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Nonlinear scalar field equations, I existence of a ground state

Nonlinear scalar field equations, I existence of a ground state,10.1007/BF00250555,Archive for Rational Mechanics and Analysis,H. Berestycki,P.-L. Lio

Nonlinear scalar field equations, I existence of a ground state   (Citations: 385)
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Journal: Archive for Rational Mechanics and Analysis - ARCH RATION MECH ANAL , vol. 82, no. 4, pp. 313-345, 1983
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    • ...Second, the use of Sobolev spaces of highly symmetric functions, which admit compact embeddings, as in 5,6...

    V. Raghavendraet al. Weak solutions for a class of semilinear elliptic equations in unbound...

    • ...[4,14] and references therein, together with a vast literature that followed these works...
    • ...In particular, it is standard to assume f is an odd function in most results on multiplicity of solutions (see [4,10,11] and references therein)...
    • ...The classical result on existence of positive solution ¯ for (P∞), which is also unique, radial and decays exponentially to zero as |x |→∞ , due to Berestycki and Lions [4] will play an important role in our...
    • ...Proof. Let ¯ ∈ N∞ be the radial positive solution of problem (P∞) given in [4]...
    • ...First, we recall for instance [12 ]a nd [4] where it is proved there are positive constants C1 and C2 such that...

    Janete S. Carvalhoet al. A note on existence of antisymmetric solutions for a class of nonlinea...

    • ...The recent mathematical literature has seen a growing interest in what we may call, borrowing a terminology from [19], the zero mass case of noncritical elliptic problems of the form −� u + V (x) u = g (u )i nR N ,N ≥ 3, (1.1)...
    • ...Berestycki and Lions in [19], where the authors probably first used the socalled double-power growth condition on g, namely, g (u) behaves as a subcritical power u q1−1 at infinity and a supercritical power u q2−1 near the origin, where q1 < 2 ∗ <q 2. A multiplicity result was also obtained in [20]...

    Marino Badialeet al. Sum of weighted Lebesgue spaces and nonlinear elliptic equations

    • ...The aim of this paper is to study the Schr ¨ odinger-Maxwell system assuming the same very general hypotheses introduced by Berestycki & Lions, in their celebrated paper [7]...
    • ...hypotheses ong are almost necessary in the sense specified in [7];...
    • ...the fact that the result is obtained for small q is not surprising for at least two reasons: first, because small q makes, in some sense, less strong the influence of the term u , which constitutes, in the first equation, a perturbation with respect to the classical nonlinear Schr ¨ odinger equation treated in [7]; second, there is a nonexistence result for largeq andg(u) = u +jujp 1u withp2]1; 2] (see [24])...
    • ...Following [7], define s0 := minfs 2 [; +1[ j g(s) = 0g (s0 = +1 if g(s)6= 0 for anys> ). We set ~ g : R! R the function such that...
    • ...Observe that, according to [7], as a consequence of (g4), there exists a functionz2 H1 r (R3) such that...

    Antonio Azzolliniet al. On the Schrödinger–Maxwell equations under the effect of a general non...

    • ...problem (1.1) has a positive, radially symmetric solution which tends to 0 at innity, which is usually called the ground state. We refer to [44, 4] for a proof...

    Manuel del Pinoet al. The Toda system and multiple-end solutions of autonomous planar ellipt...

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