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

Liping Xu, Haibo Chen

DOI: http://dx.doi.org/10.12775/TMNA.2018.015

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.

Keywords


Biharmonic equations; ground state solutions; concentration-compactness principle; Pohožaev manifold; Nehari manifold

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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 fourthorder elliptic equation, Nonlinear Anal. 68 (2008), 3325–3331.

T. Bartsch and Z. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN , 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 RN , 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 R3 , Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 787–809.

A. Lazer and P. McKenna, Large-amplitude periodic oscillations in suspension bridges: 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 R3 , 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 RN , 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 fourthorder elliptic equations in RN , 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 RN , 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.


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