Non-Nehari manifold method for a semilinear Schrödinger equation with critical Sobolev exponent

Volume 9, Issue 5, pp 3018--3030 http://dx.doi.org/10.22436/jnsa.009.05.94

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Authors

Huxiao Luo - School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, P. R. China. Xianhua Tang - School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, P. R. China. Jianhua Chen - School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, P. R. China. Jian Zhang - School of Mathematics and Statistics, Hunan University of Commerce, Changsha, Hunan, 410205, P. R. China.

Abstract

We consider the semilinear Schrödinger equation \[ \begin{cases} -\Delta u + V (x)u = K(x)|u|^{2^*-2}u + f(x; u);\, x\in R^N,\\ u \in H^1(R^N), \end{cases} \] where \(N \geq 4, 2^* := 2N/(N - 2)\) is the critical Sobolev exponent, V;K; f is 1-periodic in \(x_j\) for \(j = 1; ... ;N, f(x; u)\) is subcritical growth. We develop a direct approach to find ground state solutions of Nehari-Pankov type for the above problem. The main idea is to find a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold by using the diagonal method.

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Keywords

• Schrödinger equation
• ground state solutions of Nehari-Pankov type
• critical Sobolev exponent
• non-Nehari-manifold method.

MSC

•  35Q55
•  35J60

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