@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}, abstractNote={Abstract We look for solutions to the Schrödinger equation $$\begin{aligned} -\Delta u + \lambda u = g(u) \quad \text {in } \mathbb {R}^N \end{aligned}$$ - Δ u + λ u = g ( u ) in R N coupled with the mass constraint $$\int _{\mathbb {R}^N}|u|^2\,dx = \rho ^2$$ ∫ R N | u | 2 d x = ρ 2 , with $$N\ge 2$$ N ≥ 2 . The behaviour of g at the origin is allowed to be strongly sublinear, i.e., $$\lim _{s\rightarrow 0}g(s)/s = -\infty $$ lim s → 0 g ( s ) / s = - ∞ , which includes the case $$\begin{aligned} g(s) = \alpha s \ln s^2 + \mu |s|^{p-2} s \end{aligned}$$ g ( s ) = α s ln s 2 + μ | s | p - 2 s with $$\alpha > 0$$ α > 0 and $$\mu \in \mathbb {R}$$ μ ∈ R , $$2 < p \le 2^*$$ 2 < p ≤ 2 ∗ properly chosen. We consider a family of approximating problems that can be set in $$H^1(\mathbb {R}^N)$$ H 1 ( 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 $$H^1(\mathbb {R}^N)$$ H 1 ( R N ) , we prove the existence of infinitely, many solutions.}, number={5}, journal={CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS}, author={Mederski, Jaroslaw and Schino, Jacopo}, year={2024}, month={Jun} }
@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
$$\nabla\times\nabla\times\mathbf{U}=h(x,\mathbf{U}),\qquad x\in\mathbb{R}^3,$$
with a nonlinear function $h:\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R}^3$ 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 Schrodinger equation
$$-\Delta u+\frac{a}{r^2}u=f(x,u),\qquad u:\mathbb{R}^N\to\mathbb{R},$$
where $x=(y, z)\in\mathbb{R}^K\times\mathbb{R}^{N-K}$, $N>K\ge2$, $r=|y|$ and $a>a_0\in(−∞, 0]$. In the subcritical case, applying a critical point theory to the Schrodinger equation above, we find infinitely many bound states for both problems. In the critical case, however, 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. $a\ne0$, 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 $a=1$.}, 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}, number={3}, journal={NONLINEARITY}, author={d'Avenia, Pietro and Pomponio, Alessio and Schino, Jacopo}, year={2023}, month={Mar}, pages={1743–1775} }