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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 look for travelling wave fields $$ E(x,y,z,t)= U(x,y) \cos(kz+\omega t)+ \widetilde U(x,y)\sin(kz+\omega t),\quad (x,y,z)\in\mathbb{R}^3,\, t\in\mathbb{R}, $$ 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., Comment: 18 pages

We study the existence of solutions $(\underline u,\lambda_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -\Delta u + \lambda u = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and the dynamics of the solutions to \[ \begin{cases} \mathrm{i} \partial_t \Psi + \Delta \Psi = f(\Psi)\\ \Psi(\cdot,0) = \psi_0 \in H^1(\mathbb{R}^N; \mathbb{C}) \end{cases} \] with $\psi_0$ 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., Comment: 27 pages, minor corrections

We look for solutions to the Schr\"odinger equation \[ -\Delta u + \lambda u = g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint $\int_{\mathbb{R}^N}|u|^2\,dx = \rho^2$, 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 \[ g(s) = \alpha s \ln s^2 + \mu |s|^{p-2} s \] with $\alpha > 0$ and $\mu \in \mathbb{R}$, $2 < p \le 2^*$ 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 are interested in the existence of normalized solutions to the problem \begin{equation*} \begin{cases} (-\Delta)^m u+\frac{\mu}{|y|^{2m}}u + \lambda u = g(u), \quad x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\ \int_{\mathbb{R}^N} |u|^2 \, dx = \rho > 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}+\lambda\mathbf{U}=f(\mathbf{U}), \quad x \in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|\mathbf{U}|^2\,dx=\rho, \end{cases} \end{equation*} which arises from the system of Maxwell equations and is of great importance in nonlinear optics.

The existence of ground states and (multiple) bound states to semilinear time-independent Maxwell and Schr\"odinger equations, with or without $L^2$-constraints, is investigated., Comment: Ph.D. thesis (discussed in September 2021). Improved versions of Chapters 6 and 7 available at arXiv:2101.03076 and arXiv:2101.02611 respectively. Chapters 3 and 4 are based on arXiv:1901.05776 and arXiv:2006.03565 respectively. The results in Chapter 8 are new

We study the following nonlinear Schr\"odinger equation with a forth order dispersion term \[ \Delta^2u-\beta\Delta u=g(u) \quad \text{in } \mathbb{R}^N \] in the positive and zero mass regimes: in the former, $N\geq 2$ and $\beta > -2\sqrt{m}$, where $m>0$ depends on $g$; in the latter, $N\geq 3$ and $\beta>0$. 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., Comment: 28 pages, revised version

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., Comment: to appear in the special issue of Minimax Theory and its Applications: The Nehari manifold: theory, applications, related topics. arXiv admin note: substantial text overlap with arXiv:1901.05776

We look for ground state solutions to the Schr\"odinger-type system \[ \begin{cases} -\Delta u_j + \lambda_j u_j = \partial_jF(u)\\ \int_{\rn} u_j^2 \, dx = a_j^2\\ (\lambda_j,u_j) \in \mathbb{R} \times H^1(\mathbb{R}^N) \end{cases} j \in \{1,\dots,M\} \] with $N,M\ge1$, where $a=(a_1,\dots,a_M) \in ]0,\infty[^M$ is prescribed and $(\lambda,u) = (\lambda_1,\dots,\lambda_M,u_1,\dots u_M)$ 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 $|u_j|_{L^2}=a_j$., Comment: To appear in Adv. Differential Equations. 21 pages, a few inaccuracies corrected (especially in Proposition 1.7). Comments are welcome

We look for least energy solutions to the cooperative systems of coupled Schr\"odinger equations \begin{equation*} \begin{cases} -\Delta u_i + \lambda_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 \rho_i^2 \end{cases} i\in\{1,\dots,K\} \end{equation*} with $G\geq 0$, where $\rho_i>0$ is prescribed and $(\lambda_i, u_i) \in \mathbb{R} \times H^1 (\mathbb{R}^N)$ 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 Poho\v{z}aev constraints intersected with the product of the closed balls in $L^2(\mathbb{R}^N)$ of radii $\rho_i$, 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., $\int_{\mathbb{R}^N} |u_i|^2 \, dx=\rho_i^2$ for $i\in\{1,2\}$., Comment: To appear in Calc. Var. Partial Differential Equations

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 Schr\"odinger equation \[ -\Delta u+\frac{a}{r^2}u=f(x,u),\qquad u\colon\mathbb{R}^3\to \mathbb{R}, \] 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 Schr\"odinger equation in higher dimensions, we find infinitely many bound states for both problems.

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 $\Phi:\mathbb{R}\to [0,\infty)$ such that $\displaystyle\lim_{s\to 0}\Phi(s)/s^6=\lim_{s\to+\infty}\Phi(s)/s^6=0$. 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 Schr\"odinger equations., Comment: to appear in Archive for Rational Mechanics and Analysis