Multiple Solutions for Nonlinear Dirichlet Problems with Concave Terms
DOI:
https://doi.org/10.7146/math.scand.a-15570Abstract
We consider a nonlinear parametric Dirichlet problem with parameter λ>0, driven by the p-Laplacian and with a concave term λ|u|q−2u, 1<q<p and a Carathéodory perturbation f(z,ζ) which is asymptotically (p−1)-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 (p≠2) 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.Downloads
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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