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 $-\Delta u=f(u)$, in $\Omega $, $u=0$, on $\partial \Omega $ has a positive solution when the nonlinearity $f$ belongs to a class which satisfies $\mu {t}^{q}\leq f(t)\leq C{t}^{p}$ at infinity and behaves like ${t}^{q}$ near the origin, where $1< q < (N+2)/(N-2)$ if $N\geq 3$ and $1< q< +\infty $ 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 $p$.