Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2019
Volume
Vol. 39
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 39 (2019)
Projects
Pages
Articles
On unique solvability of a Dirichlet problem with nonlinearity depending on the derivative
(Wydawnictwa AGH, 2019) Bełdziński, Michał; Galewski, Marek
In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems.
Some remarks on the coincidence set for the Signorini problem
(Wydawnictwa AGH, 2019) Benito Delgado, Miguel de; Díaz, Jesus Ildefonso
We study some properties of the coincidence set for the boundary Signorini problem, improving some results from previous works by the second author and collaborators. Among other new results, we show here that the convexity assumption on the domain made previously in the literature on the location of the coincidence set can be avoided under suitable alternative conditions on the data.
Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions
(Wydawnictwa AGH, 2019) Bonanno, Gabriele; D'Aguì, Giuseppina; Sciammetta, Angela
In this paper, a nonlinear differential problem involving the $p$-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results.
Infinitely many solutions for some nonlinear supercritical problems with break of symmetry
(Wydawnictwa AGH, 2019) Candela, Anna Maria; Salvatore, Addolorata
In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem $\begin{cases}-\operatorname{div}(a(x,u,\nabla u))+A_t(x,u,\nabla u) = g(x,u)+h(x)&\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega,\end{cases}$
where $\Omega \subset \mathbb{R}^N$ is an open bounded domain, $N\geq 3$, and $A(x,t,\xi)$, $g(x,t)$, $h(x)$ are given functions, with $AA_t = \frac{\partial A}{\partial t}$, $a = \nabla_{\xi} A$, such that $A(x,\cdot,\cdot)$ is even and $g(x,\cdot)$ is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if $A(x,t,\xi)$ grows fast enough with respect to $t$, then the nonlinear term related to $g(x,t)$ may have also a supercritical growth.
Existence results and a priori estimates for solutions of quasilinear problems with gradient terms
(Wydawnictwa AGH, 2019) Filippucci, Roberta; Lini, Chiara
In this paper we establish a priori estimates and then an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in $\mathbb{R}^N$ with a nonlinearity involving gradient terms. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz. In particular our existence theorem extends a result by Lorca and Ubilla in two directions, namely by considering a nonlinearity which includes in the gradient term a power of u and by removing the growth condition for the nonlinearity $f$ at $u=0$.