Double phase problems with variable growth Cencelj, Matija; Rădulescu, Vicenţiu D.; Repovš, Dušan D.
Nonlinear analysis,
December 2018, 2018-12-00, 20181201, Volume:
177
Journal Article
Peer reviewed
Open access
We consider a class of double phase variational integrals driven by nonhomogeneous potentials. We study the associated Euler equation and we highlight the existence of two different Rayleigh ...quotients. One of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues. The notion of eigenvalue is understood in the sense of pairs of nonlinear operators, as introduced by Fučík, Nečas, Souček, and Souček. The analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the p(x)-Laplace operator, the generalized mean curvature operator, or the capillarity differential operator with variable exponent. The results contained in this paper complement the pioneering contributions of Marcellini, Mingione et al. in the field of variational integrals with unbalanced growth.
In this work, we study the existence and multiplicity results for the following nonlocal p(x)-Kirchhoff problem: (0.1)−a−b∫Ω1p(x)|∇u|p(x)dxdiv(|∇u|p(x)−2∇u)=λ|u|p(x)−2u+g(x,u) in Ω,u=0, on ∂Ω,where ...a≥b>0 are constants, Ω⊂RN is a bounded smooth domain, p∈C(Ω¯) with N>p(x)>1, λ is a real parameter and g is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.
We establish the existence of at least two solutions of the
Prandtl–Batchelor
like elliptic problem driven by a power nonlinearity and a singular term. The associated energy functional is ...nondifferentiable, and hence the usual variational techniques do not work. We shall use a novel approach in tackling the associated energy functional by a sequence of
C
1
functionals and a
cutoff function
. Our main tools are fundamental elliptic regularity theory and the mountain pass theorem.
Nonlinear nonhomogeneous singular problems Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D.; Repovš, Dušan D.
Calculus of variations and partial differential equations,
02/2020, Volume:
59, Issue:
1
Journal Article
Peer reviewed
Open access
We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order
(
p
-
1
)
near
+
∞
and with a reaction which has the competing effects of a ...parametric singular term and a
(
p
-
1
)
-superlinear perturbation which does not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational tools, together with suitable truncation and strong comparison techniques, we prove a “bifurcation-type” theorem that describes the set of positive solutions as the parameter
λ
moves on the positive semiaxis. We also show that for every
λ
>
0
, the problem has a smallest positive solution
u
λ
∗
and we demonstrate the monotonicity and continuity properties of the map
λ
↦
u
λ
∗
.
We consider the following convective Neumann systems:
(
S
)
{
−
Δ
p
1
u
1
+
|
∇
u
1
|
p
1
u
1
+
δ
1
=
f
1
(
x
,
u
1
,
u
2
,
∇
u
1
,
∇
u
2
)
in
Ω
,
−
Δ
p
2
u
2
+
|
∇
u
2
|
p
2
u
2
+
δ
2
=
f
2
(
x
,
u
...1
,
u
2
,
∇
u
1
,
∇
u
2
)
in
Ω
,
|
∇
u
1
|
p
1
−
2
∂
u
1
∂
η
=
0
=
|
∇
u
2
|
p
2
−
2
∂
u
2
∂
η
on
∂
Ω
,
where Ω is a bounded domain in
R
N
(
N
≥
2
) with a smooth boundary
∂
Ω,
δ
1
,
δ
2
>
0
are small parameters,
η
is the outward unit vector normal to
∂
Ω,
f
1
,
f
2
:
Ω
×
R
2
×
R
2
N
→
R
are Carathéodory functions that satisfy certain growth conditions, and
Δ
p
i
(
1
<
p
i
<
N
,
i
=
1
,
2
) are the
p
-Laplace operators
Δ
p
i
u
i
=
div
(
|
∇
u
i
|
p
i
−
2
∇
u
i
)
for
u
i
∈
W
1
,
p
i
(
Ω
)
. To prove the existence of solutions to such systems, we use a subsupersolution method. We also obtain nodal solutions by constructing appropriate subsolution and supersolution pairs. To the best of our knowledge, such systems have not been studied yet.
In this paper, we are concerned with the Neumann problem for a class of quasilinear stationary Kirchhoff-type potential systems, which involves general variable exponents elliptic operators with ...critical growth and real positive parameter. We show that the problem has at least one solution, which converges to zero in the norm of the space as the real positive parameter tends to infinity, via combining the truncation technique, variational method, and the concentration–compactness principle for variable exponent under suitable assumptions on the nonlinearities.
We consider a parametric nonlinear nonhomogeneous elliptic equation, driven by the sum of two differential operators having different structure. The associated energy functional has unbalanced growth ...and we do not impose any global growth conditions to the reaction term, whose behavior is prescribed only near the origin. Using truncation and comparison techniques and Morse theory, we show that the problem has multiple solutions in the case of high perturbations. We also show that if a symmetry condition is imposed to the reaction term, then we can generate a sequence of distinct nodal solutions with smaller and smaller energies.
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