On existence of solution for a class of semilinear elliptic equations with nonlinearities that lies between different powers
Summary: 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 \(\mu t^q\leq f(t)\leq Ct^p\) 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 required by the blowup method. Furthermore, we do not impose any restriction on the growth of \(p\).
scientific article; zbMATH DE number 5313191