That is, must be superlinear with respect to at infinity.
That is, f x, t) must be superlinear with respect to t at infinity.
That is, f ( x, t ) must be superlinear with respect to | t | p − 2 t at positive infinity.
It is easy to see that the condition (AR) implies that lim u → + ∞ F ( x, u ) u 2 = + ∞, that is, f ( x, u ) must be superlinear with respect to u at infinity.
First, we just assume that the nonlinear term (f u)) is superlinear with respect to u at infinity instead of the asymptotically linear condition or Ambrosetti-Rabinowitz condition, which is completely different from those appearing in the literature.
Condition (H4) means that the function (f t,u,v,w,z)) is superlinear with respect to v. Condition (H5) means that the function (f t,u,v,w,z)) is singular at (v=0) and it is stronger than (5).
Condition (H5) means that the function (f t,u,v,w,y,z)) is superlinear with respect to w. Condition (H6) means that the function (f t,u,v,w,y,z)) is singular at (w=theta), and it is stronger than (6).
However, the computational effort for this approach is superlinear with respect to the number of homologs being compared and substantial for all the query segments in a large sample, even using fast techniques for MSA construction and tree inference.
Noticing our condition the nonlinear term is asymptotically linear, not superlinear, with respect to at infinity, which means that the usual condition (AR) cannot be assumed in our case.
It is worth to note that, in our work, we do not assume that f is superlinear at 0 and sublinear at 1 with respect to (phi(s)).
When (alpha=R=1), f is superlinear at 0 and sublinear at 1 with respect to (phi(s):=s/sqrt{1-{s}^{2}}), the authors obtained the existence of classical positive radial solutions of the problem by using the Leray Schauder degree arguments.
