論文

We study existence, uniqueness, and qualitative property of solutions for a class of nonlinear elliptic equations of sublinearity. We also study the asymptotic behavior of a ground state solution for sublinear equations in RN and give estimates for a radius of its support set as various parameters involved, including the sublinearity power approaching to the limiting cases.

In this paper, we prove new existence and multiplicity results for critical points of lower semicontinuous functionals in Banach spaces, complementing the nonsmooth critical point theory set forth by Szulkin and avoiding the need of the Palais–Smale condition. We apply our abstract results to get entire solutions with finite energy to Born–Infeld type autonomous equations. More precisely, under almost optimal conditions on the nonlinearity, we construct a positive solution and infinitely many solutions both in the classes of radially symmetric functions and nonradiallly symmetric ones.

We consider a class of non–linear and non–local functionals giving rise to the Choquard equation with a suitably regular interaction potential, modelling, i.a., gases with impurities and axion stars. We study how existence of minimizers depends on the coupling constant, and find that there is a critical interaction strength needed for the minimizers to exist, both in dimensions two and three. In d=3, a minimizer exists also at the critical coupling but none do in d=2 under suitable assumptions on the potential. We also establish that in d=3 there exist other critical points beyond the global minimizer.

We study normalized solutions (µ, u) ∈ R × H1(RN) to nonlinear Schrödinger equations −∆u + µu = g(u) in RN21 ZRN u2dx = m, where N ≥ 2 and the mass m > 0 is given. Here, g has an L2-critical growth, both at the origin and at infinity, that is, g(s) ∼ |s|p−1s as s ∼ 0 and s ∼ ∞, where p = 1 + N4 . We continue the analysis started in [11], where we found two (possibly distinct) minimax values b ≤ 0 ≤ b of the Lagrangian functional. In this paper, we furnish explicit examples of g satisfying b < 0 < b, b = 0 < b, and b < 0 = b; notice that b = 0 = b in the power case g(t) = |t|p−1t. Moreover, we deal with the existence and non-existence of a solution with minimal energy. Finally, we discuss the assumptions required on g to obtain the existence of a positive solution for perturbations of g.

This paper treats the existence of positive solutions of - Δ u+ V(x) u= λ f(u) in RN. Here N≥ 1 , λ > 0 is a parameter and f(u) satisfies conditions only in a neighborhood of u= 0. We shall show the existence of positive solutions with potential of trapping type or G-symmetric potential where G⊂ O(N). Our results extend previous results (Adachi and Watanabe in J Math Anal Appl 507:125765, 2022; Costa and Wang in Proc Am Math Soc 133(3):787–794, 2005; do Ó et al. in J Math Anal Appl 342:432–445, 2008) as well as we also study the asymptotic behavior of a family (uλ)λ≥λ0 of positive solutions as λ → ∞.

Let (M, g) be a three-dimensional Riemannian manifold. The goal of the paper is to show that if P0 ∈ M is a nondegenerate critical point of the scalar curvature, then a neighborhood of P0 is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than 32π; moreover, it is regular in the sense that a suitable rescaling smoothly converges to a round sphere in the Euclidean three-dimensional space. We also establish generic multiplicity of foliations and the 1st multiplicity result for area-constrained Willmore spheres with prescribed (small) area in a closed Riemannian manifold. The topic has strict links with the Hawking mass.

In this paper, the precompactness of minimizing sequences under multiconstraint conditions are discussed. This minimizing problem is related to a coupled nonlinear Schrodinger system which appears in the field of nonlinear optics. As a consequence of the compactness of each minimizing sequence, the orbital stability of the set of all minimizers is obtained.

In this paper, we consider the following minimizing problem with two constraints: inf{E(u)|u = (u1,u2),u1L22 = α 1,u2L22 = α 2}, where α1,α2 > 0 and E(u) is defined by E(u):= RN 1 2i=12(|u i|2 + V i(x)|ui|2) - i=12 μi 2pi + 2|ui|2pi+2 - β p3 + 1|u1|p3+1|u 2|p3+1 dx. Here N ≥ 1, μ1,μ2,β > 0 and Vi(x) (i = 1, 2) are given functions. For Vi(x), we consider two cases: (i) both of V1 and V2 are bounded, (ii) one of V1 and V2 is bounded. Under some assumptions on Vi and pj, we discuss the compactness of any minimizing sequence.

In this paper, we study the existence and non-existence of maximizers for the Moser–Trudinger type inequalities in R N of the form DN,α(a,b):=supu∈W1,N(RN),‖∇u‖LN(RN)a+‖u‖LN(RN)b=1∫RNΦN(α|u|N′)dx.Here N≥2,N′=NN-1,a,b>0,α∈(0,αN] and ΦN(t):=et-∑j=0N-2tjj! where αN:=NωN-11/(N-1) and ω N-1 denotes the surface area of the unit ball in R N . We show the existence of the threshold α ∗ = α ∗ (a, b, N) ∈ [0 , α N ] such that D N,α (a, b) is not attained if α∈ (0 , α ∗ ) and is attained if α∈ (α ∗ , α N ). We also provide the conditions on (a, b) in order that the inequality α ∗ < α N holds.

