408 research outputs found
On a Kirchhoff type problems with potential well and indefinite potential
In this paper, we study the following Kirchhoff type problem:%
\left\{\aligned&-\bigg(\alpha\int_{\bbr^3}|\nabla u|^2dx+1\bigg)\Delta
u+(\lambda a(x)+a_0)u=|u|^{p-2}u&\text{ in }\bbr^3,\\%
&u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{\alpha,\lambda})}% where
, and are two positive parameters, a_0\in\bbr is a
(possibly negative) constant and is the potential well. By the
variational method, we investigate the existence of nontrivial solutions to
. To our best knowledge, it is the first time
that the nontrivial solution of the Kirchhoff type problem is found in the
indefinite case. We also obtain the concentration behaviors of the solutions as
.Comment: 1
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 λ on RN is proved by using variational method. We will show that if the nonlinearities are asymptotically linear at infinity and λ>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
Multiple solutions for a class of nonhomogeneous fractional Schr\"odinger equations in
This paper is concerned with the following fractional Schr\"odinger equation
\begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u+u= k(x)f(u)+h(x)
\mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in }
\mathbb{R}^{N}, \end{array} \right. \end{equation*} where , , is the fractional Laplacian, is a bounded positive
function, , is nonnegative and
is either asymptotically linear or superlinear at infinity.\\ By using the
-harmonic extension technique and suitable variational methods, we prove the
existence of at least two positive solutions for the problem under
consideration, provided that is sufficiently small
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