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# 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

#### 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