A singular perturbed problem with critical Sobolev exponent
DOI:
https://doi.org/10.12775/TMNA.2020.067Keywords
Multi-peak solutions, Lyapunov-Schmidt reduction, local uniqueness, local Pohozaev identityAbstract
This paper deals with the following nonlinear elliptic problem \begin{equation}\label{eq0.1} -\varepsilon^2\Delta u+\omega V(x)u=u^{p}+u^{2^{*}-1},\quad u> 0\quad\text{in}\ \R^N, \end{equation} where $\omega\in\R^{+}$, $N\geq 3$, $p\in (1,2^{*}-1)$ with $2^{*}={2N}/({N-2})$, $\varepsilon> 0$ is a small parameter and $V(x)$ is a given function. Under suitable assumptions, we prove that problem (\ref{eq0.1}) has multi-peak solutions by the Lyapunov-Schmidt reduction method for sufficiently small $\varepsilon$, which concentrate at local minimum points of potential function $V(x)$. Moreover, we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.References
T. Akahori, S. Ibrahim, N. Ikoma, H. Kikuchi and H. Nawa, Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies, Calc. Var. Partial Differential Equations 58:120 (2019), 32 pp.
T. Akahori, S. Ibrahim and H. Kikuchi, Linear instability and nondegeneracy of ground state for combined power-type nonlinear scalar field equations with the Sobolev critical exponent and large frequency parameter, arXiv:1810.12363.
T. Akahori, S. Ibrahim, H. Kikuchi and H. Nawa, Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equations with energycritical growth at low frequencies, Mem. Amer. Math. Soc. (to appear).
C. Alves, M. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differential Equations 43 (2012), 537–554.
A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on RN , Basel, Birkhäuser Verlag, 2006.
A. Ambrosetti, A. Malchiodi and W. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres I, Comm. Math. Phys. 235 (2003), 427–466.
A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal. 140 (1997), 285–300.
A. Ambrosetti, J. Garcia Azorero and I. Peral, Perturbation of ∆u+u(N +2)(N −2) = 0, the scalar curvature problem in RN , and related topics, J. Funct. Anal. 165 (1999), 117–149.
H. Berestycki and P. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313–345.
H. Berestycki and P. Lions, Nonlinear scalar field equations II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82 (1983), 347–375.
D. Cao and H. Heinz, Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations, Math. Z. 243 (2003), 599–642.
D. Cao, S. Li and P. Luo, Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015), 4037–4063.
M. Coles and S. Gustafson, Solitary waves and dynamics for subcritical perturbations of energy critical NLS, arXiv:1707.07219.
J. Dávila, M. del Pino and I. Guerra, Non-uniqueness of positive ground states of non-linear Schrödinger equations, Proc. Lond. Math. Soc. 106 (2013), 318–344.
M. del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 127–149.
M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann. 324 (2002), 1–32.
Y. Deng, C. Lin and S. Yan, On the prescribed scalar curvature problem in RN , local uniqueness and periodicity, J. Math. Pures Appl. 104 (2015), 1013–1044.
Y. Ding and F. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations 30 (2007), 231–249.
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), 397–408.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, second edition, Springer, Berlin, 1983.
L. Glangetas, Uniqueness of positive solutions of a nonlinear elliptic equation involving the critical exponent, Nonlinear Anal. 20 (1993), 571–603.
M. Grossi, Some results on a class of nonlinear Schrödinger equations, Math. Z. 235 (2000), 687–705.
M. Grossi, On the number of single-peak solutions of the nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 261–280.
Y. Guo, S. Peng and S. Yan, Local uniqueness and periodicity induced by concentration, Proc. Lond. Math. Soc. 114 (2017), 1005–1043.
M. Kwong, Uniqueness of positive solutions of ∆u − u + up = 0 in RN , Arch. Rational Mech. Anal. 105 (1989), 243–266.
Y. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V )α , Comm. Partial Differential Equations 13 (1988), 1499–1519.
Y. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys. 131 (1990), 223–253.
P. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic equations in the exponential case, Indiana Univ. Math. J. 47 (1998), 529–539.
P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.
J. Wei and S. Yan, New solutions for nonlinear Schrödinger equations with critical nonlinearity, J. Differential Equations 237 (2007), 446–472.
J. Zhang and W. Zou, The critical case for a Berestycki–Lions theorem, Sci. China Math. 57 (2014), 541–554.
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