Topological Methods in Nonlinear Analysis

Existence of positive ground solutions for biharmonic equations via Pohožaev-Nehari manifold

Liping Xu and Haibo Chen

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Abstract

We investigate the following nonlinear biharmonic equations with pure power nonlinearities: \begin{equation*} \begin{cases} \triangle^2u-\triangle u+V(x)u= u^{p-1}u & \text{in } \mathbb{R}^N,\\ u>0 &\text{for } u\in H^2(\mathbb{R}^N), \end{cases} \end{equation*} where $2 < p< 2^*={2N}/({N-4})$. Under some suitable assumptions on $V(x)$, we obtain the existence of ground state solutions. The proof relies on the Pohožaev-Nehari manifold, the monotonic trick and the global compactness lemma, which is possibly different to other papers on this problem. Some recent results are extended.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 2 (2018), 541-560.

Dates
First available in Project Euclid: 6 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1541473235

Digital Object Identifier
doi:10.12775/TMNA.2018.015

Mathematical Reviews number (MathSciNet)
MR3915650

Zentralblatt MATH identifier
07051679

Citation

Xu, Liping; Chen, Haibo. Existence of positive ground solutions for biharmonic equations via Pohožaev-Nehari manifold. Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 541--560. doi:10.12775/TMNA.2018.015. https://projecteuclid.org/euclid.tmna/1541473235


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