Contents Online
Methods and Applications of Analysis
Volume 22 (2015)
Number 2
Constant sign and nodal solutions for nonlinear elliptic equations with combined nonlinearities
Pages: 221 – 248
DOI: https://dx.doi.org/10.4310/MAA.2015.v22.n2.a5
Authors
Abstract
We study a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is “concave” (i.e., $(p-1)-\mathrm{sublinear}$) near zero and “convex” (i.e., $(p-1)-\mathrm{sublinear}$) near $\pm \infty$. Using variational methods combined with truncation and comparison techniques, we show that for all small values of the parameter $\lambda \gt 0$, the problem has at least five nontrivial smooth solutions (four of constant sign and the fifth nodal). In the Hilbert space case ($p = 2$), using Morse theory, we produce a sixth nontrivial smooth solution but we do not determine its sign.
Keywords
nodal solutions, nonlinear regularity, local minimizer, extremal solutions, critical groups, superlinear reaction, concave term
2010 Mathematics Subject Classification
35J20, 35J60, 35J92, 58E05
Published 1 June 2015