科研之友
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摘要
In this paper, we study the existence and asymptotic properties of solutions to the following fractional Schrodinger equation: @@@ (-Delta)(sigma)u = lambda u + vertical bar u vertical bar(q-2)u+mu(I-alpha*vertical bar u vertical bar p vertical bar u vertical bar(p-2)u in R-N @@@ under the normalized constraint @@@ integral(RN)u(2) = a(2), @@@ where N >= 2, sigma is an element of (0, 1), alpha is an element of (0, N), q is an element of (2 + 4 sigma/N, 2N/N - 2 sigma], p is an element of [2, 1 + 2 sigma + alpha/N), a > 0, mu > 0, I-alpha(x) = vertical bar x vertical bar(alpha-N) and lambda is an element of R appears as a Lagrange multiplier. In the Sobolev subcritical case q is an element of (2 + 4 sigma/N, 2N/N - 2 sigma), we show that the problem admits at least two solutions under suitable assumptions on alpha, a, and mu. In the Sobolev critical case q = 2N/N-2 sigma, we prove that the problem possesses at least one ground state solution. Furthermore, we give some stability and asymptotic properties of the solutions. We mainly extend the results of S. Bhattarai published in 2017 on J. Differ. Equ. and B. H. Feng et al published in 2019 on J. Math. Phys. concerning the above problem from L-2-subcritical and L-2-critical setting to L-2-supercritical setting with respect to q, involving Sobolev critical case especially.
