In this paper we study the nonlinear Chern–Simons–Schrödinger systems with an external potential. We show the existence, non-existence, and multiplicity of standing waves to this problem with ...asymptotically linear nonlinearities, which do not hold the Ambrosetti–Rabinowitz condition.
This paper is concerned with the existence of two nonnegative radial solutions of following nonlinear Schrödinger equation with fractional Laplacian(-Δ)αu+u=f(u)inRN,u∈Hα(RN),where 0<α<1. Under ...certain assumptions, we obtain that the above problem has at least two nontrivial radial solutions without assuming the Ambrosetti–Rabinowitz condition by variational methods and concentration compactness principle. The result extends one of the main results of Felmer et al. (2012).
This paper is concerned with the following fractional Schrödinger equations involving critical exponents: (−Δ)αu+V(x)u=k(x)f(u)+λ|u|2α∗−2uinRN, where (−Δ)α is the fractional Laplacian operator with ...α∈(0,1), N≥2, λ is a positive real parameter and 2α∗=2N/(N−2α) is the critical Sobolev exponent, V(x) and k(x) are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti–Rabinowitz condition on the subcritical nonlinearity.
We consider nonlinear problems governed by the fractional p-Laplacian in presence of nonlocal Neumann boundary conditions and we show three different existence results: the first two theorems deal ...with a p-superlinear term, the last one with a source having p-linear growth. For the p-superlinear case we face two main difficulties. First: the p-superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the linking theorem, the nonlocal nature of the associated eigenfunctions prevents the use of such a classical theorem. For these reasons, we are led to adopt another approach, relying on the notion of linking over cones.
We consider the generalized quasilinear Schrödinger equations
(P)
where
,
is an even differentiable function,
satisfies the Carathéodory condition, the potential
is continuous and λ is a parameter. ...The intention of the article is to determine the precise positive interval of λ when the problem possesses at least two nontrivial solutions. The main tool is the abstract critical point which is based on the Ekeland variational principle. Moreover, the existence of one solution and infinite solutions has been studied by the mountain pass theorem and the symmetric mountain pass theorem.
In this paper we study the nonlinear Kirchhoff equations on the whole space. We show the existence, non-existence, and multiplicity of solutions to this problem with asymptotically linear ...nonlinearities. This result can be regarded as an extension of the result in Li et al. (2012).
In this paper, we deal with the existence of solutions to the nonuniformly elliptic equation of the form
(0.1)
−
div
(
a
(
x
,
∇
u
)
)
+
V
(
x
)
|
u
|
N
−
2
u
=
f
(
x
,
u
)
|
x
|
β
+
ε
h
(
x
)
in
R
N
...when
f
:
R
N
×
R
→
R
behaves like
exp
(
α
|
u
|
N
/
(
N
−
1
)
)
when
|
u
|
→
∞
and satisfies the Ambrosetti–Rabinowitz condition. In particular, in the case of
N-Laplacian, i.e.,
a
(
x
,
∇
u
)
=
|
∇
u
|
N
−
2
∇
u
, we obtain multiplicity of weak solutions of
(0.1). Moreover, we can get the nontriviality of the solution in this case when
ε
=
0
. Finally, we show that the main results remain true if one replaces the Ambrosetti–Rabinowitz condition on the nonlinearity by weaker assumptions and thus we establish the existence and multiplicity results for a wider class of nonlinearity, see Section 7 for more details.
We consider a nonlinear elliptic equation driven by the sum of a p-Laplacian, where 1<q\leq 2\leq p<\infty -superlinear Carathéodory reaction term which doesn't satisfy the usual ...Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory together with techniques from Morse theory, we show that the problem has at least three nontrivial solutions; among them one is positive and one is negative.
In this paper, we consider the quasilinear elliptic problem with potential
where Ω is a smooth bounded domain in
(
), V is a given function in a generalized Lebesgue space
, and
is a Carathéodory ...function satisfying suitable growth conditions. Using variational arguments, we study the existence of weak solutions for
in the framework of Musielak-Sobolev spaces. The main difficulty here is that the nonlinearity
considered does not satisfy the well-known Ambrosetti-Rabinowitz condition.
By using a generalization of the Struwe-Jeanjean monotonicity trick we prove the existence of a non-trivial weak solution for the following problem
where
is a smooth bounded open set in
,
,
,
,
is ...the fractional p-Laplace operator and
is a continuous function with
and there are
and
such that
whenever
.
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