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On the existence of multiple positive solutions for a certain class of elliptic problems

100%
Orpel A.
Banach Center Publications
|
2004
|
tom 66
|
nr 1
251-262
EN
We investigate the existence of solutions for the Dirichlet problem including the generalized balance of a membrane equation. We present a duality theory and variational principle for this problem. As one of the consequences of the duality we obtain some numerical results which give a measure of a duality gap between the primal and dual functional for approximate solutions.
2
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Dirichlet problems without convexity assumption

100%
Orpel A.
Annales Polonici Mathematici
|
2005
|
tom 85
|
nr 3
193-210
EN
We deal with the existence of solutions of the Dirichlet problem for sublinear and superlinear partial differential inclusions considered as generalizations of the Euler-Lagrange equation for a certain integral functional without convexity assumption. We develop a duality theory and variational principles for this problem. As a consequence of the duality theory we give a numerical version of the variational principles which enables approximation of the solution for our problem.
3
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The Dirichlet problem with sublinear nonlinearities

100%
Orpel A.
Annales Polonici Mathematici
|
2002
|
tom 78
|
nr 2
131-140
EN
We investigate the existence of solutions of the Dirichlet problem for the differential inclusion $0 ∈ Δ x(y) + ∂_{x}G(y,x(y))$ for a.e. y ∈ Ω, which is a generalized Euler-Lagrange equation for the functional $J(x) = ∫_{Ω}{ 1/2|∇x(y)|² - G(y,x(y))}dy$. We develop a duality theory and formulate the variational principle for this problem. As a consequence of duality, we derive the variational principle for minimizing sequences of J. We consider the case when G is subquadratic at infinity.
4
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Semilinear elliptic problems in unbounded domains

100%
Orpel A.
Applicationes Mathematicae
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2006
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tom 33
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nr 3-4
345-354
EN
We investigate the existence of positive solutions and their continuous dependence on functional parameters for a semilinear Dirichlet problem. We discuss the case when the domain is unbounded and the nonlinearity is smooth and convex on a certain interval only.
5
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Systems of singular BVPs - existence of solutions and their properties

100%
Orpel A.
Banach Center Publications
|
2014
|
tom 101
|
nr 1
133-143
EN
We discuss the existence and properties of solutions for systems of singular second-order ODEs in both sublinear and superlinear cases. Our approach is based on the variational method enriched by some topological ideas. We also investigate the continuous dependence of solutions on functional parameters.
6
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Continuous dependence on function parameters for superlinear Dirichlet problems

100%
Orpel A.
Colloquium Mathematicum
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2005
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tom 103
|
nr 1
131-148
EN
We discuss the existence of solutions for a certain generalization of the membrane equation and their continuous dependence on function parameters. We apply variational methods and consider the PDE as the Euler-Lagrange equation for a certain integral functional, which is not necessarily convex and coercive. As a consequence of the duality theory we obtain variational principles for our problem and some numerical results concerning approximation of solutions.
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