Abstract
We consider a class of fractional logarithmic Schrödinger equation in bounded domains. First, by means of the constraint variational method, quantitative deformation lemma and some new inequalities, the positive ground state solutions and ground state sign-changing solutions are obtained. These inequalities are derived from the special properties of fractional logarithmic equations and are critical for us to obtain our main results. Moreover, we show that the energy of any sign-changing solution is strictly larger than twice the ground state energy. Finally, we obtain that the equation has infinitely many nontrivial solutions. Our result complements the existing ones to fractional Schrödinger problems when the nonlinearity is sign-changing and satisfies neither the monotonicity condition nor Ambrosetti-Rabinowitz condition.
Highlights
In this paper, we consider the following fractional Schrödinger equation with logarithmic nonlinearity:(−∆)αu + V(x)u = |u|p−2u ln u2, x ∈ Ω, u = 0, x ∈ RN \ Ω, (1.1) where α ∈ (0, 1), N > 2α and < p 2∗α
When α = 1, Chen et al [5] proved the existence of ground state sign-changing solutions of problem (1.2) with f (x, u) = Q(x)|u|p−2u ln u2
In 2014, Chang et al [4] proved the existence of a nodal solution of (1.2) with V(x) = 0 in bounded domain
Summary
We consider the following fractional Schrödinger equation with logarithmic nonlinearity:. When α = 1, Chen et al [5] proved the existence of ground state sign-changing solutions of problem (1.2) with f (x, u) = Q(x)|u|p−2u ln u2. In 2014, Chang et al [4] proved the existence of a nodal solution of (1.2) with V(x) = 0 in bounded domain They assume that the nonlinearity f (x, t) satisfies the following Ambrosetti– Rabinowitz condition and monotonicity condition:. When f (x, u) satisfies a monotonicity condition, Deng et al [10] dealt with the least energy sign-changing solutions for fractional elliptic equations (1.2) in bounded domain. Ji [15] concerned with the existence of the least energy sign-changing solutions for a class of fractional Schrödinger–Poisson system when f (x, t) satisfies the following monotonicity condition:. Throughout this paper, the symbol S denote unit sphere, the C, C1, C2, . . . represent several different positive constants
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