In this paper we consider the following Schrödinger equation:
{
−
Δ
u
+
V
(
x
)
u
=
g
(
x
,
u
)
for
x
∈
R
N
,
u
(
x
)
→
0
as
|
x
|
→
∞
,
where
V
(
x
)
and
g
(
x
,
u
)
are periodic with respect to
x ...and 0 is a boundary point of the spectrum
σ
(
−
Δ
+
V
)
. Replacing the classical Ambrosetti–Rabinowitz superlinear assumption on
g
(
x
,
u
)
by a general super-quadratic condition, we are able to obtain the existence of nontrivial solutions.
We consider the following
-harmonic problem
where
> 0 is a constant,
> 2
≥ 4 and
uniformly in
, which implies that
,
) does not satisfy the Ambrosetti-Rabinowitz type condition. By showing the ...Pohozaev identity for weak solutions to the limited problem of the above p-harmonic equation and using a variant version of Mountain Pass Theorem, we prove the existence and nonexistence of nontrivial solutions to the above equation. Moreover, if
,
) ≡
), the existence of a ground state solution and the nonexistence of nontrivial solutions to the above problem is also proved by using artificial constraint method and the Pohozaev identity.
In this paper we look for weak solutions of the quasilinear elliptic model problem −div(A(x,u)∇u)+12At(x,u)|∇u|2=g(x,u)+h(x)in Ω,u=0on ∂Ω,where Ω⊂RN is a bounded domain, N≥2, the real terms A(x,t), ...At(x,t)=∂A∂t(x,t) and g(x,t) are Carathéodory functions on Ω×R and h:Ω→R is a given measurable map.
We prove that, even if At(x,t)≢0, under suitable assumptions infinitely many solutions exist in spite of the lack of symmetry. A suitable supercritical growth is allowed for the nonlinear term g(x,t).
We use a variant of the variational perturbation techniques introduced by Rabinowitz in Rabinowitz (1982) but by means of a weak version of the Cerami–Palais–Smale condition.
We are concerned with the existence and multiplicity of nontrivial solutions to the following double phase problems:
-
div
(
|
∇
u
|
p
-
2
∇
u
+
α
(
x
)
|
∇
u
|
q
-
2
∇
u
)
+
V
(
x
)
|
u
|
γ
-
2
u
=
...f
(
x
,
u
)
,
in
Ω
,
u
=
0
,
on
∂
Ω
,
applying the mountain pass theorem and fountain theorem. The Ambrosetti—Rabinowitz condition as well as the monotonicity of
f
(
x
,
t
)
/
|
t
|
q
-
1
are not assumed.
Starting from a new sum decomposition of W1,p(RN)∩W1,q(RN) and using a variational approach, we investigate the existence of multiple weak solutions of a (p,q)-Laplacian equation on RN, for 1<q<p<N, ...with a sign-changing potential and a Carathéodory reaction term satisfying the celebrated Ambrosetti–Rabinowitz condition. Our assumptions are mild and different from those used in related papers and moreover our results improve or complement previous ones for the single p-Laplacian.
In this note, we study the existence and multiplicity of solutions for a system of coupled elliptic equations. We introduce a revised Ambrosetti–Rabinowitz condition, and show that the system has a ...nontrivial solution or even infinitely many solutions.
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