This paper and [20] treat the existence and nonexistence of stable (resp. outside stable) weak solutions to a fractional Hardy-Henon equation (-Δ)su = jxj'jujp-1u in RN, where 0 < s < 1, ' > -2s, p > 1, N ≥ 1 and N > 2s. In this paper, the nonexistence part is proved for the Joseph-Lundgren subcritical case.

The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In Akahori et al. (Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equation with energy-critical growth at low frequencies, preprint), semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schrödinger and Klein–Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four.

In this paper we study the uniqueness of nontrivial positive solutions for the following second order nonlinear elliptic system: [GRAPHICS] We prove that for sufficiently small beta > 0, the above system has a unique nontrivial positive solution.

In this paper, we consider the existence (and nonexistence) of solutions to −Mλ,Λ±(u″)+V(x)u=f(u)inR where Mλ,Λ+ and Mλ,Λ− denote the Pucci operators with 0<λ≤Λ<∞ V(x) is a bounded function, f(s) is a continuous function and its typical example is a power-type nonlinearity f(s)=|s|p−1s (p>1). In particular, we are interested in positive solutions which decay at infinity, and the existence (and nonexistence) of such solutions is proved.

In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under N ≥ 2. We also show the existence of least energy solution (with the Pohozaev identity) and its mountain pass characterization. For nonradial solutions, we prove the existence of at least one nonradial solution under N ≥ 4 and infinitely many nonradial solutions under either N = 4 or N ≥ 6. We treat both of the zero mass and the positive mass cases.

In this paper, we study the following semilinear elliptic equation Δu=φ(V(x)u-f(x,u(x)))inRN,u∈H1(RN)where N≥ 1 and φ(s) , V(x), f(x, s) are given functions. Under some conditions on φ(s) , V(x) , f(x, s) , we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.

In this paper, we investigate semi-classical bound states for logarithmic Schrödinger type equations with a potential function which has a finite number of singularities of at most logarithmic strength. We construct localized solutions concentrating at a logarithmic type singular point of the potential, and we also characterize the asymptotic limiting profile of the localized solutions. To accomplish these, we develop new penalization techniques for treating the difficulties associated with the non-smoothness of the variational formulation and the singularity of the potentials.

The existence and nonexistence of the minimizer of the L2-constraint minimization problem e(α):=inf{E(u)|u∈H1(RN),∥u∥L2(RN)2=α} are studied. Here, E(u):=12∫RN|∇u|2+V(x)|u|2dx-∫RNF(|u|)dx,V(x) ∈ C(RN) , 0 ≢ V(x) ≤ 0 , V(x) → 0 (| x| → ∞) and F(s)=∫0sf(t)dt is a rather general nonlinearity. We show that there exists α≥ 0 such that e(α) is attained for α> α and e(α) is not attained for 0 < α< α. We study differences between the cases V(x) ≢ 0 and V(x) ≡ 0 , and obtain sufficient conditions for α= 0. In particular, if N= 1 , 2 , then α= 0 , and hence e(α) is attained for all α> 0.

In this paper we study the existence and concentration behaviors of positive solutions to the Kirchhoff type equations -epsilon M-2(epsilon(2-N) integral(RN) vertical bar del u vertical bar(2) dx) Delta u + V(x)u = f(u) in R-N, u is an element of H-1(R-N), N >= 1, where M and V are continuous functions. Under suitable conditions on M and general conditions on f, we construct a family of positive solutions (u epsilon)epsilon is an element of(0,(epsilon) over tilde] which concentrates at a local minimum of V after extracting a subsequence (epsilon(k)).

In this paper, we study the existence of ground state solutions to the nonlinear Kirchhoff type equations -m (parallel to del u parallel to(2)(L2(RN))) Delta u + V(x)u = vertical bar u vertical bar(p-1)u in R-N; u is an element of H-1(R-N); N >= 1 where 1 < p < infinity when N = 1, 2, 1 < p < ( N + 2)/(N - 2) when N >= 3, m : [0, infinity) -> (0, infinity) is a continuous function and V : R-N -> R a smooth function. Under suitable conditions on m(s) and V, it is shown that a ground state solution to the above equation exists.

The existence and nonexistence of minimizers for two minimizing problems are considered. These problems are related to three coupled nonlinear Schrödinger equations which appear in the field of nonlinear optics. We observe the relationship between two minimizing problems and also study the asymptotic behavior when the coefficient standing for the nonlocal nonlinearity tend to 0.