# Ground states solutions for modified fourth-order elliptic systems with steep well potential

Volume 11, Issue 3, pp 323--334 Publication Date: February 09, 2018
• 491 Views

### Authors

Liuyang Shao - School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P. R. China
Haibo Chen - School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P. R. China

### Abstract

In this paper, we study the following modified quasilinear fourth-order elliptic systems $\left\{\begin{array}{lll} &\triangle^{2}u-\triangle u+(\lambda\alpha(x)+1)u-\frac{1}{2}\triangle(u^{2})u=\frac{p}{p+q}|u|^{p-2}|v|^{q}u,~~&\mbox{in} \;~\mathbb{R}^{N}, \\ & \triangle^{2}v-\triangle v+(\lambda\beta(x)+1)v-\frac{1}{2}\triangle(v^{2})v=\frac{q}{p+q}|u|^{p}|v|^{q-2}v,~~&\mbox{in} \;~\mathbb{R}^{N},\end{array} \right.$ where $\triangle^{2}=\triangle(\triangle)$ is the biharmonic operator, $\lambda>0$, and $2<p, 2<q,$ $4<p+q<22^{\ast\ast}$, $2^{\ast\ast}=\frac{2N}{N-4} \ (N\leq5)$ $(\mbox{if}~N\leq4, 2^{\ast\ast}=\infty)$ is the critical Sobolev exponent for the embedding $W^{2,2}(\mathbb{R}^{N})\hookrightarrow L^{2^{\ast\ast}}(\mathbb{R}^{N})$. Under some appropriate assumptions on $\alpha(x)$ and $\beta(x)$, we obtain that the above problem has nontrivial ground state solutions via the variational methods. We also explore the phenomenon of concentration of solutions.

### Keywords

• Fourth-order elliptic
• variational methods
• ground state solutions
• concentration

### References

• [1] P. Alvarez-Caudevilla, E. Coloradoa, V. A. Galaktionov , Existence of solutions for a system of coupled nonlinear stationary bi-harmonic Schrödinger equations , Nonlinear Anal., 23 (2015), 78–93.

• [2] T. Bartsch, Z. Q. Wang , Existence and multiplicity results for superlinear elliptic problems on $R^N$ , Comm. Partial Differential Equations, 20 (1995), 1725–1741.

• [3] H. Brézis, E. Lieb , A relation between point convergence of functions and convergence of functionals , Proc. Amer. Math. Soc., 88 (1983), 486–490.

• [4] P. Candito, L. Li, R. Livrea, Infinitely many solutions for a perturbed nonlinear Navier boundary value problem involving the p-biharmonic, Nonlinear Anal., 75 (2012), 6360–6369.

• [5] S. Chen, J. Liu, X. Wu , Existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth-order elliptic equations on $R^N$, Appl. Math. Comput., 248 (2014), 593–608.

• [6] Y. Chen, P. J. McKenna , Traveling waves in a nonlinear suspension beam: theoretical results and numerical observations, J. Differ. Equ., 136 (1997), 325–355.

• [7] Q.-H. Choi, T. Jung , Multiplicity of solutions and source terms in a fourth order nonlinear elliptic equation , Acta Math. Sci., 19 (1999), 361–374.

• [8] Z.-Y. Deng, Y.-S. Huang , Symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents, Appl. Math. Comput., 264 (2015), 323–334.

• [9] M. Hajipour, A. Malek, High accurate NRK and MWENO scheme fornonlinear degenerate parabolic PDEs, Appl. Math. Model., 36 (2012), 4439–4451.

• [10] M. Hajipour, A. Malek , High accurate modified WENO method for the solution of Black-Scholes equation, Comput. Appl. Math., 34 (2015), 125–140.

• [11] Y.-S. Huang, X.-Q. Liu , Sign-changing solutions for p-biharmonic equations with Hardy potential in the half-space , J. Math. Anal. Appl., 444 (2016), 1417–1437.

• [12] A. C. Lazer, P. J. McKenna , Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis , SIAM Rev., 32 (1990), 537–578.

• [13] A. C. Lazer, P. J. McKenna , Global bifurcation and a theorem of Tarantello , J. Math. Anal. Appl., 181 (1994), 648–655.

• [14] H. Liu, H. Chen , Least energy nodal solution for quasilinear biharmonic equations with critical exponent in $R^N$, Appl. Math. Lett., 48 (2015), 85–90.

• [15] H. Liu, H. Chen , Ground-state solution for a class of biharmonic equations including critical exponent, Z. Angew. Math. Phys., 66 (2015), 3333–3343.

• [16] D. Lü , Multiple solutions for a class of biharmonic elliptic systems with Sobolev critical exponent,, Nonlinear Anal., 74 (2011), 6371–6382.

• [17] P. J. McKenna, W. Walter , Traveling waves in a suspension bridge , SIAM J. Appl. Math., 50 (1990), 703–715.

• [18] M. Willem , Minimax Theorems, Birkhäuser, Berlin (1996)

• [19] M. T. O. Pimenta, S. H. M. Soares , Existence and concentration of solutions for a class of biharmonic equations, J. Math. Anal. Appl., 390 (2012), 274–289.

• [20] L.-Y. Shao, H. Chen, Multiple solutions for Schrödinger-Poisson systems with sign-changing potential and critical nonlinearity , Electron. J. Differential Equations, 2016 (2016), 8 pages.

• [21] L.-Y. Shao, H. Chen , Existence and concentration result for a quasilinear Schrödinger equation with critical growth, Z. Angew. Math. Phys., 2017 (2017 ), 16 pages.

• [22] H. Shi, H. Chen , Existence and multiplicity of solutions for a class of generalized quasilinear Schrödinger equations, J. Math. Anal. Appl., 452 (2017), 578–594.

• [23] J. Sun, J. Chu, T.-F. Wu , Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian , J. Differential Equations, 262 (2017), 945–977.

• [24] A. Szulkin, T. Weth , Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802– 3822.

• [25] N. S. Trudinger , On Harnack type inequalities and their application to quasilinear elliptic equations , Comm. Pure Appl. Math., 20 (1967), 721–747.

• [26] F. Wang, M. Avci, Y. An, Existence of solutions for fourth order elliptic equations of Kirchhoff type, J. Math. Anal. Appl., 409 (2014), 140–146.

• [27] Y.-J. Wang, Y.-T. Shen , Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations, 246 (2009), 3109–3125.

• [28] H. Xiong, Y.-T. Shen , Nonlinear biharmonic equations with critical potential , Acta Math. Sci., 21 (2005), 1285–1294.

• [29] J. Zhang, S. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems , Nonlinear Anal., 60 (2005), 221–230.

• [30] W. Zhang, X. Tang, J. Zhang , Infinitely many solutions for fourth-order elliptic equations with general potentials , J. Math. Anal. Appl., 407 (2013), 359–368.

• [31] J. Zhang, X. Tang, W. Zhang , Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential , J. Math. Anal. Appl., 420 (2014), 1762–1775.

• [32] W. Zhang, X. Tang, J. Zhang , Infinitely many solutions for fourth-order elliptic equations with sign-changing potential , Taiwanese J. Math., 18 (2014), 645–659.

• [33] W. Zhang, X. Tang, J. Zhang , Existence and concentration of solutions for sublinear fourth-order elliptic equations, Electron. J. Differential Equations, 2015 (2015 ), 9 pages.

• [34] W. Zou, M. Schechter , Critical Point Theory and its Applications, Springer, New York (2006)