Page 225 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 225
TOPOLOGICAL METHODS IN DIFFERENTIAL EQUATIONS (MS-13)
applied in the definition of the variational functional and minimizers are shown to satisfy De
Giorgi type conditions.
Multiplicity of finite energy solutions for singular elliptic equations
Aleksandra Orpel, aleksandra.orpel@wmii.uni.lodz.pl
University of Łódz´, Poland
Our main goal is to investigate the following class of nonlinear elliptic equations with sub-
quadratic growth with respect to the gradient
div(a(||x||)∇u(x)) + h(x, u(x), ∇u(x)) = 0 in ΩR
where ΩR = {x ∈ Rn, ||x|| > R}, n > 2 and h(x, u, z) = f (x, u) − b(x, u)||z||β + g(x)x · z
with β ∈ (0, 2 . We consider the case when b may be singular at zero and describe conditions
guaranteeing the existence of a large number of positive solutions such that for a certain A > 0,
u(x) ≤ A||x||2−n in ΩR1, where R1 is sufficiently large. The rate of decay of ∇u is also
discussed. We present the approach based on the subsolution and supersolution method for
bounded subdomains and a certain convergence procedure. These results cover both sublinear
and superlinear f .
References
[1] S.Cui, Existence and nonexitence of positive solutions of singular semilinear elliptic bound-
ary value problems, Nonlinear Analysis,41 (2000), 149-176.
[2] A. Orpel, Positive evanescent solutions of singular elliptic problems in exterior domains,
Electronic Journal of Qualitative Theory of Differential Equations No. 36 (2016), 1-12, doi:
10.14232/ejqtde.2016.1.36.
[3] A. Orpel, "Multiplicity of positive solutions for singular elliptic problems" , Mathematische
Nachrichten, accepted
Periodic solutions to a forced Kepler problem in the plane
Duccio Papini, duccio.papini@uniud.it
Università degli Studi di Udine, Italy
Coauthors: Alberto Boscaggin, Walter Dambrosio
We investigate the following forced Kepler problem in the plane:
x¨ = −x + ∇xU (t, x), x ∈ R2 \ {(0, 0)},
|x|3
where U (t, x) is T -periodic in the first variable and satisfies U (t, x) = O(|x|α) for some α ∈
(0, 2) as |x| → ∞. We look for a T -periodic solution which minimimizes the corresponding
action functional on a space of loops which are not null-homotopic in the punctured plane.
On one hand, we do not impose further symmetry conditions on the perturbation’s potential
U . On the other, the solution we find is generalised, according to the definition given in the
223
applied in the definition of the variational functional and minimizers are shown to satisfy De
Giorgi type conditions.
Multiplicity of finite energy solutions for singular elliptic equations
Aleksandra Orpel, aleksandra.orpel@wmii.uni.lodz.pl
University of Łódz´, Poland
Our main goal is to investigate the following class of nonlinear elliptic equations with sub-
quadratic growth with respect to the gradient
div(a(||x||)∇u(x)) + h(x, u(x), ∇u(x)) = 0 in ΩR
where ΩR = {x ∈ Rn, ||x|| > R}, n > 2 and h(x, u, z) = f (x, u) − b(x, u)||z||β + g(x)x · z
with β ∈ (0, 2 . We consider the case when b may be singular at zero and describe conditions
guaranteeing the existence of a large number of positive solutions such that for a certain A > 0,
u(x) ≤ A||x||2−n in ΩR1, where R1 is sufficiently large. The rate of decay of ∇u is also
discussed. We present the approach based on the subsolution and supersolution method for
bounded subdomains and a certain convergence procedure. These results cover both sublinear
and superlinear f .
References
[1] S.Cui, Existence and nonexitence of positive solutions of singular semilinear elliptic bound-
ary value problems, Nonlinear Analysis,41 (2000), 149-176.
[2] A. Orpel, Positive evanescent solutions of singular elliptic problems in exterior domains,
Electronic Journal of Qualitative Theory of Differential Equations No. 36 (2016), 1-12, doi:
10.14232/ejqtde.2016.1.36.
[3] A. Orpel, "Multiplicity of positive solutions for singular elliptic problems" , Mathematische
Nachrichten, accepted
Periodic solutions to a forced Kepler problem in the plane
Duccio Papini, duccio.papini@uniud.it
Università degli Studi di Udine, Italy
Coauthors: Alberto Boscaggin, Walter Dambrosio
We investigate the following forced Kepler problem in the plane:
x¨ = −x + ∇xU (t, x), x ∈ R2 \ {(0, 0)},
|x|3
where U (t, x) is T -periodic in the first variable and satisfies U (t, x) = O(|x|α) for some α ∈
(0, 2) as |x| → ∞. We look for a T -periodic solution which minimimizes the corresponding
action functional on a space of loops which are not null-homotopic in the punctured plane.
On one hand, we do not impose further symmetry conditions on the perturbation’s potential
U . On the other, the solution we find is generalised, according to the definition given in the
223
