## 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

### Keywords:

semilinear elliptic equation; positive solution
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### References:

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