solutions of two classes of semilinear elliptic boundary value
problems with variational structures: one is the semilinear elliptic equations
with the homogeneous Dirichlet boundary condition and another is the linear
elliptic equations with semilinear Neumann boundary conditions. Extensive
numerical results indicate that our approach is very effective and speeds up
the LMMs significantly.Comment: 32 pages, 7 figures; Accepted by Journal of Computational Mathematics
on January 3, 202
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