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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.

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., 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 look for ground state solutions to the Schr��dinger-type system \[ \begin{cases} -��u_j + ��_j u_j = \partial_jF(u)\\ \int_{\rn} u_j^2 \, dx = a_j^2\\ (��_j,u_j) \in \mathbb{R} \times H^1(\mathbb{R}^N) \end{cases} j \in \{1,\dots,M\} \] with $N,M\ge1$, where $a=(a_1,\dots,a_M) \in ]0,\infty[^M$ is prescribed and $(��,u) = (��_1,\dots,��_M,u_1,\dots u_M)$ 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 $|u_j|_{L^2}=a_j$., To appear in Adv. Differential Equations. 21 pages, a few inaccuracies corrected (especially in Proposition 1.7). Comments are welcome

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 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 has subcritical growth at infinity or is critical in $\mathbb{R}^3$, i.e. $h(x,\mathbf{U})=|\mathbf{U}|^4\mathbf{U}$. If $h$ is radial in $\mathbf{U}$, $N=3$, $K=2$ 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��dinger equation \[ -��u+\frac{a}{r^2}u=f(x,u),\qquad u\colon\mathbb{R}^N\to \mathbb{R}, \] where $x=(y,z)\in \mathbb{R}^K\times \mathbb{R}^{N-K}$, $N>K\geq 2$, $r=|y|$ and $a>a_0\in(-\infty,0]$. 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 case $N=3$, $K=2$ and $a=1$. In the subcritical case, instead, applying a critical point theory to the Schr��dinger equation above, we find infinitely many bound states for both problems.