In this letter, we study the existence and multiplicity of solutions for a class of p-Laplacian equations. We introduce a revised Ambrosetti–Rabinowitz condition, and show that the equation has a ...nontrivial solution and infinitely many solutions, respectively.
In this paper, we prove the existence of nontrivial nonnegative solutions to a class of elliptic equations and systems which do not satisfy the Ambrosetti–Rabinowitz (AR) condition where the ...nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth at ∞. The known results without the AR condition in the literature only involve nonlinear terms of polynomial growth. We will use suitable versions of the Mountain Pass Theorem and Linking Theorem introduced by Cerami (Istit. Lombardo Accad. Sci. Lett. Rend. A, 112(2):332–336,
1978
Ann. Mat. Pura Appl., 124:161–179,
1980
). The Moser–Trudinger inequality plays an important role in establishing our results. Our theorems extend the results of de Figueiredo, Miyagaki, and Ruf (Calc. Var. Partial Differ. Equ., 3(2):139–153,
1995
) and of de Figueiredo, do Ó, and Ruf (Indiana Univ. Math. J., 53(4):1037–1054,
2004
) to the case where the nonlinear term does not satisfy the AR condition. Examples of such nonlinear terms are given in Appendix
A
. Thus, we have established the existence of nontrivial nonnegative solutions for a wider class of nonlinear terms.
The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type
where
is an open bounded domain,
,
and
,
are
...-Carathéodory functions on
with partial derivatives
, respectively
, while
,
are given Carathéodory maps defined on
which are partial derivatives of a function
. We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional
, related to problem (P), admits at least one critical point in the “right” Banach space
. Moreover, if
is even, then (P) has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition, a “good” decomposition of the Banach space
and suitable generalizations of the Ambrosetti–Rabinowitz Mountain Pass Theorems.
In this paper, we consider the existence of strong solutions of the following p(x)-Laplacian Dirichlet problem via critical point theory: {−div(∣∇u∣p(x)−2∇u)=f(x,u), in Ω,u=0, on ∂Ω. We give a ...new growth condition, under which, we use a new method to check the Cerami compactness condition. Hence, we prove the existence of strong solutions of the problem as above without the growth condition of the well-known Ambrosetti–Rabinowitz type and also give some results about multiplicity of the solutions.
In this paper, we study the following superlinear
p
-Kirchhoff-type equation:
{
M
(
∫
R
2
N
|
u
(
x
)
−
u
(
y
)
|
p
|
x
−
y
|
N
+
p
s
d
x
d
y
)
(
−
△
)
p
s
u
(
x
)
−
λ
|
u
|
p
−
2
u
=
g
(
x
,
u
)
in
...Ω
,
u
=
0
in
R
N
∖
Ω
,
Under suitable assumptions on
g
(
x
,
u
)
without the (AR) condition, the existence of infinitely many solutions for the Kirchhoff equation of a fractional
p
-Laplacian is obtained by using the fountain theorem. Our conclusions generalize and extend some existing results.
In this paper, we consider the following coupled gradient-type quasilinear elliptic system
-
div
(
a
(
x
,
u
,
∇
u
)
)
+
A
t
(
x
,
u
,
∇
u
)
=
G
u
(
x
,
u
,
v
)
in
Ω
,
-
div
(
b
(
x
,
v
,
∇
v
)
)
+
B
...t
(
x
,
v
,
∇
v
)
=
G
v
x
,
u
,
v
in
Ω
,
u
=
v
=
0
on
∂
Ω
,
where
Ω
is an open bounded domain in
R
N
,
N
≥
2
. We suppose that some
C
1
–Carathéodory functions
A
,
B
:
Ω
×
R
×
R
N
→
R
exist such that
a
(
x
,
t
,
ξ
)
=
∇
ξ
A
(
x
,
t
,
ξ
)
,
A
t
(
x
,
t
,
ξ
)
=
∂
A
∂
t
(
x
,
t
,
ξ
)
,
b
(
x
,
t
,
ξ
)
=
∇
ξ
B
(
x
,
t
,
ξ
)
,
B
t
(
x
,
t
,
ξ
)
=
∂
B
∂
t
(
x
,
t
,
ξ
)
, and that
G
u
(
x
,
u
,
v
)
,
G
v
(
x
,
u
,
v
)
are the partial derivatives of a
C
1
–Carathéodory nonlinearity
G
:
Ω
×
R
×
R
→
R
. Roughly speaking, we assume that
A
(
x
,
t
,
ξ
)
grows at least as
(
1
+
|
t
|
s
1
p
1
)
|
ξ
|
p
1
,
p
1
>
1
,
s
1
≥
0
, while
B
(
x
,
t
,
ξ
)
grows as
(
1
+
|
t
|
s
2
p
2
)
|
ξ
|
p
2
,
p
2
>
1
,
s
2
≥
0
, and that
G
(
x
,
u
,
v
) can also have a supercritical growth related to
s
1
and
s
2
. Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.
In this paper we prove the existence of at least one nonnegative nontrivial weak solution in D1,p(RN)∩D1,q(RN) for the equation −Δpu−Δqu+a(x)|u|p−2u+b(x)|u|q−2u=f(x,u),x∈RN, where ...1<q<p<q⋆:=NqN−q,p<N,Δmu:=div(|∇u|m−2∇u) is the m-Laplacian operator, the coefficients a and b are continuous, coercive and positive functions, and the nonlinearity f is a Carathéodory function satisfying some hypotheses which do not include the Ambrosetti–Rabinowitz condition.
We consider a class of pseudo-relativistic Hartree equations in presence of general nonlinearities not satisfying the Ambrosetti–Rabinowitz condition. Using variational methods based on critical ...point theory, we show the existence of two non trivial signed solutions, one positive and one negative.
We consider the semilinear elliptic problem
where a is a continuous function which may change sign and f is superlinear but does not satisfy the standard Ambrosetti-Rabinowitz condition. We show that ...if f is regularly varying of index one at infinity then the above problem has a positive solution, provided α satisfies some additional assumptions. Our proof uses an abstract theorem due to L. Jeanjean on critical points of functionals with mountain-pass structure, and it relies on the obtention of a priori bounds for positive solutions..
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