In this paper we establish the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains. The p-polyharmonic operators \Delta^L_p were recently introduced by Colasuonno and Pucci in 2011 for all orders L and independently in the same volume of the journal by Lubyshev only for L even. In Section 4 the results are then extended to non-degenerate p(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given by Colasuonno, Pucci and Varga in 2012. Several useful properties of the underlying functional solution space [W^{L,p}_0(\Omega)]^d, endowed with the natural norm arising from the variational structure of the problem, are also proved both in the homogeneous case p=Const. and in the non-homogeneous case p=p(x). In the latter some sufficient conditions on the variable exponent p are given to prove the positivity of the the first eigenvalue of the p(x)-polyharmonic operator \Delta^L_{p(x)}.

On the existence of stationary solutions for higher order p-Kirchhoff problems / Autuori, Giuseppina; F., Colasuonno; Pucci, Patrizia. - In: COMMUNICATIONS IN CONTEMPORARY MATHEMATICS. - ISSN 0219-1997. - STAMPA. - 16:5(2014). [10.1142/S0219199714500023]

On the existence of stationary solutions for higher order p-Kirchhoff problems

AUTUORI, GIUSEPPINA;
2014-01-01

Abstract

In this paper we establish the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains. The p-polyharmonic operators \Delta^L_p were recently introduced by Colasuonno and Pucci in 2011 for all orders L and independently in the same volume of the journal by Lubyshev only for L even. In Section 4 the results are then extended to non-degenerate p(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given by Colasuonno, Pucci and Varga in 2012. Several useful properties of the underlying functional solution space [W^{L,p}_0(\Omega)]^d, endowed with the natural norm arising from the variational structure of the problem, are also proved both in the homogeneous case p=Const. and in the non-homogeneous case p=p(x). In the latter some sufficient conditions on the variable exponent p are given to prove the positivity of the the first eigenvalue of the p(x)-polyharmonic operator \Delta^L_{p(x)}.
2014
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/301570
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