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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 \(p\).

MSC:

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

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