Multiplicity theorems for resonant and superlinear nonhomogeneous elliptic equations. (English) Zbl 1368.35088

Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D.

Topol. Methods Nonlinear Anal. 48, No. 1, 283-320 (2016).

The interesting paper deals with multiplicity issues for the solutions to the Dirichlet problem \[ \begin{cases} -\Delta_pu(z)-\Delta u(z)=f(z,u(z)) & \text{ in }\;\Omega,\\ u=0 & \text{ on }\;\partial\Omega, \end{cases} \] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with \(C^2\)-smooth boundary and \(\Delta_p\) is the \(p\)-Laplace operator with \(p>2.\)
The authors consider two distinct cases of the above problem.
In the first one, the reaction term \(f(z,\cdot)\) is \((p-1)\)-linear near \(\pm\infty\) and resonant with respect to a nonprincipal variational eigenvalue of \(\big(-\Delta_{p},W_{0}^{1,p}(\Omega)\big).\) A multiplicity theorem is derived in this case that produces three nontrivial solutions.
In the second case, the reaction term \(f(z,\cdot)\) is \((p-1)\)-superlinear but it does not satisfy the Ambrosetti-Rabinowitz condition. The authors obtain two multiplicity results now. In the first one six nontrivial solutions are produced all with sign information, while in the second theorem the existence of five nontrivial solutions is given. The approach employed combines variational methods with Morse theory, truncation methods, and comparison techniques.

Reviewer: Dian K. Palagachev (Bari)

MSC:

35J20	Variational methods for second-order elliptic equations
35J60	Nonlinear elliptic equations
35J92	Quasilinear elliptic equations with \(p\)-Laplacian

Keywords:

\(p\)-Laplacian; Dirichlet problem