Discontinuous elliptic problems in \(\mathbb R^N\) without monotonicity assumptions. (English) Zbl 1115.35043

The authors consider the semilinear elliptic problem \[ \begin{cases} -\Delta u = f(u) \qquad \text{in} \;\mathbb R^N,\\ \lim _{| x| \rightarrow \infty } u(x) = 0, \end{cases} \] where \(N\geq 3\) and \(f\) is a nonnegative function with an upward “jump-discontinuity” at some point \(a>0\). The existence of a positive, radial solution is proved. The main tool is an approximating argument which does not require any monotonicity assumptions on the nonlinearity.


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
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