On Some Nonlinear Fractional Equations Involving the Bessel Operator

Abstract

Under different assumptions on the potential functions b and c, we study the fractional equation (I − Δ) α u = λb(x)|u| p−2 u + c(x)|u| q−2 u in R N . Our existence results are based on compact embedding properties for weighted spaces.

Existence of solutions of scalar field equations with fractional operator

“…Second, we deal with the case f (x, s) depends on x. Here, exploiting the optimal path and characterization by the mountain pass value, we show the existence of positive solution of (1), which generalizes the result in [23] and enables us to find a simpler sufficient condition than that of [34] in some case. See the comments after Remark 1.4.…”

“…Clearly, Theorem 1.1 improves these results. Furthermore, in [4,34], the authors raise a question that one can prove the existence of least energy solution and infinitely many solutions of (4) with general nonlinearity. Theorem 1.1 answers this question.…”

“…On the other hand, in [34], the author obtains a nontrivial solution of (1) with f (x, s) = λb(x)|u| p−1 u + c(x)|u| q−1 u under different conditions on b(x), c(x), p, q where λ > 0 is a constant. Among other things, under 1 < p, q < 2 * α − 1 and some strict inequality for the mountain pass value (the infimum of the functional on the 2010 Mathematics Subject Classification.…”

“…This paper is especially motivated by two papers [23] and [34]. In [23], the existence of positive solution of (1) is proved under the following conditions on f (x, s): f (x, t)dt.…”

“…This paper is especially motivated by two papers [23] and [34]. In [23], the existence of positive solution of (1) is proved under the following conditions on f (x, s):(vi) There exist continuous functionsf (s) and a(x) such thatf satisfies (i)-(v) and 0On the other hand, in [34], the author obtains a nontrivial solution of (1) with f (x, s) = λb(x)|u| p−1 u + c(x)|u| q−1 u under different conditions on b(x), c(x), p, q where λ > 0 is a constant. Among other things, under 1 < p, q < 2 * α − 1 and some strict inequality for the mountain pass value (the infimum of the functional on the 2010 Mathematics Subject Classification.…”