Multiple Solutions for Nonlinear Dirichlet Problems with Concave Terms

Authors

  • Leszek Gasiński
  • Nikolaos S. Papageorgiou

DOI:

https://doi.org/10.7146/math.scand.a-15570

Abstract

We consider a nonlinear parametric Dirichlet problem with parameter λ>0, driven by the p-Laplacian and with a concave term λ|u|q2u, 1<q<p and a Carathéodory perturbation f(z,ζ) which is asymptotically (p1)-linear at infinity. Using variational methods combined with Morse theory and truncation techniques, we show that there is a critical value λ>0 of the parameter such that for λ(0,λ) the problem has five nontrivial smooth solutions, four of constant sign (two positive and two negative) and the fifth nodal. In the semilinear case (p=2), we show that there is a sixth nontrivial smooth solution, but we cannot provide information about its sign. Finally for the critical case λ=λ, we show that the nonlinear problem (p2) still has two nontrivial constant sign smooth solutions and the semilinear problem (p=2) has three nontrivial smooth solutions, two of which have constant sign.

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Published

2013-12-01

How to Cite

Gasiński, L., & Papageorgiou, N. S. (2013). Multiple Solutions for Nonlinear Dirichlet Problems with Concave Terms. MATHEMATICA SCANDINAVICA, 113(2), 206–247. https://doi.org/10.7146/math.scand.a-15570

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Articles