Please use this identifier to cite or link to this item: http://repositorio.yachaytech.edu.ec/handle/123456789/621
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dc.contributor.advisorMayorga Zambrano, Juan Ricardo-
dc.contributor.authorNarváez Pistala, Iván Andrés-
dc.date.accessioned2023-05-30T14:43:42Z-
dc.date.available2023-05-30T14:43:42Z-
dc.date.issued2023-05-
dc.identifier.urihttp://repositorio.yachaytech.edu.ec/handle/123456789/621-
dc.descriptionIn this thesis it's proved the This project deals with the multiplicity of solutions of the problem: %following \begin{equation}\label{eq_} \left\{\begin{array}{rll} -\varepsilon^4 \Delta^2 u(x)+V(x) u(x)-|u(x)|^{p-1} u(x) & =0, & \quad x \in \mathbb{ R}^N, \\ u(x) & \longrightarrow 0, & \text { as }|x| \longrightarrow +\infty, \end{array}\right. \end{equation} where $\Delta^2= \Delta\circ\Delta $, and $1< p+1<2^*$ with %against $$ 2^*=\left\{\begin{array}{cc} \displaystyle{\frac{2 N}{N-2}}, & \text { if } N\ge 3 ; \\ +\infty, & \text { if } N=1,2 , \end{array}\right. $$ under the following assumptions:\\ %\begin{enumerate} %\item (V1) $V \in C\left(\mathbb{R}^N\right)$ is a nonnegative function,\\ %\item (V2) $V(x) \longrightarrow +\infty$, when $|x| \longrightarrow +\infty$,\\ %\item (V3) $\{V=\inf(V)=0 \}\neq \emptyset.$ \\ %\end{enumerate} By a suitable scaling, with $v(x)=u\left(\varepsilon^{\alpha} x\right)$ and $V_{\varepsilon}(x)=V\left(\varepsilon^{ -\alpha} x\right)$, for $\varepsilon>0$ and $\alpha= -1$, we deal with the following equivalent version of (1): $$ (P_{\varepsilon}')\left\{\begin{aligned} -\Delta^2 u(x)+V_{\varepsilon}(x)u(x)-|u(x)|^{ p-1} u(x) &=0, & x \in \mathbb{R }^N, \\ u(x) &\longrightarrow 0, & \text { as }|x| \longrightarrow +\infty. \end{aligned}\right. $$ We consider the associated functional $ I_{\varepsilon}: \mathcal{M}_{\varepsilon} \subseteq \mathrm{H}^2_{\varepsilon} \longrightarrow \mathbb{R}$, given by \begin{equation*}\label{eq_2} I_{\varepsilon}(u)=\frac{1}{2} \int_{\mathbb{R}^N}\left[|\Delta u(x)|^2+V_{\varepsilon}(x) |u(x)|^2\right] d x, \end{equation*} defined on the manifold% variety $$ \mathcal{M}_{\varepsilon}=\left\{u \in \mathrm{H}^2_{\varepsilon} \big/ \int_{\mathbb{R}^N}|u(x)|^{p+1} d x=1\right\}, $$ where $$ \mathrm{H}^2_{ \varepsilon}=\left\{u \in \mathrm{H}^2\left(\mathbb{R}^N\right)\big/\|u\|_{\varepsilon} =\left(\int_{\mathbb{R}^N}\left[|\Delta u(x)|^2+V_{\varepsilon}(x)|u(x)|^2\right] d x\right)^{1 / 2}<+\infty\right\} . $$ We apply a Ljusternik-Schnirelman scheme to show that the set of critical points has infinite elements, this is done through scaling for get weak solutions of $(P_{\varepsilon}')es
