503 research outputs found
Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity
We study asymptotic behavior of positive ground state solutions of the
nonlinear Kirchhoff equation as and ,
where or , , is the Sobolev
critical exponent, , are constants and is a
parameter. In particular, we prove that in the case , as , after a suitable rescaling the ground state solutions of the problem
converge to the unique positive solution of the equation
and as , after another rescaling the ground state solutions
of the problem converge to a particular solution of the critical Emden-Fowler
equation . We establish a sharp asymptotic
characterisation of such rescalings, which depends in a non-trivial way on the
space dimension and . We also discuss a connection of our results
with a mass constrained problem associated to the Kirchhoff equation with the
mass normalization constraint .Comment: 40 page
Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations on R
The multiplicity of positive solutions for Kirchhoff type equations depending on a nonnegative parameter Ξ» on RN is proved by using variational method. We will show that if the nonlinearities are asymptotically linear at infinity and Ξ»>0 is sufficiently small, the Kirchhoff type equations have at least two positive solutions. For the perturbed problem, we give the result of existence of three positive solutions
Normalized Solutions to Nonautonomous Kirchhoff Equation
In this paper, we study the existence of normalized solutions to the
following Kirchhoff equation with a perturbation: where , . We treat three cases.
(i)When , we obtain the existence of global
constraint minimizers.
(ii)When , we prove the existence of mountain
pass solution.
(iii)When , we establish the existence of
bound state solutions.Comment: arXiv admin note: text overlap with arXiv:2301.07926 by other author
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