In this paper, we consider the following coupled Schrödinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis-Nirenberg ...problem{−Δu+λ1u=μ1u5+βu2v3,x∈Ω,−Δv+λ2v=μ2v5+βv2u3,x∈Ω,u=v=0,x∈∂Ω, where Ω is a ball in R3, −λ1(Ω)<λ1,λ2<−14λ1(Ω), μ1,μ2>0 and β>0. Here λ1(Ω) is the first eigenvalue of −Δ with Dirichlet boundary condition in Ω. We show that the problem has at least one positive solution for all 0<β<2μ1μ2. In particular, when λ1=λ2, we prove the existence of positive synchronized solutions for all β>0.
Non-smooth variational problems and applications in mechanics Kovtunenko, Victor A.; Itou, Hiromichi; Khludnev, Alexander M. ...
Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences,
07/2024, Letnik:
382, Številka:
2275
Journal Article
In this paper we shall show an existence result of a positive solution for a Kirchhoff problem type in a bounded domain of RN, that is, for the problem −M(∫Ω|∇u|2dx)Δu=λf(x,u)+|u|2∗−2uin Ω,u=0on ∂Ω. ...We shall study the asymptotic behavior of this solution when λ converges to infinity. Our approach is based on the variational method, an appropriated truncated argument, and a priori estimates to obtain the solution.
We study the existence, multiplicity and concentration behavior of positive solutions for the nonlinear Kirchhoff type problem
{
−
(
ε
2
a
+
ε
b
∫
R
3
|
∇
u
|
2
)
Δ
u
+
V
(
x
)
u
=
f
(
u
)
in
R
3
,
u
...∈
H
1
(
R
3
)
,
u
>
0
in
R
3
,
where
ε
>
0
is a parameter and
a
,
b
>
0
are constants;
V is a positive continuous potential satisfying some conditions and
f is a subcritical nonlinear term. We relate the number of solutions with the topology of the set where
V attains its minimum. The results are proved by using the variational methods.
A mistake in the assumptions of case
(H+)$$ \left({H}_{+}\right) $$ in Theorem 1.1 is corrected. This requires a slight refinement of Lemma 3.6 leading to the multiplicity result.
In this article, we study the existence and multiplicity of high energy solutions to the problem proposed as a model for the dynamics of galaxies: −Δu+V(x)u=|u|2∗−2u|y|,x=(y,z)∈Rm×Rn−m,where n>4, ...2≤m<n, 2∗≔2(n−1)n−2 and potential function V(x):Rn→R. Benefiting from a global compactness result, we show that there exist at least two positive high energy solutions. Our proofs are based on barycenter function, quantitative deformation lemma and Brouwer degree theory.
We explain how to derive a network element for the linear elasticity problem. After presenting sufficient conditions on the network for the validity of a discrete Korn inequality, we also propose ...several variations of the presented method and in particular we explain how it can be used on meshes to derive schemes that remain stable while keeping the stencil as compact as possible. Numerical examples illustrate the good behavior of the method, in both the mesh-based and truly meshless contexts.
We prove the existence multiple solitary waves for a generalized Kadomtsev-Petviashvili equation with a potential in dimension two. The number of waves correspond to the number of global minimum ...points of the potential when a parameter is small enough.
In this paper we study the following (p,q)-Laplacian equation with nonlocal Choquard reaction −Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)=1|x|μ∗F(u)f(u)inRN,where 1<p<q<N, 0<μ<N, V is the absorption potential and ...the nonlinear term F is the primitive function of f. Under some suitable conditions on the potential and nonlinearity, the new existence result of nontrivial solutions is obtained via variational methods.
In this survey paper, by using variational methods, we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type (ProQuest: Formulae and/or non-USASCII text ...omitted) where Omega is a bounded or an exterior domain of dbl-struck R super(N) and q is a continuous positive function. The results presented in this paper extend several contributions concerning the Lane-Emden equation and we focus on new phenomena which are due to the presence of variable exponents.
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