dc.description.abstractEstudiamos la multiplicidad de soluciones del siguiente problema: \begin{equation}\label{eq_} \left\{\begin{array}{rll} -\varepsilon^4 \Delta^2 u(x)+V(x) u(x)-|u(x)|^{p-1} u(x) & =0, & x \in \mathbb{R}^N, \\ u(x) & \longrightarrow 0, & \text { cuando }|x| \longrightarrow +\infty, \end{array}\right. \end{equation} donde $\Delta^2 = \Delta\circ\Delta $, y $1< p+1<2^*$ con $$ 2^*=\left\{\begin{array}{cc} \displaystyle{\frac{2 N}{N-2}}, & \text { if } N\ge 3 ; \\ +\infty, & \text { if } N=1,2 , \end{array}\right. $$ %$$2^*=\displaystyle{\frac{2 N}{N-2}}, \quad \text{if} \quad N\ge 3,$$ bajo los siguientes supuestos:\\ %\item (V1) $V \in C\left(\mathbb{R}^N\right)$ es una funci\'on no negativa,\\ %\item (V2) $V(x) \longrightarrow +\infty$, cuando $|x| \longrightarrow +\infty$,\\ %\item (V3) $\{V=\inf (V)=0\}\neq \emptyset.$\\%$\inf\{Z\}\neq \emptyset.$ %\end{enumerate}%\\ Via un escalamiento adecuado, con $v(x)=u\left(\varepsilon^{\alpha} x\right)$ y $V_{\varepsilon}(x)=V\left(\varepsilon^{ -\alpha} x\right)$, para $\varepsilon>0$ y $\alpha= -1$, tratamos con la siguiente versi\'on equivalente de (2): %(\ref{eq_}): $$ (P_{\varepsilon}')\left\{\begin{aligned} -\Delta^2 u(x)+V_{\varepsilon}(x)u(x)-|u(x)|^{ p-1} u(x) & =0, & x \in \mathbb{R}^N, \\ u(x) &\longrightarrow 0, & \text { cuando }|x| \longrightarrow +\infty . \end{aligned}\right. $$ Consideramos el funcional asociado $ I_{\varepsilon}: \mathcal{M}_{\varepsilon} \subseteq \mathrm{H}^2_{\varepsilon} \longrightarrow \mathbb{R}$, dado por \begin{equation*}\label{eq_2} I_{\varepsilon}(u)=\frac{1}{2} \int_{\mathbb{R}^N}\left[|\Delta u(x)|^2+V_{\varepsilon}(x)|u(x)|^2\right] d x, \end{equation*} definido sobre la variedad $$ \mathcal{M}_{\varepsilon}=\left\{u \in \mathrm{H}^2_{\varepsilon} \big/ \int_{\mathbb{R}^N}|u(x)|^{p+1} d x=1\right\}, $$ donde $$ \mathrm{H}^2_{ \varepsilon}=\left\{u \in \mathrm{H}^2\left(\mathbb{R}^N\right)\big/\|u\|_{\varepsilon}=\left(\int_{\mathbb{R}^N}\left[|\Delta u(x)|^2+V_{\varepsilon}(x)|u(x)|^2\right] d x\right)^{1 / 2}< +\infty\right\} . $$ %nosotros Aplicamos un esquema de Ljusternik-Schnirelman para mostrar que el conjunto de puntos críticos tiene infinitos elementos, esto es realizado a través de escalamiento para obtener soluciones débiles de $(P_{\varepsilon}')es
dc.language.isoenges
dc.publisherUniversidad de Investigación de Tecnología Experimental Yachayes
dc.rightsopenAccesses
dc.subjectEcuación no lineal de Schrödingeres
dc.subjectOperador biarmónico de Laplacees
dc.subjectMultiplicidad de solucioneses
dc.subjectEcuaciones diferenciales parcialeses
dc.subjectNonlinear Schrödinger equationes
dc.subjectBiharmonic Laplace operatores
dc.subjectMultiplicity of solutionses
dc.subjectPartial differential equationses
dc.titleMultiplicity of solutions for the nonlinear biharmonic Schrödinger equation with critical frequencyes
dc.typebachelorThesises
dc.description.degreeMatemático/aes
dc.pagination.pages62 hojases
Appears in Collections:Matemática

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