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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.

35J10 Schrödinger operator, Schrödinger equation
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI
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