## Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity.(English)Zbl 1389.35071

The authors discuss decay properties of weak solutions for problems $$-\Delta u+Vu=f(u)$$, $$u\in H^1(\mathbb{R}^N)\cap L^\infty (\mathbb{R}^N)$$, where the potential $$V$$ is Hölder continuous, positive, bounded away from zero, and the nonlinearity $$f$$ is continuous, $$|f(u)|\leq C|u|^q$$ near $$u=0$$ for some $$q>1$$.
In Section 2 it is shown that weak solutions have exponential decay at infinity, with decay at least as fast as that of $$H(x):=G(x,0)$$, where $$G$$ is the Green’s function of the operator $$T:=-\Delta +V$$.
Under additional conditions on $$f$$ (such as analyticity) exact rate of decay for positive solutions is established. Also, analyticity of $$f$$ and periodicity of the potential $$V$$ are used to prove local path connectivity of sets solutions and discrete critical values at low energy levels.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35J20 Variational methods for second-order elliptic equations
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