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: http://dx.doi.org/10.4310/MAA.2015.v22.n2.a5

Authors

Sergiu Aizicovici (Department of Mathematics, Ohio University, Athens, Ohio, U.S.A.)

Nikolaos S. Papageorgiou (Department of Mathematics, National Technical University, Zografou Campus, Athens, Greece)

Vasile Staicu (Department of Mathematics, CIDMA, University of Aveiro, Campus Universitário de Santiago, Aveiro, Portugal)

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

Full Text (PDF format)