Futoshi Takahashi

Blow up Points and the Morse Indices of Solutions to the Liouville Equation in Two-Dimension

Abstract We consider the Liouville equation −Δu = λeu in Ω, u =0 on∂Ω, on a smooth bounded domain Ω in ℝ2, where λ > 0 is a parameter. Let {un} be an m-point blowing up solution sequence of the problem for λ = λn ↓ 0, which satisfies for m ∈ℕ. We prove that the number of blow up points m is less than or equal to the Morse index of un for n sufficiently large. As a corollary, we show that if a solution un of Morse index one has the property that , then the number of blow up points of the sequence is exactly one. Note that in the last result, we do not need any geometrical assumption such as the convexity of the domain.

Extremal solutions to Liouville–Gelfand type elliptic problems with nonlinear Neumann boundary conditions

Consider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞) is a smooth, strictly positive, convex, increasing function with superlinear at +∞, and λ > 0 is a parameter. In this paper, after introducing a suitable notion of weak solutions, we prove several properties of extremal solutions u* corresponding to λ = λ*, called an extremal parameter, such as regularity, uniqueness, and the existence of weak eigenfunctions associated to the linearized extremal problem.

Chapter 4 - Nonlinear Eigenvalue Problem with Quantization

This chapter describes nonlinear eigenvalues problem with quantization. The chapter explains energy quantization, mass quantization, and eigenvalues problem. Quantized blowup mechanism is widely observed in nonlinear problems derived from physical principles and phenomena. Energy quantization is a typical case described by the Dirichlet norm. Energy quantization introduces several examples and studies the harmonic map case. Energy quantization arises if the problem is provided with the scaling invariance of energy—for example, harmonic map, semilinear elliptic equation involved by the critical Sobolev exponent, and H-systems. Energy identity with bubbling occurs to the noncompact solution sequence in these cases. Mass quantization is concerned with the L 1 norm. It is associated with a different exponent from that of energy quantization but it is derived from the scaling invariance of the problem similarly, where the location of blowup points are prescribed by the linear part.

Asymptotic Nondegeneracy of Least Energy Solutions to an Elliptic Problem with Critical Sobolev Exponent

Abstract Consider the problem -∆u = N(N - 2)up + εk(x)u in Ω, u > 0 in Ω, u|∂Ω = 0 where Ω is a smooth bounded domain in ℝN (N ≥ 6), p = (N +2)/(N - 2), ε > 0 and k ∈ C2(Ω̅). Under certain assumptions, we prove the nondegeneracy of least energy solutions to the above problem as Ɛ →0. This is an extension of the recent work of Grossi [9].

Improved Rellich Type Inequalities in {\mathbb {R}}^{N}

We consider the second or higher-order Rellich inequalities on the whole space \({{\mathbb {R}}}^N\). In spite of the lack of the Poincare inequality on the whole space, we show that the higher-order Rellich inequalities with optimal constants can be improved, by adding explicit remainder terms to the inequalities.

Non-degeneracy of least-energy solutions for an elliptic problem with nearly critical nonlinearity

We consider the problem −Δ u = c 0 K ( x ) u pe , u > 0 in Ω , u = 0 on δΩ , where Ω is a smooth, bounded domain in ℝ N , N ≥ 3, c 0 = N ( N − 2), pe = ( N + 2)/( N − 2) − e and K is a smooth, positive function on . We prove that least-energy solutions of the above problem are non-degenerate for small e > 0 under some assumptions on the coefficient function K . This is a generalization of the recent result by Grossi for K ≡ 1, and needs precise estimates and a new argument.

Asymptotic behavior of least energy solutions for a biharmonic problem with nearly critical growth

as ε → +0, where Ω is a smooth bounded domain in RN (N 5), c0 = (N − 4)(N − 2)N (N + 2), ε > 0 is a small positive parameter, pε = p − ε, p = (N + 4)/(N − 4) is the critical Sobolev exponent from the view point of the Sobolev embedding H2 ∩ H1 0 (Ω) ↪→ Lp+1(Ω), and K ∈ C2(Ω) is a given positive function. Boundary condition of (Pε,K) is called as the Navier boundary condition. For the second-order Laplacian-case problem ⎧⎨ ⎩ −Δu = N (N − 2)K(x)u(N+2)/(N −2)−ε in Ω, u > 0 in Ω, u = 0 on ∂Ω,

Finsler Hardy–Kato's inequality

We prove an improved version of the trace-Hardy inequality, so-called Katos inequality, on the half-space in Finsler context. The resulting inequality extends the former one obtained by \cite{AFV} in Euclidean context. Also we discuss the validity of the same type of inequalities on open cones.

Weighted Hardy's inequality in a limiting case and the perturbed Kolmogorov equation

ABSTRACT In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant on the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein–Uhlenbeck operator perturbed by a critical singular potential in a two-dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the limiting case of results which are established in the work of Goldstein et al. [Weighted Hardys inequality and the Kolmogorov equation perturbed by an inverse-square potential. Appl Anal. 2012;91(11):2057–2071] and Hauer and Rhandi [A weighted Hardy inequality and nonexistence of positive solutions. Arch Math. 2013;100:273–287] in the non-critical cases, and are also considered as extensions of a result in Cabré and Martel [Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potential singulier. C R Acad Sci Paris Sér I Math. 1999;329:973–978] to the Kolmogorov operator case perturbed by a critical singular potential.

Nonexistence of Multi-bubble Solutions for a Higher Order Mean Field Equation on Convex Domains

In this note, we are concerned with the blowing-up behavior of solutions to the 2p-th order mean field equation under the Navier boundary condition: Open image in new window where Ω is a smooth bounded domain in ℝ2p for p∈ℕ. By using a new Pohozaev type identity for the Green function of (−Δ) p under the Navier boundary condition, we show that the set of blow up points for any blowing-up solution sequence must be a singleton on convex domains, under some assumptions on the weight function V.