## Tohoku Mathematical Journal

- Tohoku Math. J. (2)
- Volume 69, Number 2 (2017), 261-286.

### Robin problems with indefinite and unbounded potential, resonant at $-\infty$, superlinear at $+\infty$

Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu

#### Abstract

We consider a semilinear Robin problem with an indefinite and unbounded potential and a reaction which exhibits asymmetric behavior as $x\rightarrow\pm\infty$. More precisely it is sublinear near $-\infty$ with possible resonance with respect to the principal eigenvalue of the negative Robin Laplacian and it is superlinear at $+\infty$. Resonance is also allowed at zero with respect to any nonprincipal eigenvalue. We prove two multiplicity results. In the first one, we obtain two nontrivial solutions and in the second, under stronger regularity conditions on the reaction, we produce three nontrivial solutions. Our work generalizes the recent one by Recova-Rumbos (Nonlin. Anal. 112 (2015), 181--198).

#### Article information

**Source**

Tohoku Math. J. (2), Volume 69, Number 2 (2017), 261-286.

**Dates**

First available in Project Euclid: 24 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tmj/1498269626

**Digital Object Identifier**

doi:10.2748/tmj/1498269626

**Mathematical Reviews number (MathSciNet)**

MR3682166

**Zentralblatt MATH identifier**

1375.35202

**Subjects**

Primary: 35J20: Variational methods for second-order elliptic equations

Secondary: 35J60: Nonlinear elliptic equations 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

**Keywords**

Indefinite and unbounded potential Robin boundary condition asymmetric reaction critical groups multiple nontrivial solutions

#### Citation

Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D. Robin problems with indefinite and unbounded potential, resonant at $-\infty$, superlinear at $+\infty$. Tohoku Math. J. (2) 69 (2017), no. 2, 261--286. doi:10.2748/tmj/1498269626. https://projecteuclid.org/euclid.tmj/1498269626