In this article, we obtain the existence and infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real ...parameters, via combining the variational method, and the concentration‐compactness principle for anisotropic variable exponent under suitable assumptions on the nonlinearities.
In this paper, our focus lies in addressing the Dirichlet problem associated with a specific class of critical anisotropic elliptic equations of Schrödinger-Kirchhoff type. These equations ...incorporate variable exponents and a real positive parameter. Our objective is to establish the existence of at least one solution to this problem.
We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find ...infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.
This paper is devoted to study some nonlinear elliptic Neumann equations of the type
in the anisotropic variable exponent Sobolev spaces, where
is a Leray-Lions operator and
,
, ∇
),
(
,
, ∇
) are ...two Carathéodory functions that verify some growth conditions. We prove the existence of renormalized solutions for our strongly nonlinear elliptic Neumann problem.
The aim of this paper is to study the existence of solutions for some quasilineare anisotropic elliptic problem associated with differential inclusion. We study the two cases of f∈ L∞(Ω) and f∈ ...L1(Ω). Moreover, we show the uniqueness of solution under some additional assumptions.
In this paper, we prove the existence of in finitely many solutions for the following system
by applying a critical point variational principle obtained by Ricceri as a consequence of a more general ...variational principle and the theory of the anisotropic variable exponent Sobolev spaces 2010 Mathematics Subject Classification. 35K05 - 35K55.
We establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic equations involving the anisotropic
-Laplace operator, on a bounded domain with smooth boundary. ...We work on the anisotropic variable exponent Sobolev spaces and our main tool is the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz.
We study a general class of anisotropic problems involving $\vec p(\cdot)$-Laplace type operators. We search for weak solutions that are constant on the boundary by introducing a new subspace of the ...anisotropic Sobolev space with variable exponent and by proving that it is a reflexive Banach space. Our argumentation for the existence of weak solutions is mainly based on a variant of the mountain pass theorem of Ambrosetti and Rabinowitz.
We study the nonlinear degenerate problem
−
∑
i
=
1
N
∂
x
i
a
i
(
x
,
∂
x
i
u
)
=
f
(
x
,
u
)
in Ω,u= 0 on ∂Ω, where Ω ⊂ ℝ
N
(N≥ 3) is a bounded domain with smooth boundary,
∑
i
=
1
N
∂
x
i
a
i
(
x
...,
∂
x
i
u
)
is a
p
→
(
·
)
- Laplace type operator and the nonlinearityfis
(
P
+
+
−
1
)
- superlinear at infinity (with
p
→
(
x
)
=
(
p
1
(
x
)
,
p
2
(
x
)
,
…
p
N
(
x
)
)
and
P
+
+
=
max
i
∈
{
1
,
…
,
N
}
{
sup
x
∈
Ω
p
i
(
x
)
}
)
. By means of the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz, we establish the existence of a sequence of weak solutions in appropriate anisotropic variable exponent Sobolev spaces.
2010Mathematics Subject Classification: 35J25, 35J62, 35D30, 46E35, 35J20.
Key words and phrases: Quasilinear elliptic equations, Multiple weak solutions, Critical point theory, Anisotropic variable exponent Sobolev spaces, Symmetric mountain-pass theorem.
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