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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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