On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers

Abstract

We prove that the semilinear elliptic equation $-\Delta u=f\left(u\right)$, in $\Omega$, $u=0$, on $\partial \Omega$ has a positive solution when the nonlinearity $f$ belongs to a class which satisfies $\mu t^q\le f\left(t\right)\le C{t}^p$ at infinity and behaves like $t^q$ near the origin, where if $N\ge 3$ and 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 $p$.