Displaying similar documents to “Non-trivial solutions for a two-point boundary value problem”

New variational principle and duality for an abstract semilinear Dirichlet problem

Marek Galewski (2003)

Annales Polonici Mathematici

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A new variational principle and duality for the problem Lu = ∇G(u) are provided, where L is a positive definite and selfadjoint operator and ∇G is a continuous gradient mapping such that G satisfies superquadratic growth conditions. The results obtained may be applied to Dirichlet problems for both ordinary and partial differential equations.

Dirichlet problems without convexity assumption

Aleksandra Orpel (2005)

Annales Polonici Mathematici

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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. ...

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

Aleksandra Orpel (2004)

Banach Center Publications

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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.

Continuous dependence on function parameters for superlinear Dirichlet problems

Aleksandra Orpel (2005)

Colloquium Mathematicae

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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.

Triple solutions for a Dirichlet boundary value problem involving a perturbed discretep(k)-Laplacian operator

Mohsen Khaleghi Moghadam, Johnny Henderson (2017)

Open Mathematics

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Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p(k)-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.

Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket

Gabriele Bonanno, Giovanni Molica Bisci, Vicenţiu Rădulescu (2012)

ESAIM: Control, Optimisation and Calculus of Variations

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Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.

Existence and stability of solutions for semilinear Dirichlet problems

Marek Galewski (2006)

Annales Polonici Mathematici

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We provide existence and stability results for semilinear Dirichlet problems with nonlinearities satisfying some general local growth conditions. We derive a general abstract result which we then apply to prove the existence of solutions, their stability and continuous dependence on parameters for a sixth order ODE with Dirichlet type boundary data.

Some Developments on Dirichlet Problems with Discontinuous Coefficients

Lucio Boccardo (2009)

Bollettino dell'Unione Matematica Italiana

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This paper, dedicated to the memory of Guido Stampacchia in the thirtieth anniversary of his death, starts from his lectures on Dirichlet problems of forty years ago. As Sergei Prokofiev named his first symphony the "Classical", since it was written in the style that Joseph Haydn would have used if he had been alive at the time, this paper strongly follows the one by Guido Stampacchia about elliptic equations with discontinuous coefficients ([8]).