Page 218 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 218
TOPOLOGICAL METHODS IN DIFFERENTIAL EQUATIONS (MS-13)
Existence results of fourth order equations with perturbed two-point
boundary conditions
Alberto Cabada, alberto.cabada@usc.gal
Universidade de Santiago de Compostela, Spain
In this talk we establish the existence and multiplicity of positive solutions for a fourth-order
boundary value problem. Integral perturbations of some kind of two-point boundary condi-
tions are considered. After the construction of a Green’s function and the study of its constant
sign, it is defined a positive cone, where to apply the Krasnoselskii compression/expansion and
Leggett-Williams fixed point theorems in cones. A generalization for a higher order case is also
considered. Some particular examples are given.
About coupled gradient-type quasilinear elliptic systems with
supercritical growth
Anna Maria Candela, annamaria.candela@uniba.it
Università degli Studi di Bari Aldo Moro, Italy
The aim of this talk is pointing out some recent results on the coupled gradient–type quasilinear
elliptic system
−div(A(x, u)|∇u|p1−2∇u) + 1 Au (x, u)|∇u|p1 = Gu(x, u, v) in Ω,
p1 in Ω,
on ∂Ω,
(P ) −div(B(x, v)|∇v|p2−2∇v) + 1 Bv (x, v)|∇v|p2 = Gv(x, u, v)
p2
u = v = 0
where Ω ⊂ RN is an open bounded domain, p1, p2 > 1 and A(x, u), B(x, v) are C1–Carathéodo-
ry functions on Ω × R with partial derivatives Au(x, u), respectively Bv(x, v). Here,
(Gu(x, u, v), Gv(x, u, v)) = ∇G(x, u, v) where G(x, u, v) is a given function on Ω × R2.
Even if the coefficients A(x, u) and B(x, v) make the variational approach more difficult,
suitable hypotheses allow us to prove that the weak bounded solutions of problem (P ) are
critical points of the functional
J (u, v) = 1 A(x, u)|∇u|p1dx + 1 B(x, v)|∇v|p2dx − G(x, u, v)dx
p1 Ω p2 Ω
Ω
in the Banach space X = X1 × X2, where Xi = W01,pi(Ω) ∩ L∞(Ω) for i = 1, 2.
Unluckily, classical variational theorems cannot apply to J in X but, following an approach
which exploits the interaction between · X and the standard norm on W01,p1(Ω) × W01,p2(Ω),
the existence of critical points of J can be proved by means of a generalized Mountain Pass
Theorem.
In particular, if the coefficients A(x, u) and B(x, v) grow in the “right” way then G(x, u, v)
can have a suitable supercritical growth and if J is even then (P ) has infinitely many weak
bounded solutions.
These results are part of joint works with Caterina Sportelli and Addolorata Salvatore.
216
Existence results of fourth order equations with perturbed two-point
boundary conditions
Alberto Cabada, alberto.cabada@usc.gal
Universidade de Santiago de Compostela, Spain
In this talk we establish the existence and multiplicity of positive solutions for a fourth-order
boundary value problem. Integral perturbations of some kind of two-point boundary condi-
tions are considered. After the construction of a Green’s function and the study of its constant
sign, it is defined a positive cone, where to apply the Krasnoselskii compression/expansion and
Leggett-Williams fixed point theorems in cones. A generalization for a higher order case is also
considered. Some particular examples are given.
About coupled gradient-type quasilinear elliptic systems with
supercritical growth
Anna Maria Candela, annamaria.candela@uniba.it
Università degli Studi di Bari Aldo Moro, Italy
The aim of this talk is pointing out some recent results on the coupled gradient–type quasilinear
elliptic system
−div(A(x, u)|∇u|p1−2∇u) + 1 Au (x, u)|∇u|p1 = Gu(x, u, v) in Ω,
p1 in Ω,
on ∂Ω,
(P ) −div(B(x, v)|∇v|p2−2∇v) + 1 Bv (x, v)|∇v|p2 = Gv(x, u, v)
p2
u = v = 0
where Ω ⊂ RN is an open bounded domain, p1, p2 > 1 and A(x, u), B(x, v) are C1–Carathéodo-
ry functions on Ω × R with partial derivatives Au(x, u), respectively Bv(x, v). Here,
(Gu(x, u, v), Gv(x, u, v)) = ∇G(x, u, v) where G(x, u, v) is a given function on Ω × R2.
Even if the coefficients A(x, u) and B(x, v) make the variational approach more difficult,
suitable hypotheses allow us to prove that the weak bounded solutions of problem (P ) are
critical points of the functional
J (u, v) = 1 A(x, u)|∇u|p1dx + 1 B(x, v)|∇v|p2dx − G(x, u, v)dx
p1 Ω p2 Ω
Ω
in the Banach space X = X1 × X2, where Xi = W01,pi(Ω) ∩ L∞(Ω) for i = 1, 2.
Unluckily, classical variational theorems cannot apply to J in X but, following an approach
which exploits the interaction between · X and the standard norm on W01,p1(Ω) × W01,p2(Ω),
the existence of critical points of J can be proved by means of a generalized Mountain Pass
Theorem.
In particular, if the coefficients A(x, u) and B(x, v) grow in the “right” way then G(x, u, v)
can have a suitable supercritical growth and if J is even then (P ) has infinitely many weak
bounded solutions.
These results are part of joint works with Caterina Sportelli and Addolorata Salvatore.
216
