論文

In this paper, the existence of least energy solution and infinitely many solutions is proved for the equation in where , and f(s) is a Berestycki-Lions type nonlinearity. The characterization of the least energy by the mountain pass value is also considered and the existence of optimal path is shown. Finally, exploiting these results, the existence of positive solution for the equation in is established under suitable conditions on f(x, s).

In this paper we study the existence of standing waves for coupled nonlinear Schrodinger equations. The interaction between equations plays an important role in our study. When the interaction is strong, the least energy solution is a solution whose both components are positive. When the interaction is weak, the least energy solution is a semitrivial solution, namely a solution of a form (u(1), 0) or (0, u(2)). Moreover, minimizing method on the Nehari type manifold with codimension 2 gives us a positive solution when the interaction is weak.

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in R-N: -Delta u = g(u) in R-N, u is an element of H-1(R-N). We give an extension of the existence results due to H. Berestycki, T. Gallouet and O. Kavian [2]. We take a mountain pass approach in H-1(R-N) and introduce a new method generating a Palais-Smale sequence with an additional property related to Pohozaev identity.

We study the existence of radially symmetric solutions u is an element of H(1)(Omega) of the following nonlinear scalar field equation -Delta u = g(vertical bar x vertical bar, u) in Omega. Here Omega = R(N) or {x is an element of R(N) vertical bar vertical bar x vertical bar > R}, N >= 2. We generalize the results of Li and Li (1993) [13] and Li (1990) [14] in which they studied the problem in R(N) and {vertical bar x vertical bar > R} with the Dirichlet boundary condition. Furthermore, we extend it to the Neumann boundary problem and we also consider the nonlinear Schrodinger equation that is the case g(r, s) = -V(r)s + (g) over tilde (s). (C) 2011 Elsevier Inc. All rights reserved.

This paper and Hasegawa et al. (2021) treat the existence and nonexistence of stable weak solutions to a fractional Hardy–Hénon equation (−Δ)su=|x|ℓ|u|p−1u in RN, where 0−2s, p>1, N≥1 and N>2s. In this paper, when p is critical or supercritical in the sense of the Joseph–Lundgren, we prove the existence of a family of positive radial stable solutions, which satisfies the separation property. We also show the multiple existence of the Joseph–Lundgren critical exponent for some ℓ∈(0,∞) and s∈(0,1), and this property does not hold in the case s=1.

Given a measure ρ on a domain Ω ⊂ Rm , we study spacelike graphs over Ω in Minkowski space with Lorentzian mean curvature ρ and Dirichlet boundary condition on ∂Ω , which solve [Figure not available: see fulltext.] The graph function also represents the electric potential generated by a charge ρ in electrostatic Born-Infeld’s theory. Even though there exists a unique minimizer uρ of the associated action Iρ(ψ)≐∫Ω(1-1-|Dψ|2)dx-⟨ρ,ψ⟩ among functions ψ satisfying | Dψ| ≤ 1 , by the lack of smoothness of the Lagrangian density for | Dψ| = 1 one cannot guarantee that uρ satisfies the Euler-Lagrange equation (BI). A chief difficulty comes from the possible presence of light segments in the graph of uρ . In this paper, we investigate the existence of a solution for general ρ . In particular, we give sufficient conditions to guarantee that uρ solves (BI) and enjoys log -improved energy and Wloc2,2 estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of ρ to ensure the solvability of (BI).

The existence of L2 –normalized solutions is studied for the equation −Δu + μu = f (x, u) in RN, ∫RN u2 dx = m. Here m > 0 and f (x, s) are given, f (x, s) has the L2-subcritical growth and (μ, u) ∈ R × H1(RN) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.

Abstract. We study the existence of L2 normalized solutions for nonlinear Schrödinger equations and systems. Under new Palais-Smale type conditions, we develop new deformation arguments for the constrained functional on (Formula Presented). As applications, we give other proofs to the results of [5, 8, 22]. As to the results of [5, 22], our deformation result enables us to apply the genus theory directly to the corresponding functional to obtain infinitely many solutions. As to the result [8], via our deformation result, we can show the existence of vector solution without using constraint related to the Pohozaev identity.

