Abstract
We study the existence of positive solutions for an elliptic equation in RN for N = 2, 3 which is related with the existence of standing (localized) waves and the existence of the ground state solutions for some physical model or systems in fluid mechanics to describe the evolution of weakly nonlinear water waves. We use a variational approach and the well-known principle of concentration-compactness due to P. Lions to obtain the existence of this type of solutions, even in the case that the nonlinear term g is a non-homogeneous function or an operator defined in H1(RN) with values in R.
Highlights
We study the existence of positive solutions for an elliptic equation in RN for N = 2, 3 which is related with the existence of standing waves and the existence of the ground state solutions for some physical model or systems in fluid mechanics to describe the evolution of weakly nonlinear water waves
The first step in the study of some interesting physical problems in acoustics, in the context of gravity waves, in fluid mechanics, or optics is to establish the existence of special positive solutions as travelling and standing waves, as happens in the case of the generalized Schrödinger equation, the generalized Davey-Stewartson type systems, the Zakharov-Rubenchik system and the generalization the Zakharov-Rubenchik system
If we look for standing wave solutions for (1) of the form ψ(x, t) = eictu(x), u satisfies the equation
Summary
The first step in the study of some interesting physical problems in acoustics, in the context of gravity waves, in fluid mechanics, or optics is to establish the existence of special positive solutions as travelling and standing waves, as happens in the case of the generalized Schrödinger equation, the generalized Davey-Stewartson type systems, the Zakharov-Rubenchik ( known Benney-Roskes) system and the generalization the Zakharov-Rubenchik system. In this paper we are interested in establishing a general existence result of positive solutions of the special elliptic equation in RN (N = 2, 3). Where g is a function or an operator defined such that g(u) = g(|u|)u In this case, we find that u satisfies the elliptic equation (2). We note that equation (3) is related with the generalized Schrödinger equation in the case g(u) = a|u|pu, with the Davey-Stewartson type systems in the case g(u) = a|u|pu + bE1(|u|2)u, with the Benney-Roskes/Zakharov-Rubenchik system in the case g(u) = a|u|2u + bE2(|u|2)u, https://rajpub.com/index.php/jam and with the generalization of the Zakharov-Rubenchik system in the case g(u) = a|u|pu + b|u|2u + cE3(|u|2)u, where Ej (j = 1, 2, 3) is a linear operator defined on H1(RN ) via a Fourier multiplier of the form. As far as our knowledge goes, the last result is new to the literature
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