1,129 research outputs found
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
Multiple solutions for a fractional -Laplacian equation with sign-changing potential
We use a variant of the fountain Theorem to prove the existence of infinitely
many weak solutions for the following fractional p-Laplace equation
(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u) in R^N, where ,,, is the fractional -Laplace operator, the
nonlinearity f is -superlinear at infinity and the potential V(x) is allowed
to be sign-changing
Ground states for superlinear fractional Schr\"odinger equations in \R^{N}
In this paper we study ground states of the following fractional
Schr\"odinger equation
(- \Delta)^{s} u + V(x) u = f(x, u) \, \mbox{ in } \, \R^{N},
u\in \H^{s}(\R^{N})
where , and is a continuous function satisfying a
suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. We
consider the cases when the potential is -periodic or has a bounded
potential well
Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition
The purpose of this paper is to study -periodic solutions to
[(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in} (0,T)^{N} (P)
u(x+Te_{i})=u(x) &\mbox{for all} x \in \R^{N}, i=1, \dots, N
where , , , and is a continuous
function, -periodic in and satisfying a suitable growth assumption
weaker than the Ambrosetti-Rabinowitz condition. The nonlocal operator
can be realized as the Dirichlet to Neumann map for a
degenerate elliptic problem posed on the half-cylinder
. By using a variant of the Linking
Theorem, we show that the extended problem in admits a
nontrivial solution which is -periodic in . Moreover, by a
procedure of limit as , we also prove the existence of a
nontrivial solution to (P) with
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
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