@article{bieganowski_mederski_schino_2024, title={Normalized Solutions to at Least Mass Critical Problems: Singular Polyharmonic Equations and Related Curl-Curl Problems}, volume={34}, ISSN={["1559-002X"]}, DOI={10.1007/s12220-024-01770-y}, abstractNote={Abstract We are interested in the existence of normalized solutions to the problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^m u+\frac{\mu }{|y|^{2m}}u + \lambda u = g(u), \quad x = (y,z) \in \mathbb {R}^K \times \mathbb {R}^{N-K}, \\ \int _{\mathbb {R}^N} |u|^2 \, dx = \rho > 0, \end{array}\right. } \end{aligned}$$ ( - Δ ) m u + μ | y | 2 m u + λ u = g ( u ) , x = ( y , z ) ∈ R K × R N - K , ∫ R N | u | 2 d x = ρ > 0 , in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the $$L^2$$ L 2 -ball. Moreover, we find also a solution to the related curl–curl problem $$\begin{aligned} {\left\{ \begin{array}{ll} \nabla \times \nabla \times \textbf{U}+\lambda \textbf{U}=f(\textbf{U}), \quad x \in \mathbb {R}^N,\\ \int _{\mathbb {R}^N}|\textbf{U}|^2\,dx=\rho ,\\ \end{array}\right. } \end{aligned}$$ ∇ × ∇ × U + λ U = f ( U ) , x ∈ R N , ∫ R N | U | 2 d x = ρ , which arises from the system of Maxwell equations and is of great importance in nonlinear optics.}, number={10}, journal={JOURNAL OF GEOMETRIC ANALYSIS}, author={Bieganowski, Bartosz and Mederski, Jaroslaw and Schino, Jacopo}, year={2024}, month={Oct} } @article{mederski_schino_2024, title={Normalized solutions to Schrödinger equations in the strongly sublinear regime}, volume={63}, ISSN={["1432-0835"]}, DOI={10.1007/s00526-024-02729-1}, number={5}, journal={CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS}, author={Mederski, Jaroslaw and Schino, Jacopo}, year={2024}, month={Jun} } @article{bociu_ftaka_nguyen_schino_2024, title={Piecewise regular solutions to scalar balance laws with singular nonlocal sources}, volume={409}, ISSN={["1090-2732"]}, url={https://doi.org/10.1016/j.jde.2024.07.004}, DOI={10.1016/j.jde.2024.07.004}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Bociu, Lorena and Ftaka, Evangelia and Nguyen, Khai T. and Schino, Jacopo}, year={2024}, month={Nov}, pages={181–222} } @article{gaczkowski_mederski_schino_2023, title={MULTIPLE SOLUTIONS TO CYLINDRICALLY SYMMETRIC CURL-CURL PROBLEMS AND RELATED SCHRODINGER EQUATIONS WITH SINGULAR POTENTIALS}, volume={55}, ISSN={["1095-7154"]}, DOI={10.1137/22M1494786}, abstractNote={.We look for multiple solutions $\mathbf{U}:\mathbb{R}^3\to \mathbb{R}^3$ to the curl-curl problem $\nabla \times \nabla \times \mathbf{U}=h(x,\mathbf{U})$ , $x\in \mathbb{R}^3$ , with a nonlinear function $h:\mathbb{R}^3\times \mathbb{R}^3\to \mathbb{R}^3$ 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: $-\Delta u+\frac{a}{r^2}u=f(x,u)$ , $u:\mathbb{R}^3\to \mathbb{R}$ , where $x=(y,z)\in \mathbb{R}^2\times \mathbb{R}$ , $r=|y|$ , and $a\geq 0$ . 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.Keywordscylindrical symmetrycurl-curl problemssingular potentialsmultiple solutionsMSC codes35Q6035J2078A25}, number={5}, journal={SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, author={Gaczkowski, Michal and Mederski, Jaroslaw and Schino, Jacopo}, year={2023}, pages={4425–4444} } @article{d'avenia_pomponio_schino_2023, title={Radial and non-radial multiple solutions to a general mixed dispersion NLS equation}, volume={36}, ISSN={["1361-6544"]}, DOI={10.1088/1361-6544/acb62d}, abstractNote={Abstract We study the following nonlinear Schrödinger equation with a fourth-order dispersion term Δ2u−βΔu=g(u)in RN in the positive and zero mass regimes: in the former, N⩾2 and β>−2m , where m > 0 depends on g ; in the latter, N⩾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.}, number={3}, journal={NONLINEARITY}, author={d'Avenia, Pietro and Pomponio, Alessio and Schino, Jacopo}, year={2023}, month={Mar}, pages={1743–1775} }