On fractional logarithmic Schrodinger equations

摘要

We study the following fractional logarithmic Schrodinger equation: @@@ (-Delta)(s)u + V(x)u = u logu(2), x is an element of R-N, @@@ where N >=, (-Delta)(s) denotes the fractional Laplace operator, 0 < s < 1 and V(x) is an element of C(R-N). Under different assumptions on the potential V(x), we prove the existence of positive ground state solution and least energy sign-changing solution for the equation. It is known that the corresponding variational functional is not well defined in H-s(R-N), and inspired by Cazenave (Stable solutions of the logarithmic Schrodinger equation, Nonlinear Anal. 7 (1983), 1127-1140), we first prove that the variational functional is well defined in a subspace of H-s(R-N). Then, by using minimization method and Lions' concentration-compactness principle, we prove that the existence results.

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