1 - 5 of 5
Number of results to display per page
Search Results
2. Impulsive boundary value problems for $p(t)$-Laplacian's via critical point theory
- Creator:
- Galewski, Marek and O´Regan, Donal
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- $p( t)$-Laplacian, impulsive condition, critical point, variational method, and Dirichlet problem
- Language:
- English
- Description:
- In this paper we investigate the existence of solutions to impulsive problems with a $p(t)$-Laplacian and Dirichlet boundary value conditions. We introduce two types of solutions, namely a weak and a classical one which coincide because of the fundamental lemma of the calculus of variations. Firstly we investigate the existence of solution to the linear problem, i.e. a problem with a fixed rigth hand side. Then we use a direct variational method and next a mountain pass approach in order to get the existence of at least one weak solution to the nonlinear problem.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Nonlinear elliptic differential equations with multivalued nonlinearities
- Creator:
- Fiacca, Antonella, Matzakos, Nikolas, Papgeorgiou, Nikolas S., and Servadei, Raffaella
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- upper solution, lower solution, order interval, truncation function, pseudomonotone operator, coercive operator, extremal solution, Yosida approximation, nonsmooth Palais-Smale condition, critical point, and eigenvalue problem
- Language:
- English
- Description:
- In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all $\mathbb{R}$. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of $\mathbb{R}$. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally, in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4. On a class of nonlinear problems involving a $p(x)$-Laplace type operator
- Creator:
- Mihăilescu, Mihai
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- $p(x)$-Laplace operator, generalized Lebesgue-Sobolev space, critical point, weak solution, and electrorheological fluid
- Language:
- English
- Description:
- We study the boundary value problem $-{\mathrm div}((|\nabla u|^{p_1(x) -2}+|\nabla u|^{p_2(x)-2})\nabla u)=f(x,u)$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain in ${\mathbb{R}} ^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x)=\max \lbrace p_1(x),p_2(x)\rbrace $ for any $x\in \overline{\Omega }$ or $m(x)<q(x)< \frac{N\cdot m(x)}{(N-m(x))}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda $ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ${\mathbb{Z}} _2$-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
5. Quasilinear elliptic problems with multivalued terms
- Creator:
- Halidias, Nikolaos and Papageorgiou, Nikolaos S.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- subdifferential, critical point, Palais-Smale condition, Mountain Pass Theorem, Saddle Point Theorem, multivalued term, Dirichlet problem, Neumann problem, p-Laplacian, and Rayleigh quotient
- Language:
- English
- Description:
- We study the quasilinear elliptic problem with multivalued terms.We consider the Dirichlet problem with a multivalued term appearing in the equation and a problem of Neumann type with a multivalued term appearing in the boundary condition. Our approach is based on Szulkin’s critical point theory for lower semicontinuous energy functionals.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public