Tohoku Mathematical Journal

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

Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu

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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


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