Multiple positive solutions for quasilinear elliptic equations of \(p(x)\)-Laplacian type with sign-changing nonlinearity. (English) Zbl 1316.35138

Ho, Ky; Kim, Chan-Gyun; Sim, Inbo

Electron. J. Differ. Equ. 2014, Paper No. 237, 12 p. (2014).

There are established sufficient conditions for the existence of multiple positive solutions to nonautonomous quasilinear elliptic equations with \(p(x)\)-Laplacian and sign-changing nonlinearity \[ -\Delta_{p(x)}u=\lambda f(x,u)\quad x\in\Omega, \qquad u(x)=0\quad \partial \Omega. \] For solving the Dirichlet problem the authors use variational and topological methods. The nonexistence of positive solutions is also studied.

Reviewer: Lubomira Softova (Aversa)

MSC:

35J92	Quasilinear elliptic equations with \(p\)-Laplacian
35J35	Variational methods for higher-order elliptic equations
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J40	Boundary value problems for higher-order elliptic equations
35B09	Positive solutions to PDEs

Keywords:

\(p(x)\)-Laplacian; variable exponent; sign-changing nonlinearity; positive solutions; multiplicity