On existence of solution for a class of semilinear elliptic equations with nonlinearities that lies between different powers. (English) Zbl 1187.35064

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 \(p\).


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
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