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Alves, Claudianor O.; Patricio, Geovany F.

Existence of solution for a class of indefinite variational problems with discontinuous nonlinearity. (English) Zbl 1535.35049

J. Math. Sci., New York 266, No. 4, Series A, 635-663 (2022).
Summary: This paper concerns the existence of a nontrivial solution for the following problem \[ (P)\qquad \begin{cases} -{\Delta} u + V(x)u \in \partial_u F(x,u)\text{ a.e. in }\mathbb{R}^N, \\u \in H^1(\mathbb{R}^N), \end{cases}\] where \(F(x,t)={{\int \limits}_0^t}f(x,s) ds\), \(f\) is a \(\mathbb{Z}^N \)-periodic measurable function and \(\lambda = 0\) does not belong to the spectrum of \(- \Delta + V\). Here, \( \partial_tF\) denotes the generalized gradient of \(F\) with respect to variable \(t\).

Cited in 2 Documents

MSC:

35J15 Second-order elliptic equations
35J61 Semilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence

Keywords:

second-order elliptic equation; discontinuous nonlinearity; existence
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References:

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