## Berestycki-Lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials.(English)Zbl 1422.35023

Summary: This paper is dedicated to studying the nonlinear Schrödinger equations of the form $\begin{cases} -\Delta u+V(x)u=f(u), \quad x\in \mathbb{R}^N, \\ u\in H^1(\mathbb{R}^N), \end{cases}$ where $$V \in\mathcal{C}^{1}(\mathbb{R}^{N}, [0, \infty))$$ satisfies some weak assumptions, and $$f\in \mathcal{C}(\mathbb{R}, \mathbb{R})$$ satisfies the general Berestycki-Lions assumptions. By introducing some new tricks, we prove that the above problem admits a ground state solution of Pohožaev type and a least energy solution. These results generalize and improve some ones in [L. Jeanjean and K. Tanaka, Indiana Univ. Math. J. 54, No. 2, 443–464 (2005; Zbl 1143.35321); Proc. Am. Math. Soc. 131, No. 8, 2399–2408 (2003; Zbl 1094.35049); H. Berestycki and P.-L. Lions, Arch. Ration. Mech. Anal. 82, 313–345 (1983; Zbl 0533.35029)] and some other related literature. In particular, our assumptions are “almost” necessary when $$V(x) \equiv V_{\infty} > 0$$, moreover, our approach could be useful for the study of other problems where radial symmetry of bounded sequence either fails or is not readily available, or where the ground state solutions of the problem at infinity are not sign definite.

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 35J65 Nonlinear boundary value problems for linear elliptic equations

### Citations:

Zbl 1143.35321; Zbl 1094.35049; Zbl 0533.35029
Full Text:

### References:

 [1] H. Berestycki, P. L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rational Mech. Anal. 82, (1983), 313-345. · Zbl 0533.35029 [2] S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Disc. Contin. Dyn. Syst. A38, (2018), 2333-2348. · Zbl 1398.35026 [3] S. T. Chen and X. H. Tang, Ground state solutions for generalized quasilinear Schrödinger equations with variable potentials and Berestycki-Lions nonlinearities, J. Math. Phys. 59, (2018), no. 081508, 1-18 · Zbl 1395.35171 [4] S. T. Chen, B. L. Zhang, X. H. Tang, Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity, Adv. Nonlinear Anal. (2018), [5] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman- Lazer-type problem set on ℝN, Proc. Roy. Soc. Edinburgh Sect. A129, (1999), 787-809. · Zbl 0935.35044 [6] L. Jeanjean, K. Tanka, A remark on least energy solutions in ℝN, Proc. Amer. Math. Soc. 131, (2003), 2399-2408. · Zbl 1094.35049 [7] L. Jeanjean, K. Tanka, A positive solution for a nonlinear Schrödinger equation on ℝN, Indiana Univ. Math. J. 54, (2005), 443-464. [8] L. Jeanjean and J. F. Toland, Bounded Palais-Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. I Math. 327, (1998), 23-28. · Zbl 0996.47052 [9] S. I. Pohožaev, Eigenfunctions of the equation au + 2f(u) = 0, Sov. Math. Doklady5, (1965), 1408-1411. · Zbl 0141.30202 [10] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43, (1992), 270-291. · Zbl 0763.35087 [11] Y. H. Sato, M. Shibata, Existence of a positive solution for nonlinear Schrödinger equations with general nonlinearity, Adv. Nonlinear Anal. 3, (2014), 55-67. [12] J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290, (1985), 701-710. · Zbl 0617.35072 [13] M. Struwe, Variational methods?, Results in Mathematics and Related Areas, 3, Springer-Verlag, Berlin, 1996. [14] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations56, (2017), no. 4, 1-25. · Zbl 1376.35056 [15] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev type for Schrödinger-Poisson problems with general potentials, Disc. Contin. Dyn. Syst. A37, (2017), 4973-5002. · Zbl 1371.35051 [16] X. H. Tang and X. Y. Lin, Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems, Sci. China Math62, (2019), . [17] X. H. Tang, X. Lin, J. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ, (2018), 1-15. . · Zbl 1414.35062 [18] X. H. Tang, S. T. Chen, Singularly perturbed choquard equations with nonlinearity satisfying Berestycki-Lions assumptions, Adv. Nonlinear Anal., [19] A. Vincenzo, Zero mass case for a fractional Berestycki-Lions-type problem, Adv. Nonlinear Anal. 7, (2018), 365-374. · Zbl 1394.35542 [20] M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston Inc., Boston, MA, 1996.
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