Author:
Leuyacc Yony Raúl Santaria
Abstract
<abstract><p>In this work, we are interested in studying the existence of nontrivial weak solutions to the following class of Schrödinger equations</p>
<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\lbrace\begin{array}{rcll} -{\rm div}(w(x)\nabla u) \ & = &\ f(x, u), &\ x \in B_1(0), \\ u \ & = &\ 0, &\ x \in \partial B_1(0), \end{array}\right. $\end{document} </tex-math></disp-formula></p>
<p>where $ w(x) = \big(\ln (1/|x|)\big)^{\beta} $ for some $ \beta \in [0, 1) $, the nonlinearity $ f(x, s) $ behaves like $ {\rm \exp}((1+h(|x|))|s|^{2/(1-\beta)}) $ and $ h $ is a continuous radial function such that $ h(r) $ tends to infinity as $ r $ tends to $ 1 $. The proof involves variational methods and a new version of Trudinger-Moser inequality.</p></abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)
Reference37 articles.
1. A. Adimurthi, Existence of positive solutions of the semilinear Dirichlet Problem with critical growth for the N-Laplacian, Ann. Sc. Norm. Sup. Pisa IV, 17 (1990), 393–413.
2. S. L. Yadava, Multiplicity results for semilinear elliptic equations in a bounded domain of $ \mathbb{R}^2$ involving critical exponent, Ann. Sc. Norm. Sup. Pisa-Classe di Scienze, 17 (1990), 481–504.
3. F. S. B. Albuquerque, C. O. Alves, E. S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in $\mathbb{R}^2$, J. Math. Anal. Appl., 409 (2014) 1021–1031. https://doi.org/10.1016/j.jmaa.2013.07.005
4. A. Alvino, A. V. Ferone, G. Trombetti, Moser-type inequalities in Lorentz spaces, Potential Anal., 5 (1996), 273–299. https://doi.org/10.1007/BF00282364
5. T. Bartsh, M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555–3561. https://doi.org/10.1016/S0362-546X(03)00020-8
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