On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers - Claudianor O. Alves; Marco A. S. Souto (Abstract and Applied Analysis)
We prove that the semilinear elliptic equation −Δu=f(u), in Ω, u=0, on ∂Ω has a positive solution when the nonlinearity f belongs to a class which satisfies μtq≤f(t)≤Ctp at infinity and behaves like tq near the origin, where 1<q<(N+2)/(N−2) if N≥3 and 1<q<+∞ if N=1,2. In our approach, we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity does not satisfy any hypotheses such those required by the blowup method. Furthermore, we do not impose any restriction on the growth of p.