Minimization of quotients with variable exponents

Alves, Claudianor O.; Ercole, Grey; Bolãnos, Mario. D. Huamán

doi:10.1016/j.jde.2018.04.010

Cited by 1 publication

(1 citation statement)

References 29 publications

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“…This phenomenon is quite different from that of the well‐studied equation whose nonlinearity has a constant exponent in the sense that the spikes of both the positive ground state and the least energy sign‐changing solution will always concentrate around the global minimum point of

V (x)

( cf. ) . (b)We also would like to refer the readers to some recent works and the references therein for the studies on some other type of more general problems with variable exponents, which contain many interesting results and show some new phenomena unique to this class of problems. For example, in , the authors proved a new continuous embedding from

H_{0}^{1} false(B_{1} (0) false)

to the following variable exponent Lebesgue space

L^{2^{*} + {| x |}^{α}} false(B_{1} (0) false)

for

0 < α < \min {N / 2, N - 2}

, where

B_{1} false(0 false)

is the unit ball in

{boldR}^{N} false(N ⩾ 3 false)

.…”

Section: Introductionmentioning

confidence: 99%

A monotone property of the ground state energy to the scalar field equation and applications

Wang

2019

J. London Math. Soc.

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We consider the ground state energy m (p, λ) for the classical nonlinear scalar field equationwhere N 3, λ > 0, 2 < p < 2 * and 2 * = 2N N −2 is the critical Sobolev exponent. It is well known that the equation has a unique ground state U p,λ in H 1 (R N ) such that U p,λ (0) = max x∈R N U p,λ (x). In this paper, we uncover a monotone property of m (p, λ) is fixed. For applications, we establish the existence and the concentration behavior of the least energy solutions (positive or sign changing) for the singularly perturbed elliptic equation with a variable exponent p(x):Our results show that the concentration behavior of the least energy solutions of the above equation is quite different from that of the semi-classical states of nonlinear Schrödinger equations with p being a constant: −ε 2 Δu + V (x)u = |u| p−2 u. Roughly speaking, for small λ, the least energy solution concentrates at the global minimum point of p(x), and for large λ, the least energy solution concentrates at the global maximum point of p(x).

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