In this article, we continue our investigation of the variable coefficients thin obstacle problem which was initiated in [20], [21]. Using a partial Hodograph-Legendre transform and the implicit function theorem, we prove the higher order Hölder regularity for the regular free boundary, if the associated coefficients are of the corresponding regularity. For the zero obstacle, this yields an improvement of a full derivative for the free boundary regularity compared to the regularity of the coefficients. In the presence of inhomogeneities, we gain three halves of a derivative for the free boundary regularity with respect to the regularity of the inhomogeneity. Further, we show analyticity of the regular free boundary for analytic coefficients. We also discuss the set-up of $W^{1,p}$ coefficients with $p>n+1$ and $L^p$ inhomogeneities. Key ingredients in our analysis are the introduction of generalized Hölder spaces, which allow to interpret the transformed fully nonlinear, degenerate (sub)elliptic equation as a perturbation of the Baouendi-Grushin operator, various uses of intrinsic geometries associated with appropriate operators, the application of the implicit function theorem to deduce (higher) regularity.

In this paper, we study the following fractional nonlinear Schrödinger system $$ \left\{ \begin{array}{ll} (-\Delta)^s u +u=\mu_1 |u|^{2p-2}u+\beta |v|^p|u|^{p-2}u, ~~x\in \mathbb R^N, \\ (-\Delta)^s v +v=\mu_2 |v|^{2p-2}v+\beta |u|^p|v|^{p-2}v, ~~x\in \mathbb R^N, \end{array} \right. $$ where $0 < s < 1, \mu_1 > 0, \mu_2 > 0, 1 < p < 2_s^*/2, 2_s^*=+\infty$ for $N\le 2s$ and $2_s^*=2N/(N-2s)$ for $N > 2s$, and $\beta \in \mathbb R$ is a coupling constant. We investigate the existence and non-degeneracy of proportional positive vector solutions for the above system in some ranges of $\mu_1,\mu_2, p, \beta$. We also prove that the least energy vector solutions must be proportional and unique under some additional assumptions.

We prove existence of suitably defined measure-valued solutions to the homogeneous Dirichlet initial-boundary value problem with a Radon measure as initial datum, for a class of degenerate parabolic equations without strong coerciveness. The notion of solution is natural, since it is obtained by a suitable approximation procedure which can be regarded as a first step towards a continuous dependence on the initial data. Moreover, we also discuss some qualitative properties of the constructed solutions concerning the evolution of their singular part.

We prove a version of the Poincaré–Hopf theorem suitable for strongly indefinite functionals and then apply it to infer a number of bifurcation results in infinite-dimensional Hilbert spaces.

We consider a Kirchhoff type elliptic problem; \begin{equation*} \begin{cases} -\left(1+\alpha \int_{\Omega}|\nabla u|^2dx\right)\Delta u =f(x,u),\ u\ge0\text{ in }\Omega,\\ u=0\text{ on }\partial \Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^2$ is a bounded domain with a smooth boundary $\partial \Omega$, $\alpha > 0$ and $f$ is a continuous function in $\overline{\Omega}\times \mathbb{R}$. Moreover, we assume $f$ has the Trudinger-Moser growth. We prove the existence of solutions of (P), so extending a former result by de Figueiredo-Miyagaki-Ruf [11] for the case $\alpha =0$ to the case $\alpha>0$. We emphasize that we also show a new multiplicity result induced by the nonlocal dependence. In order to prove this, we carefully discuss the geometry of the associated energy functional and the concentration compactness analysis for the critical case.