arXiv subjects
Publications and source records attributed to Jacopo Schino.
Given $m \in \mathbb{N} \setminus \{0\}$ and $ρ> 0$, we find solutions $(λ,u)$ to the problem \begin{equation*} \begin{cases} \bigl(-\frac{\mathrm{d}^2}{\mathrm{d} x^2}\bigr)^m u + λG'(u) = F'(u)\\ \int_{\mathbb{R}} K(u) \, \mathrm{d}x = ρ\end{cases} \end{equation*} in the following cases: $m=1$ or $2G(s) = K(s) = s^2$. In the former, we follow a bifurcation argument; in the latter, we use variational methods.
In this paper, we search for normalized solutions to a fractional, nonlinear, and possibly strongly sublinear Schrödinger equation $$(-Δ)^s u + μu = g(u) \quad \hbox{in $\mathbb{R}^N$},$$ under the mass constraint $\int_{\mathbb{R}^N} u^2 \, \mathrm{d}x = m>0$; here, $N\geq 2$, $s \in (0,1)$, and $μ$ is a Lagrange multiplier. We study the case of $L^2$-subcritical nonlinearities $g$ of Berestycki--Lions type, without assuming that $g$ is superlinear at the origin, which allows us to include examples like a logarithmic term $g(u)= u\log(u^2)$ or sublinear powers $g(u)=u^q-u^r$, $0<r<1<q$. Due to the generality of $g$ and the fact that the energy functional might be not well-defined, we implement an approximation process in combination with a Lagrangian approach and a new Pohožaev minimization in the product space, finding a solution for large values of $m$. In the sublinear case, we are able to find a solution for each $m$. Several insights on the concepts of minimality are studied as well. We highlight that some of the results are new even in the local setting $s=1$ or for $g$ superlinear.
Via a new inequality à la Gagliardo--Nirenberg, we prove the existence and nonexistence of solutions to \begin{equation*} \begin{cases} (-Δ)^s u + \fracμ{|y|^{2s}} u + λu = f(u), \quad \mathbb{R}^N \ni x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\ \int_{\mathbb{R}^N} u^2 \, \mathrm{d}x = ρ\end{cases} \end{equation*} in the mass-critical and mass-subcritical regimes, where $s>0$, $N \ge K \ge 2$, $μ\in \mathbb{R}$ belongs to a specific range, $ρ>0$ is given a priori, and $λ\in \mathbb{R}$ is unknown. Additionally, we obtain similar results for the problem above with $μ=0$ and $N \ge 1$ as well as a related curl-curl equation. Finally, we provide a thorough insight into the threshold for $ρ$ that divides the scenarios of negative and zero least energy.
We consider the singular elliptic problem of the form \[ -Δu + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), \] where the coefficients are allowed to have low regularity. Under natural spectral assumptions on $-Δ+V$, geometric assumptions on the manifold $M$ ensuring the Sobolev embedding $H^1(M)\hookrightarrow L^{2^*}(M)$, and a suitable global integrability/smallness condition involving $\mathcal{A}$, $\mathcal{B}$, and a function $ψ\in H^1(M)$, we prove the existence of a nonnegative finite-energy supersolution. If, in addition, the Ricci curvature is nonnegative and $\mathcal{B}\ge 0$, we obtain a positive finite-energy solution. The proof relies on a family of $\varepsilon$-regularized problems, mountain pass arguments, and a limiting procedure in which Harnack's inequality plays a crucial role in handling the singular term on noncompact manifolds. We also prove a nonexistence result showing that the global integrability condition on $\mathcal{A}$ is, in a precise sense, necessary for the existence of nonnegative supersolutions.
In the paper, we prove the existence of a positive and essentially bounded solution to a Lichnerowicz equation in the Einstein-scalar field theory on a closed manifold with non-constant mean curvature. In particular, the non-constant mean curvature gives rise to supercritical terms in the equation, on top of singular ones. We employ a recent fixed-point argument, which involves sub- and supersolutions. Additionally, we provide several conditions on the coefficients in the equation that prevent the existence of positive classical solutions.
We look for travelling wave fields $$ E(x,y,z,t)= U(x,y) \cos(kz+ωt)+ \widetilde U(x,y)\sin(kz+ωt),\quad (x,y,z)\in\mathbb{R}^3,\, t\in\mathbb{R}, $$ satisfying Maxwell's equations in a nonlinear and cylindrically symmetric medium. We obtain a sequence of solutions with diverging energy that is different from that obtained by McLeod, Stuart, and Troy. In addition, we consider a more general nonlinearity, controlled by an \textit{N}-function.
