Soliton Solutions for a Generalized Quasilinear Elliptic Problem
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Soliton Solutions for a Generalized Quasilinear Elliptic Problem Marcelo F. Furtado1 · Edcarlos D. Silva2 · Maxwell L. Silva2 Received: 13 September 2018 / Accepted: 6 September 2019 / © Springer Nature B.V. 2019
Abstract We establish existence and multiplicity of solutions for the elliptic quasilinear Schr¨odinger equation −div(g 2 (u)∇u) + g(u)g (u)|∇u|2 + V (x)u = h(x, u), x ∈ RN , where g is a suitable function, V is a coercive like potential and the nonlinearity h is superlinear at infinity and at the origin. In the proofs, we apply minimization on the Nehari manifold and Ljusternik-Schnirelmann theory. Keywords Variational methods · Generalized Schr¨odinger equation · Quasilinear elliptic problems · Superlinear elliptic problems Mathematics Subject Classification (2010) Primary 35J20; Secondary 35J25 · 35J60
1 Introduction Let us consider the nonlinear Schr¨odinger equation i∂t z = −z + W (x)z − l(|z|2 )l (|z|2 )z − h(x, z),
x ∈ RN , t > 0,
Marcelo F. Furtado was partially supported by CNPq/Brazil and FAPDF/Brazil. Edcarlos D. Silva was partially supported by CNPq grants 429955/2018-9 Edcarlos D. Silva
[email protected] Marcelo F. Furtado [email protected] Maxwell L. Silva [email protected] 1
Departamento de Matem´atica, Universidade de Bras´ılia, 70910-900, Bras´ılia DF, Brazil
2
Instituto de Matem´atica e Estat´ıstica, Universidade Federal de Goi´as, 74001-970, Goiˆania GO, Brasil
M.F. Furtado et al.
where W ∈ C(RN , R) is a potential, l ∈ C(R+ , R), h ∈ C(RN × R, R) is a given nonlinearity and we look for solutions z ∈ C(R × RN , C) with finite energy. This equation has been accepted as a model in physical phenomena depending on the function l. For instance, if l(t) = 1 we have the classical semilinear Schr¨odinger equation [16]. When l(t) = t, the equation arises from fluid mechanics, plasma physics and dissipative quantum mechanics, see [11, 14, 19, 26]. We also refer to [4, 13, 15] for further physical applications. If we are interested in solitary wave solutions, namely solutions with the special form z(t, x) = exp(−i E t)u(x), with E ∈ R and u being a real valued function, we are lead to consider the equation − u + V (x)u − [l(u2 )]l (u2 )u = h(x, u),
x ∈ RN ,
(1.1)
with V (x) = W (x) + E . In the simplest case l(t) = 1, we have a semilinear equation and there exist a lot of papers concerning existence, non-existence, multiplicity and concentration behavior of solutions (see [5, 7, 17, 28] and its references). In the superfluid film case, namely l(t) = t α/2 , for α > 0, the problem also has been extensively studied during the last years, see [21, 24, 25, 27, 31, 33]. In order to present the object of study of this paper we notice that, if we set g(t) = 1 + 2(tl (t 2 ))2 , then the problem (1.1) can be written as −div(g 2 (u)∇u) + g(u)g (u)|∇u|2 + V (x)u = h(x, u), (P ) u ∈ H 1 (RN ).
x ∈ RN ,
It will be considered in a general framework, by assuming that the function g verifies (g0 )
g ∈ C 1 (R, R) is positive, even, non-decreasing in (0, +∞) and satisfies g(t) ∈ (0, +∞),
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