It is established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in R N with critical growth. Applying a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in H 1 (R N ) and satisfy the geometric hypotheses of the Mountain Pass Theorem. The Concentration-Compactness Principle and a comparison argument allow to verify that the problem possesses a nontrivial solution.
ResumoEste artigo tem como objetivo apresentar uma possibilidade de trabalho com o tema simetria na sala de aula do Ensino Fundamental. A proposta baseia-se na pesquisa desenvolvida no Mestrado Profissional em Ensino de Ciências e Matemática realizada segundo uma abordagem qualitativa. Na pesquisa, foi realizado um trabalho de campo que consistiu no desenvolvimento de uma sequência didática planejada sob a perspectiva do ensino de Matemática através da resolução de problemas com alunos de uma turma de sétimo ano do Ensino Fundamental. Na elaboração da sequência didática, foram levados em consideração aspectos como a intuição e a visualização, relevantes ao estudo da geometria. Durante a realização da sequência didática os diálogos entre os alunos e * Doutorando em Ensino de Ciências e Matemática pela Universidade Cruzeiro do Sul (UNICSUL), São Paulo, SP, Brasil. Docente do Ensino Fundamental na Rede Municipal de Educação, São José dos Campos, SP, Brasil. Endereço para correspondência: Rua Passadena, 355, ap. 63-A,
Using a variational approach, we study the existence of solutions for the following class of quasilinear Schrödinger equations:
78.0pt−normalΔu+V(x)u−normalΔtrue(false|ufalse|2βtrue)false|ufalse|2β−2u=gfalse(ufalse)|x|a1emin1emR2,\begin{equation*} \hspace*{6.5pc}-\Delta u+V(x)u-\Delta \big (|u|^{2\beta }\big )|u|^{2\beta -2}u=\frac{g(u)}{|x|^a}\quad \mbox{in}\quad \mathbb {R}^2,\hspace*{-6.5pc} \end{equation*}where β>1/2$\beta >1/2$, a∈false[0,2false)$a\in [0,2)$, Vfalse(xfalse)$V(x)$ is a positive potential bounded away from zero and can be “large” at infinity, the nonlinearity gfalse(sfalse)$g(s)$ is allowed to satisfy the exponential critical growth with respect to the Trudinger–Moser inequality. Precisely, gfalse(sfalse)$g(s)$ behaves like exptrue(α0false|sfalse|4βtrue)$\exp \big (\alpha _0 |s|^{4 \beta }\big )$ as false|sfalse|→∞$|s| \rightarrow \infty$ for some α0>0$\alpha _0 >0$. This model of equation has been proposed in the theory of superfluid films in plasma physics. As for as we know, this the first work involving this class of operators and singular nonlinearities with exponential critical growth. Moreover, we are able to deal with exponents β>1/2$\beta > 1/2$.
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