We study the eigenvalue problem for positively homogeneous, of degree one, elliptic ODE on finite intervals and PDE on balls. We establish the existence and completeness results for principal and higher eigenpairs, i.e., pairs of an eigenvalue and its corresponding eigenfunction. (c) 2012 Elsevier Masson SAS. All rights reserved.

This is a continuation of Ikoma and Ishii (Ann Inst H Poincar, Anal Non Lin,aire 29:783-812, 2012) and we study the eigenvalue problem for fully nonlinear elliptic operators, positively homogeneous of degree one, on finite intervals or balls. In the multi-dimensional case, we consider only radial eigenpairs. Our eigenvalue problem has a general first-order boundary condition which includes, as a special case, the Dirichlet, Neumann and Robin boundary conditions. Given a nonnegative integer n, we prove the existence and uniqueness, modulo multiplication of the eigenfunction by a positive constant, of an eigenpair whose eigenfunction, as a radial function in the multi-dimensional case, has exactly n zeroes. When an eigenfunction has n zeroes, we call the corresponding eigenvalue of nth order. Furthermore, we establish results concerning comparison of two eigenvalues, characterizations of nth order eigenvalues via differential inequalities, the maximum principle for the boundary value problem in connection with the principal eigenvalue, and existence of a solution having n zeroes, as a radial function in the multi-dimensional case, of the boundary value problem with an inhomogeneous term.

We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Mobius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new geometric expansions of exponentiated small symmetric Clifford tori and analyze the sharp asymptotic behaviour of degenerating tori under the action of the Mobius group. In this first work we prove two existence results by minimizing or maximizing a suitable reduced functional, in particular we obtain embedded area-constrained Willmore tori (or, equivalently, toroidal critical points of the Hawking mass under area-constraint) in compact 3-manifolds with constant scalar curvature and in the double Schwarzschild space. In a forthcoming paper new existence theorems will be achieved via Morse theory.

This is the second part of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three -manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the Mobius degeneration of the tori. In the first paper the construction was performed via minimization, here by Morse Theory. To this aim we establish new geometric expansions of the derivative of the Willmore functional on small Clifford tori (in geodesic normal coordinates) which degenerate to small geodesic spheres with a small handle under the action of the Mobius group. By using these sharp asymptotics we give sufficient conditions, in terms of the ambient curvature tensors and Morse inequalities, for having existence/multiplicity of embedded tori which are stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area.

We consider a singular perturbation problem for a system of nonlinear Schrodinger equations: -epsilon(2)Delta v(1) + V(1)(x)v(1) = mu(1)v(1)(3) + beta v(1)v(2)(2) in R(N), -epsilon(2)Delta v(2) + V(2)(x)v(2) = mu(2)v(2)(3) + beta v(1)(2)v(2) in R(N), v(1)(x), v(2)(x) > 0 in R(N), (*) v(1)(x), v(2)(x) is an element of H(1)(R(N)). where N = 2, 3, mu(1), mu(2), beta > 0 and V(1)(x), V(2)(x) : R(N) -> (0,infinity) are positive continuous functions. We consider the case where the interaction beta > 0 is relatively small and we define for P is an element of R(N) the least energy levelm(P) for non- trivial vector solutions of the rescaled " limit" problem: -Delta v(1) + V(1)(P)v(1) = mu(1)v(1)(3) + beta v(1)v(2)(2) in R(N), -Delta v(2) + V(2)(P)v(2) = mu(2)v(2)(3) + beta v(1)(2)v(2) in R(N), v(1)(x), v(2)(x) > 0 in R(N), (**) v(1)(x), v(2)(x) is an element of H(1)(R(N)). We assume that there exists an open bounded set Lambda subset of R(N) satisfying inf(P is an element of Lambda) m(P) < inf(P is an element of partial derivative Lambda) m(P). We show that (*) possesses a family of non-trivial vector positive solutions {(v(1)epsilon(x), v(2 epsilon) (x))}(epsilon.(0, epsilon 0]) which concentrates-after extracting a subsequence epsilon(n) -> 0-to a point P(0) is an element of Lambda with m(P(0)) = inf(P is an element of Lambda)m(P). Moreover (v(1 epsilon)(x), v(2 epsilon)(x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.