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A generalisation of the Gagliardo--Nirenberg Inequality with applications to mass-critical and mass-subcritical elliptic equations
Bartosz Bieganowski, Jacopo Schino
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), \…
Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds
Bartosz Bieganowski, Pietro d'Avenia, Jacopo Schino
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…
A Lichnerowicz equation in the Einstein-scalar field theory on non-CMC closed manifolds
Bartosz Bieganowski, Pietro d'Avenia, Jacopo Schino +1
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-con…
A Pohožaev minimization for normalized solutions: fractional sublinear equations of logarithmic type
Marco Gallo, Jacopo Schino
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}…
Normalized solutions to polyharmonic equations with Hardy-type potentials and exponential critical nonlinearities
Bartosz Bieganowski, Olímpio Hiroshi Miyagaki, Jacopo Schino
Via a constrained minimization, we find a solution 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…
Piecewise regular solutions to scalar balance laws with singular nonlocal sources
Lorena Bociu, Evangelia Ftaka, Khai T. Nguyen +1
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.…