971 research outputs found
Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness
We study the following Kirchhoff equation A
special feature of this paper is that the nonlinearity and the potential
are indefinite, hence sign-changing. Under some appropriate assumptions on
and , we prove the existence of two different solutions of the equation
via the Ekeland variational principle and Mountain Pass Theorem
Existence and multiplicity of nontrivial solutions to the modified Kirchhoff equation without the growth and Ambrosetti-Rabinowitz conditions
The paper focuses on the modified Kirchhoff equation a + b Z RN |∇u| 2 dx� ∆u − u∆(u 2 ) + V(x)u = λ f(u), x ∈ R N, where a, b > 0, V(x) ∈ C(RN, R) and λ < 1 is a positive parameter. We just assume that the nonlinearity f(t) is continuous and superlinear in a neighborhood of t = 0 and at infinity. By applying the perturbation method and using the cutoff function, we get existence and multiplicity of nontrivial solutions to the revised equation. Then we use the Moser iteration to obtain existence and multiplicity of nontrivial solutions to the above original Kirchhoff equation. Moreover, the nonlinearity f(t) may be supercritical
Kirchhoff equations in generalized Gevrey spaces: local existence, global existence, uniqueness
In this note we present some recent results for Kirchhoff equations in
generalized Gevrey spaces. We show that these spaces are the natural framework
where classical results can be unified and extended. In particular we focus on
existence and uniqueness results for initial data whose regularity depends on
the continuity modulus of the nonlinear term, both in the strictly hyperbolic
case, and in the degenerate hyperbolic case.Comment: 20 pages, 4 tables, conference paper (7th ISAAC congress, London
2009
Mountain pass solutions for the fractional Berestycki-Lions problem
We investigate the existence of least energy solutions and infinitely many
solutions for the following nonlinear fractional equation (-\Delta)^{s} u =
g(u) \mbox{ in } \mathbb{R}^{N}, where , ,
is the fractional Laplacian and is an
odd function satisfying Berestycki-Lions type
assumptions. The proof is based on the symmetric mountain pass approach
developed by Hirata, Ikoma and Tanaka in \cite{HIT}. Moreover, by combining the
mountain pass approach and an approximation argument, we also prove the
existence of a positive radially symmetric solution for the above problem when
satisfies suitable growth conditions which make our problem fall in the so
called "zero mass" case
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