In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem \\begin{cases}-\operatorname{div}(a(x,u,\nabla u))+A_t(x,u,\nabla u) = g(x,u)+h(x) ...&\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega,\end{cases}\ where \(\Omega \subset \mathbb{R}^N\) is an open bounded domain, \(N\geq 3\), and \(A(x,t,\xi)\), \(g(x,t)\), \(h(x)\) are given functions, with \(A_t = \frac{\partial A}{\partial t}\), \(a = \nabla_{\xi} A\), such that \(A(x,\cdot,\cdot)\) is even and \(g(x,\cdot)\) is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if \(A(x,t,\xi)\) grows fast enough with respect to \(t\), then the nonlinear term related to \(g(x,t)\) may have also a supercritical growth.
In this paper, we consider a semilinear Neumann problem with an indefinite linear part and a Carathéodory nonlinearity which is superlinear near infinity and near zero, but does not satisfy the ...Ambrosetti–Rabinowitz condition. Using an abstract existence theorem for
C
1-functions having a local linking at the origin, we establish the existence of at least one nontrivial smooth solution.
We show the incompleteness of a usually used version of the generalized Ambrosetti–Rabinowitz condition in superlinear problems, also used in the paper cited in the title, and we propose a complete ...one.
In this paper, we consider the existence of multiple solutions of the homogeneous Dirichlet problem for a (
p
,
q
)-elliptic system with nonlinear product term as follows:
{
−
Δ
p
u
=
λ
α
(
x
)
|
u
|
...α
(
x
)
−
2
u
|
v
|
β
(
x
)
+
F
u
(
x
,
u
,
v
)
in
Ω
,
−
Δ
q
v
=
λ
β
(
x
)
|
u
|
α
(
x
)
|
v
|
β
(
x
)
−
2
v
+
F
v
(
x
,
u
,
v
)
in
Ω
,
u
=
0
=
v
on
∂
Ω
.
We emphasize that the potential
F
(
x
,
u
,
v
)
might contain a nonlinear product term which includes
F
(
x
,
u
,
v
)
=
|
u
|
θ
1
(
x
)
|
v
|
θ
2
(
x
)
ln
(
1
+
|
u
|
)
ln
(
1
+
|
v
|
)
as a prototype, and does not require
F
(
x
,
u
,
v
)
→
+
∞
as
|
u
|
+
|
v
|
→
+
∞
. With novel growth conditions on
F
(
x
,
u
,
v
)
, we develop a new method to check the Cerami compactness condition. Through arguments of critical point theory, we prove the existence of multiple constant-sign solutions for our elliptic system without requiring the well-known Ambrosetti–Rabinowitz condition. Moreover, we also give a result guaranteeing the existence of infinitely many solutions.
We consider a nonlinear Dirichlet problem driven by the
p
-Laplacian differential. The right-hand-side nonlinearity, exhibits a
(
p
−
1
)
-sublinear term of the form
m
(
z
)
|
x
|
r
−
2
x
,
r
<
p
...(concave term), and a Carathéodory term
f
(
z
,
x
)
which is
(
p
−
1
)
-superlinear near
+
∞
. However, it does not satisfy the usual Ambrosetti–Rabinowitz condition (AR-condition). Instead we employ a more general condition. Using a variational approach based on the critical point theory and the Ekeland variational principle, we show the existence of two nontrivial positive smooth solutions and then the existence of two nontrivial negative smooth solutions.
This article concerns the existence and multiplicity of solutions to a class of p(x)-Laplacian equations. We introduce a revised Ambrosetti-Rabinowitz condition, and show that the problem has a ...nontrivial solution and infinitely many solutions.
In this paper, we study the existence of multiple solutions to the following nonlinear elliptic boundary value problem of
p
-Laplacian type:
(*)
where 1 <
p
< ∞, Ω ⊆ ℝ
N
is a bounded smooth domain, Δ
...p
u
= div(|
Du
|
p
−2
Du
) is the
p
-Laplacian of
u
and
f
: Ω × ℝ → ℝ satisfies
uniformly with respect to
x
∈ Ω, and
l
is not an eigenvalue of −Δ
p
in
W
0
1,
p
(Ω) but
f
(
x
,
t
) dose not satisfy the Ambrosetti-Rabinowitz condition. Under suitable assumptions on
f
(
x
,
t
), we have proved that (*) has at least four nontrivitial solutions in
W
0
1,
p
(Ω) by using Nonsmooth Mountain-Pass Theorem under (
C
)
c
condition. Our main result generalizes a result by N. S. Papageorgiou, E. M. Rocha and V. Staicu in 2008 (Calculus of Variations and Partial Differential Equations, 33: 199–230(2008)) and a result by G. B. Li and H. S. Zhou in 2002 (Journal of the London Mathematical Society, 65: 123–138(2002)).
Let Ω be a bounded domain in ℝ
. In this paper, we consider the following nonlinear elliptic equation of N-Laplacian type:
when f is of subcritical or critical exponential growth. This nonlinearity ...is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N−Laplacian in ℝ
when the nonlinear term f has the exponential growth only deal with the case when f satisfies the (AR) condition. Our approach is based on a suitable version of the Mountain Pass Theorem introduced by G. Cerami 9, 10, 21. Examples of f and comparison with earlier assumptions in the literature are given.
In this paper, a class of nonperiodic discrete wave equations with Dirichlet boundary conditions are obtained by using the center-difference method. It is a strongly indefinite discrete Hamiltonian ...system. By using a variant and generalized weak linking theorem, the existence of the nontrivial time homoclinic solutions for the system will be obtained. The obtained main results here allow the classical Ambrosetti-Rabinowitz superlinear condition to be replaced by a general superquadratic condition. Such a method cannot be used for the corresponding continuous wave equations, however, it is valid for some general discrete Hamiltonian systems. Similarly, the existence of homoclinic periodic solutions can also be considered.
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