Via a constrained minimization, we find a solution $(λ,u)$ to the problem \begin{equation*} \begin{cases} (-Δ)^m u+\fracμ{|x|^{2m}}u + λu = ηu^3 + g(u)\\ \int_{\mathbb{R}^{2m}} u^2 \, dx = ρ\end{cases} \end{equation*} with $1 \le m \in \mathbb{N}$, $μ,η\ge 0$, $ρ> 0$, and $g$ having exponential critical growth at infinity and mass supercritical growth at zero.
We study the existence of solutions $(\underline u,λ_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -Δu + λu = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and the dynamics of the solutions to \[ \begin{cases} \mathrm{i} \partial_t Ψ+ ΔΨ= f(Ψ)\\ Ψ(\cdot,0) = ψ_0 \in H^1(\mathbb{R}^N; \mathbb{C}) \end{cases} \] with $ψ_0$ close to $\underline u$. Here, the nonlinear term $f$ has mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution, the orbital stability of all such solutions, the existence of a second solution with higher energy, and the strong instability of such a solution.
The present paper establishes a local well-posed result for piecewise regular solutions with single shock of scalar balance laws with singular integral of convolution type kernels. In a neighborhood of the shock curve, a detailed description of the solution is provided for a general class of initial data.
We are interested in the existence of normalized solutions to the problem \begin{equation*} \begin{cases} (-Δ)^m u+\fracμ{|y|^{2m}}u + λu = g(u), \quad x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\ \int_{\mathbb{R}^N} |u|^2 \, dx = ρ> 0, \end{cases} \end{equation*} in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the $L^2$-ball. Moreover, we find also a solution to the related curl-curl problem \begin{equation*} \begin{cases} \nabla\times\nabla\times\mathbf{U}+λ\mathbf{U}=f(\mathbf{U}), \quad x \in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|\mathbf{U}|^2\,dx=ρ, \end{cases} \end{equation*} which arises from the system of Maxwell equations and is of great importance in nonlinear optics.
We look for solutions to the Schrödinger equation \[ -Δu + λu = g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint $\int_{\mathbb{R}^N}|u|^2\,dx = ρ^2$, with $N\ge2$. The behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\lim_{s\to0}g(s)/s = -\infty$, which includes the case \[ g(s) = αs \ln s^2 + μ|s|^{p-2} s \] with $α> 0$ and $μ\in \mathbb{R}$, $2 < p \le 2^*$ properly chosen. We consider a family of approximating problems that can be set in $H^1(\mathbb{R}^N)$ and the corresponding least-energy solutions, then we prove that such a family of solutions converges to a least-energy one to the original problem. Additionally, under certain assumptions about $g$ that allow us to work in a suitable subspace of $H^1(\mathbb{R}^N)$, we prove the existence of infinitely many solutions.
We look for multiple solutions $\mathbf{U}\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem \[ \nabla\times\nabla\times\mathbf{U}=h(x,\mathbf{U}),\qquad x\in\mathbb{R}^3, \] with a nonlinear function $h\colon\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R}^3$ which is critical in $\mathbb{R}^3$, i.e., $h(x,\mathbf{U})=|\mathbf{U}|^4\mathbf{U}$, or has subcritical growth at infinity. If $h$ is radial in $\mathbf{U}$ and $a=1$ below, then we show that the solutions to the problem above are in one-to-one correspondence with the solutions to the following Schrödinger equation \[ -Δu+\frac{a}{r^2}u=f(x,u),\qquad u\colon\mathbb{R}^3\to \mathbb{R}, \] where $x=(y,z)\in \mathbb{R}^2\times \mathbb{R}$, $r=|y|$ and $a \ge 0$. In the critical case, the multiplicity problem for the latter equation has been studied only in the autonomous case $a=0$ and the available methods seem to be insufficient for the problem involving the singular potential, i.e., $a\neq 0$, due to the lack of conformal invariance. Therefore we develop methods for the critical curl-curl problem and show the multiplicity of bound states for both equations. In the subcritical case, instead, studying the Schrödinger equation in higher dimensions, we find infinitely many bound states for both problems.
