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We are concerned with the following fractional $p$-Laplacian inclusion: \begin{equation*} (-\Delta)_p^su+V(x)|u|^{p-2}u\in \lambda[\underline{f}(x,u(x)),\, \overline{f}(x,u(x))] \quad \textmd{in} \ \ \mathbb R^{N}, \end{equation*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $0<s<1<p<+\infty$, $sp<N$, and $f:\mathbb R^{N}\times\mathbb R \to \mathbb R$ is measurable with respect to each variable separately. We show that our problem with the discontinuous nonlinearity $f$ admits at least one or two nontrivial weak solutions. In order to do this, the main tool is the Berkovits-Tienari degree theory for weakly upper semicontinuous set-valued operators. In addition, our main assertions continue to hold when $(-\Delta)_p^su$ is replaced by any non-local integro-differential operator.
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참고문헌 (26)
참고문헌 신청
A. Ambrosetti / 1989 / The dual variational principle and elliptic problems with discontinuous nonlinearities / J. Math. Anal. Appl 140(2):363~373
M. Badiale / 1997 / Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities / Nonlinear Anal 29(6):639~677
B. Barrios / 2012 / On some critical problems for the fractional Laplacian operator / J. Differential Equations 252(11):6133~6162
T. Bartsch / 1995 / Existence and multiplicity results for some superlinear elliptic problems on RN / Comm. Partial Differential Equations 20(9-10):1725~1741
J. Berkovits / 1996 / Topological degree theory for some classes of multis with applications to hyperbolic and elliptic problems involving discontinuous nonlinearities / Dynam. Systems Appl 5(1):1~18
M. Badiale / 1997 / Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities / Nonlinear Anal 29(6):639~677
B. Barrios / 2012 / On some critical problems for the fractional Laplacian operator / J. Differential Equations 252(11):6133~6162
T. Bartsch / 1995 / Existence and multiplicity results for some superlinear elliptic problems on RN / Comm. Partial Differential Equations 20(9-10):1725~1741
J. Berkovits / 1996 / Topological degree theory for some classes of multis with applications to hyperbolic and elliptic problems involving discontinuous nonlinearities / Dynam. Systems Appl 5(1):1~18
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