By using the penalization method and the Ljusternik–Schnirelmann theory, we investigate the multiplicity of positive solutions of the following fractional Schrödinger equation $$ arepsilon^{2s}(-Delta)^{s} u + V(x)u= f(u) mbox{ in } mathbb{R}^{N} $$ where $arepsilon>0$ is a parameter, $sin (0, 1)$, $N>2s$, $(-Delta)^{s}$ is the fractional Laplacian, $V$ is a positive continuous potential with local minimum, and $f$ is a superlinear function with subcritical growth. we also obtain a multiplicity result when $f(u)= |u|^{q-2}u + lambda |u|^{r-2}u$ with $2<2^{*}_{s}leq r$ and $lambda >0$, by combining a truncation argument and a Moser-type iteration.

Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method / Ambrosio, Vincenzo. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 196:6(2017), pp. 2043-2062. [10.1007/s10231-017-0652-5]

Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method

Ambrosio, Vincenzo
2017-01-01

Abstract

By using the penalization method and the Ljusternik–Schnirelmann theory, we investigate the multiplicity of positive solutions of the following fractional Schrödinger equation $$ arepsilon^{2s}(-Delta)^{s} u + V(x)u= f(u) mbox{ in } mathbb{R}^{N} $$ where $arepsilon>0$ is a parameter, $sin (0, 1)$, $N>2s$, $(-Delta)^{s}$ is the fractional Laplacian, $V$ is a positive continuous potential with local minimum, and $f$ is a superlinear function with subcritical growth. we also obtain a multiplicity result when $f(u)= |u|^{q-2}u + lambda |u|^{r-2}u$ with $2<2^{*}_{s}leq r$ and $lambda >0$, by combining a truncation argument and a Moser-type iteration.
2017
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/264920
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