We study the following nonlinear Schrödinger equation with a forth order dispersion term \[ Δ^2u-βΔu=g(u) \quad \text{in } \mathbb{R}^N \] in the positive and zero mass regimes: in the former, $N\geq 2$ and $β> -2\sqrt{m}$, where $m>0$ depends on $g$; in the latter, $N\geq 3$ and $β>0$. In either regimes, we find an infinite sequence of solutions under rather generic assumptions about $g$; if $N=2$ in the positive mass case, or $N=4$ in the zero mass case, we need to strengthen such assumptions. Our approach is variational.
The existence of ground states and (multiple) bound states to semilinear time-independent Maxwell and Schrödinger equations, with or without $L^2$-constraints, is investigated.
We look for ground state solutions to the Schrödinger-type system \[ \begin{cases} -Δu_j + λ_j u_j = \partial_jF(u)\\ \int_{\rn} u_j^2 \, dx = a_j^2\\ (λ_j,u_j) \in \mathbb{R} \times H^1(\mathbb{R}^N) \end{cases} j \in \{1,\dots,M\} \] with $N,M\ge1$, where $a=(a_1,\dots,a_M) \in ]0,\infty[^M$ is prescribed and $(λ,u) = (λ_1,\dots,λ_M,u_1,\dots u_M)$ is the unknown. We provide generic assumptions about the nonlinearity $F$ which correspond to the $L^2$-subcritical and $L^2$-critical cases, i.e., when the energy is bounded from below for all or some values of $a$. Making use of a recent idea, we minimize the energy over the constraint $\Set{\left|u_j\right|_{L^2}\le a_j \text{ for all } j}$ and then provide further assumptions that ensure $|u_j|_{L^2}=a_j$.
We look for least energy solutions to the cooperative systems of coupled Schrödinger equations \begin{equation*} \begin{cases} -Δu_i + λ_i u_i = \partial_iG(u)\quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, u_i \in H^1(\mathbb{R}^N), \int_{\mathbb{R}^N} |u_i|^2 \, dx \leq ρ_i^2 \end{cases} i\in\{1,\dots,K\} \end{equation*} with $G\geq 0$, where $ρ_i>0$ is prescribed and $(λ_i, u_i) \in \mathbb{R} \times H^1 (\mathbb{R}^N)$ is to be determined, $i\in\{1,\dots,K\}$. Our approach is based on the minimization of the energy functional over a linear combination of the Nehari and Pohožaev constraints intersected with the product of the closed balls in $L^2(\mathbb{R}^N)$ of radii $ρ_i$, which allows to provide general growth assumptions about $G$ and to know in advance the sign of the corresponding Lagrange multipliers. We assume that $G$ has at least $L^2$-critical growth at $0$ and admits Sobolev critical growth. The more assumptions we make about $G$, $N$, and $K$, the more can be said about the minimizers of the corresponding energy functional. In particular, if $K=2$, $N\in\{3,4\}$, and $G$ satisfies further assumptions, then $u=(u_1,u_2)$ is normalized, i.e., $\int_{\mathbb{R}^N} |u_i|^2 \, dx=ρ_i^2$ for $i\in\{1,2\}$.
We survey recent results concerning ground states and bound states $u\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times u)+V(x)u= f(x,u) \quad\hbox{ in } \mathbb{R}^3,$$ which originates from the nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\nabla\times(\nabla\times \cdot)$. The growth of the nonlinearity $f$ is superlinear and subcritical at infinity or purely critical and we demonstrate a variational approach to the problem involving the generalized Nehari manifold. We also present some refinements of known results.
We look for ground states and bound states $E:\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times E)= f(x,E) \qquad\hbox{in } \mathbb{R}^3$$ which originates from nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\nabla\times(\nabla\times \cdot)$. The growth of the nonlinearity $f$ is controlled by an $N$-function $Φ:\mathbb{R}\to [0,\infty)$ such that $\displaystyle\lim_{s\to 0}Φ(s)/s^6=\lim_{s\to+\infty}Φ(s)/s^6=0$. We prove the existence of a ground state, i.e. a least energy nontrivial solution, and the existence of infinitely many geometrically distinct bound states. We improve previous results concerning ground states of curl-curl problems. Multiplicity results for our problem have not been studied so far in $\mathbb{R}^3$ and in order to do this we construct a suitable critical point theory. It is applicable to a wide class of strongly indefinite problems, including this one and Schrödinger equations.