Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the p-Laplacian

Giovanni Anello; Giuseppe Cordaro

doi:10.1017/S030821050000175X

Abstract

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In this paper we present a result of existence of infinitely many arbitrarily small positive solutions to the following Dirichlet problem involving the p-Laplacian, where Ω ∈ RN is a bounded open set with sufficiently smooth boundary ∂Ω, p > 1, λ > 0, and f: Ω × R → R is a Carathéodory function satisfying the following condition: there exists t̄ > 0 such that Precisely, our result ensures the existence of a sequence of a.e. positive weak solutions to the above problem, converging to zero in L∞(Ω).

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Bonanno, Gabriele and Livrea, Roberto 2003. Multiplicity theorems for the Dirichlet problem involving the p-Laplacian. Nonlinear Analysis: Theory, Methods & Applications, Vol. 54, Issue. 1, p. 1.

Obersnel, Franco and Omari, Pierpaolo 2006. Positive solutions of elliptic problems with locally oscillating nonlinearities. Journal of Mathematical Analysis and Applications, Vol. 323, Issue. 2, p. 913.

Motreanu, D. Motreanu, V. V. and Papageorgiou, N. S. 2007. A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations. manuscripta mathematica, Vol. 124, Issue. 4, p. 507.

Afrouzi, G.A. and Heidarkhani, S. 2007. Three solutions for a Dirichlet boundary value problem involving the -Laplacian. Nonlinear Analysis: Theory, Methods & Applications, Vol. 66, Issue. 10, p. 2281.

Kristály, Alexandru Moroşanu, Gheorghe and Tersian, Stepan 2007. Quasilinear elliptic problems in RN involving oscillatory nonlinearities. Journal of Differential Equations, Vol. 235, Issue. 2, p. 366.

Anello, Giovanni and Cordaro, Giuseppe 2007. Perturbation from Dirichlet problem involving oscillating nonlinearities. Journal of Differential Equations, Vol. 234, Issue. 1, p. 80.

Filippakis, Michael E. and Papageorgiou, Nikolaos S. 2008. Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian. Journal of Differential Equations, Vol. 245, Issue. 7, p. 1883.

Afrouzi, G.A. and Heidarkhani, S. 2008. Three solutions for a quasilinear boundary value problem. Nonlinear Analysis: Theory, Methods & Applications, Vol. 69, Issue. 10, p. 3330.

He, Xiaoming and Zou, Wenming 2009. Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Analysis: Theory, Methods & Applications, Vol. 70, Issue. 3, p. 1407.

Wang, Libo and Ge, Weigao 2009. Infinitely many positive solutions of a singular dirichlet problem involving the p-Laplacian. Communications in Nonlinear Science and Numerical Simulation, Vol. 14, Issue. 11, p. 3784.

Afrouzi, G.A. and Heidarkhani, S. 2009. Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the (p1,…,pn) -Laplacian. Nonlinear Analysis: Theory, Methods & Applications, Vol. 70, Issue. 1, p. 135.

Heidarkhani, Shapour and Tian, Yu 2010. Multiplicity results for a class of gradient systems depending on two parameters. Nonlinear Analysis: Theory, Methods & Applications, Vol. 73, Issue. 2, p. 547.

Wang, Libo Ge, Weigao and Pei, Minghe 2010. Infinitely many solutions of a second-order p-Laplacian problem with impulsive condition. Applications of Mathematics, Vol. 55, Issue. 5, p. 405.

Kristály, Alexandru 2010. On singular elliptic equations involving oscillatory terms. Nonlinear Analysis: Theory, Methods & Applications, Vol. 72, Issue. 3-4, p. 1561.

Afrouzi, G. A. and Heidarkhani, S. 2010. Multiplicity results for a two-point boundary value double eigenvalue problem. Ricerche di Matematica, Vol. 59, Issue. 1, p. 39.

Afrouzi, G.A. and Heidarkhani, S. 2010. Multiplicity theorems for a class of Dirichlet quasilinear elliptic systems involving the -Laplacian. Nonlinear Analysis: Theory, Methods & Applications, Vol. 73, Issue. 8, p. 2594.

Wang, Libo and Pei, Minghe 2011. Infinitely many solutions of a Sturm-Liouville system with impulses. Journal of Applied Mathematics and Computing, Vol. 35, Issue. 1-2, p. 577.

D’Aguì, Giuseppina and Sciammetta, Angela 2012. Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions. Nonlinear Analysis: Theory, Methods & Applications, Vol. 75, Issue. 14, p. 5612.

Han, Qi 2015. Infinitely many positive harmonic functions with an oscillating boundary condition on exterior regions. Complex Variables and Elliptic Equations, Vol. 60, Issue. 8, p. 1106.

Faraci, Francesca and Farkas, Csaba 2016. New Conditions for the Existence of Infinitely Many Solutions for a Quasi-Linear Problem. Proceedings of the Edinburgh Mathematical Society, Vol. 59, Issue. 3, p. 655.

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Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the p-Laplacian

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