Sep 2026· Annali di Matematica Pura ed Applicata· 0 citations· 22 references
Abstract
We study the existence and multiplicity of sign-changing solutions of the semilinear elliptic equation
$$ -\Delta _g u + u = f(u) \quad \text {on } \mathbb {S}^2, $$
where
$$(\mathbb {S}^2,g)$$
denotes the two-dimensional unit sphere endowed with a smooth Riemannian metric,
$$\Delta _g$$
is the Laplace–Beltrami operator, and
f
may exhibit critical exponential growth in the sense of the Trudinger-Moser inequality. Our method combines a variational construction on geodesic lunes with a global reflection scheme. We first solve a Dirichlet problem on a fixed lune and show that the corresponding mountain-pass level remains strictly below the Trudinger-Moser threshold. Reflecting this solution across meridians and alternating its sign yields a sequence of global solutions with prescribed nodal patterns and quantized energy levels when
f
is odd. For small nonsymmetric perturbations of the nonlinearity, we employ a Lyapunov-Schmidt reduction to adjust the resulting multi-lune configurations, thereby proving the persistence of these sign-changing solutions.
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
We study the following curvature equation with two singular sources on a flat torus $$ \Delta u+e^u=8\pi(\delta_p+\delta_{-p})\quad\text{on } E_\tau:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau), $$ where $\delta_p$ denotes the Dirac measure at $p$. The case $\wp'(2p)=0$ was solved by Kuo (J. Differential Geom. 2026). In thi...
In this paper, we investigate the analytic properties of the Hardy-type operator $-\Delta + \mu H_0$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, where $\mu \geq -1$ and $H_0$ is a critical Hardy potential at infinity--arising naturally from discrete Hardy inequalities. We derive sharp fixed-pole Green...
We prove that the energy functional associated with surfaces of prescribed mean curvature in $\mathbb{R}^3$ exhibits a convexity property when restricted to maps from the unit $2$-disk having small Dirichlet energy and a fixed boundary value, provided that the $3$-form $H \cdot \text{vol}_{\mathbb{R}^3}$ has a primitiv...
In this paper, we consider the inequality $\Delta_pu+\Delta_qu+u^s|\nabla u|^t\leq 0$ on geodesically complete noncompact Riemannian manifolds. By a test function argument, we establish Liouville-type theorems under the upper bound of volume of geodesic ball. In the Euclidean space $\mathbb{R}^n$, we obtain new nonexis...
Let $\Omega\subset\mathbb R^2$ be a bounded smooth domain. We prove that the weak Dirichlet-to-Neumann map for $-\Delta+V$ uniquely determines every complex-valued potential $V\in L^p(\Omega)$, $p>1$, provided that zero is not a Dirichlet eigenvalue. This extends the previously known range $p>4/3$ to all $p>1$. The pro...
Cătălin I. Cârstea, Jenn-Nan Wang· 0 citations
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