Ground state solutions for fractional Schrödinger equation with variable potential and Berestycki–Lions type nonlinearity

Abstract

We consider the following nonlinear fractional Schrödinger equation: $$ (-\triangle )^{s} u+V(x)u=g(u) \quad \text{in } \mathbb{R} ^{N}, $$ ( − △ ) s u + V ( x ) u = g ( u ) in R N , where $s\in (0, 1)$ s ∈ ( 0 , 1 ) , $N>2s$ N > 2 s , $V(x)$ V ( x ) is differentiable, and $g\in C ^{1}(\mathbb{R} , \mathbb{R} )$ g ∈ C 1 ( R , R ) . By exploiting the minimization method with a constraint over Pohoz̆aev manifold, we obtain the existence of ground state solutions. With the help of Pohoz̆aev identity we also process the existence of the least energy solutions for the above equation. Our results improve the existing study on this nonlocal problem with Berestycki–Lions type nonlinearity to the one that does not need the oddness assumption.

Asymptotic behaviour of the least energy solutions of fractional semilinear Neumann problem

“…In [7], authors have obtained a weak solution w ∈ H s (R n ) of (1.8) with least energy among all other solutions. In particular, they have proved the following result.…”

“…Proof of Theorem 1.2. The proof of the first inequality of Theorem 1.2 is fairly standard and simple, which can be seen in the literature, for instance, see Theorem 1.1 [7]. Since it is short, for the sake of completeness, we include it here.…”

“…where 1 < p < n+2s n−2s , n > max {1, 2s} , 0 < s < 1 is thoroughly studied, see for instance [7,15,20,21] and the references therein.…”

Ground state solutions of Nehari–Pohožaev type for a kind of nonlinear problem with general nonlinearity and nonlocal convolution term

“…There is also other work about ground state solutions for (1.2); we refer to [20,21]. Motivated by the above work and [22][23][24][25][26][27][28][29][30], in the present paper, we shall extend the results concerning the existence of ground state solutions for (1.2) in [23] to (1.1). Compared with (1.2), it is more difficult to deal with (1.1) for the reason that q ∈ [1 + α 3 , 3 + α).…”

Nonlinear conservation laws for the Schrödinger boundary value problems of second order

“…Bound state solutions of sublinear Schrödinger equations with lack of compactness were studied in [8]. The existence of ground state solutions for nonlinear fractional Schrödinger equation was obtained in [11] by applying the minimization method with a constraint over a Pohožaev manifold. A diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator was considered in [48].…”