## Topological Methods in Nonlinear Analysis

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

#### 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

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

#### References

• V. Alexiades, A. Elcrat and P. Schaefer, Existence theorems for some nonlinear fourth-order elliptic boundary value problems, Nonlinear Anal. 4 (1980), no. 4, 805–813.
• C. Alves, J.M. do Ó and O. Miyagaki, On a class of singular biharmonic problems involving critical exponents, J. Math. Anal. Appl. 277 (2003), 12–26.
• Y. An and R. Liu, Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation, Nonlinear Anal. 68 (2008), 3325–3331.
• T. Bartsch and Z. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\R^N$, Comm. Partial Differential Equations 20 (1995), 1725–1741.
• P. Carrião, R. Demarque and O. Miyagaki, Nonlinear biharmonic problems with singular potentials, Commun.Pure Appl. Anal. 13 (2014), 2141–2154.
• J. Chabrowski and J.M. do Ó, On some fourth-order semilinear elliptic problems in $\R^N$, Nonlinear Anal. 49 (2002), 861–884.
• G. Che and H. Chen, Nontrivial solutions and least energy nodal solutions for a class of fourth-order elliptic equations, J. Appl. Math. Comput., DOI 10.1007/s12190-015-0956-9.
• Y. Chen and P. McKenna, Traveling waves in a nonlinear suspension beam: theoretical results and numerical observations, J. Differential Equations 135 (1997), 325–355.
• Y. Deng, Q. Gao and L. Jin, On the existence of nontrivial solutions for p-harmonic equations on unbounded domain, Nonlinear Anal. 69 (2008), 4713–4731.
• A. Harrabi, Fourth-order elliptic equations, Adv. Nonlinear Stud. 14 (2014), 593–604.
• S. Hu and L. Wang, Existence of nontrivial solutions for fourth-order asymptotically linear elliptic equations, Nonlinear Anal. 94 (2014), 120–132.
• L. Jeanjean, On the existence of bounded Palais–Smale sequence and application to a Landesman–Lazer type problem set on $\R^3$, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 787–809.
• A. Lazer and P. McKenna, Large-amplitude periodic oscillations in suspension bridges\rom: some new connections with nonlinear analysis, SIAM Rev. 32 (1990), 537–578.
• G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\R^3$, J. Differential Equations 257 (2014), 566–600.
• T. Li, J. Sun and T. Wu, Existence of homoclinic solutions for a fourth order differential equation with a parameter, Appl. Math. Comput. 251 (2015), 499–506.
• S. Liang, J. Zhang and Y. Luo, Existence of solutions for a class of biharmonic equations with critical nonlinearity in $\R^N$, Rev. R. Acad. Cien. Serie A. Mat. 110 (2016), 681–693.
• P. Lions, The concentration compactness principle in the calculus of variations: The locally compact case, Part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145; The concentration compactness principle in the calculus of variations: The locally compact case, Part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1984), 223–283.
• J. Liu, S. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\R^N$, J. Math. Anal. Appl. 395 (2012), 608–615.
• Z. Liu and S. Guo, On ground state solutions for the Schrödinger–Poisson equations with critical growth, J. Math. Anal. Appl. 412 (2014), 435-448.
• Z. Liu and S. Guo, Existence of positive ground state solutions for Kirchhoff type equation with general critical growth, Topol. Methods Nonlinear Anal. 43 (2014), 1–99.
• P. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math. 50 (1990), 703–715.
• D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), 655–674.
• J. Sun and T. Wu, Two homoclinic solutions for a nonperiodic fourth order differential equation with a perturbation, J. Math. Anal. Appl. 413 (2014), 622–632.
• Y. Wang and Y. Shen, Infinitely many sign-changing solutions for a class of biharmonic equation without symmetry, Nonlinear Anal. 71 (2009), 967–977.
• M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
• M. Yang and Z. Shen, Infinitely many solutions for a class of fourth order elliptic equations in $\R^N$, Acta Math. Sin. (Engl. Ser.) 24 (2008), 1269–1278.
• Y. Ye and C. Tang, Infinitely many solutions for fourth-order elliptic equations, J. Math. Anal. Appl. 394 (2012), 841–854.
• Y. Yin and X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal.Appl. 375 (2011), 699–705.
• W. Zhang, X. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with general potentials, J. Math. Anal. Appl. 407 (2013), 359–368.
• L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger–Poisson equations, J. Math. Anal. Appl. 346 (2008), 155–169.
• J. Zhou and X. Wu, Sign-changing solutions for some fourth-order nonlinear elliptic problems, J. Math. Anal. Appl. 342 (2008), 542–558.