Page 226 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 226
TOPOLOGICAL METHODS IN DIFFERENTIAL EQUATIONS (MS-13)
paper [Boscaggin, Ortega, Zhao, Periodic solutions and regularization of a Kepler problem with
time-dependent perturbation, Trans. Amer. Math. Soc. 372 (2018), 677–703]. In particular,
such solution may have a finite number of collisions with the origin in each period, while its
energy and bouncing directions behave in a regular way at each collision time.
Existence of radial bounded solutions for some quasilinear elliptic
equation in RN
Addolorata Salvatore, addolorata.salvatore@uniba.it
Università degli studi di Bari Aldo Moro, Italy
We study the quasilinear equation
(P ) − div(A(x, u)|∇u|p−2∇u) + 1 At(x, u)|∇u|p + |u|p−2u = g(x, u) in RN ,
p
with N ≥ 3, p > 1, where A(x, t), At(x, t) = ∂A (x, t) and g(x, t) are Carathéodory functions
∂t
on RN × R.
Under suitable assumptions on A(x, t) and g(x, t) the problem has a good variational struc-
ture, i.e. the weak bounded solutions of problem (P ) are critical points of the C1 functional
J (u) = 1 A(x, u)|∇u|pdx + 1 |u|pdx − G(x, u)dx,
p RN p RN
RN
on the Banach space X = W 1,p(RN ) ∩ L∞(RN ), with G(x, t) = t g(x, s)ds.
0
In order to overcome the lack of compactness, we assume that the problem has radial sym-
metry, then we look for critical points of J restricted to Xr, subspace of the radial functions in
X.
Following an approach which exploits the interaction between · X and the norm on
W 1,p(RN ), we prove the existence of at least one weak bounded radial solution of (P ) by
applying a generalized version of the Ambrosetti–Rabinowitz Mountain Pass Theorem.
The result is contained in a joint work with Anna Maria Candela.
Solutions to problem driven by A-Laplacian operator
Calogero Vetro, calogero.vetro@unipa.it
University of Palermo, Italy
We give sufficient conditions for existence of weak solutions to quasilinear elliptic Dirichlet
problem driven by A-Laplace operator in a bounded domain Ω. The techniques, based on
a variant of the symmetric mountain pass theorem, exploit variational methods. This study
extends and complements various qualitative results for some standard cases of Laplace-type
operators. We also provide information about the asymptotic behavior of the solutions as a
suitable parameter goes to 0+.
References
[1] J. P. Gossez, Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, Non-
linear Analysis, Function Spaces and Applications, Leipzig: BSB B.G. Teubner Verlags-
gesellschaft, 1979, 59–94.
224
paper [Boscaggin, Ortega, Zhao, Periodic solutions and regularization of a Kepler problem with
time-dependent perturbation, Trans. Amer. Math. Soc. 372 (2018), 677–703]. In particular,
such solution may have a finite number of collisions with the origin in each period, while its
energy and bouncing directions behave in a regular way at each collision time.
Existence of radial bounded solutions for some quasilinear elliptic
equation in RN
Addolorata Salvatore, addolorata.salvatore@uniba.it
Università degli studi di Bari Aldo Moro, Italy
We study the quasilinear equation
(P ) − div(A(x, u)|∇u|p−2∇u) + 1 At(x, u)|∇u|p + |u|p−2u = g(x, u) in RN ,
p
with N ≥ 3, p > 1, where A(x, t), At(x, t) = ∂A (x, t) and g(x, t) are Carathéodory functions
∂t
on RN × R.
Under suitable assumptions on A(x, t) and g(x, t) the problem has a good variational struc-
ture, i.e. the weak bounded solutions of problem (P ) are critical points of the C1 functional
J (u) = 1 A(x, u)|∇u|pdx + 1 |u|pdx − G(x, u)dx,
p RN p RN
RN
on the Banach space X = W 1,p(RN ) ∩ L∞(RN ), with G(x, t) = t g(x, s)ds.
0
In order to overcome the lack of compactness, we assume that the problem has radial sym-
metry, then we look for critical points of J restricted to Xr, subspace of the radial functions in
X.
Following an approach which exploits the interaction between · X and the norm on
W 1,p(RN ), we prove the existence of at least one weak bounded radial solution of (P ) by
applying a generalized version of the Ambrosetti–Rabinowitz Mountain Pass Theorem.
The result is contained in a joint work with Anna Maria Candela.
Solutions to problem driven by A-Laplacian operator
Calogero Vetro, calogero.vetro@unipa.it
University of Palermo, Italy
We give sufficient conditions for existence of weak solutions to quasilinear elliptic Dirichlet
problem driven by A-Laplace operator in a bounded domain Ω. The techniques, based on
a variant of the symmetric mountain pass theorem, exploit variational methods. This study
extends and complements various qualitative results for some standard cases of Laplace-type
operators. We also provide information about the asymptotic behavior of the solutions as a
suitable parameter goes to 0+.
References
[1] J. P. Gossez, Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, Non-
linear Analysis, Function Spaces and Applications, Leipzig: BSB B.G. Teubner Verlags-
gesellschaft, 1979, 59–94.
224
