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Si studiano soluzioni positive dell’equazione $-{ϵ}^2\mathrm{\Delta }u+u=u^p$ in $\mathrm{\Omega }$, dove $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ , $p>1$ ed $ϵ$ è un piccolo parametro positivo. Si impongono in genere condizioni al bordo di Neumann. Quando $ϵ$ tende a zero, dimostriamo esistenza di soluzioni che si concentrano su curve o varietà.

We prove existence of positive solutions for the equation $-{\mathrm{△}}_{g_0}u+u=\left(1+ϵK\left(x\right)\right)u^{2^{*}-1}$ on $S^n$, arising in the prescribed scalar curvature problem. is the Laplace-Beltrami operator on $S^n$, $2^{\mathrm{\ast }}$ is the critical Sobolev exponent, and $ϵ$ is a small parameter. The problem can be reduced to a finite dimensional study which is performed with Morse theory.

On the unit disk $B_1\subset {ℝ}^2$ we study the Moser-Trudinger functional $E\left(u\right)={\int }_{B_1}\left(e^{u^2}-1\right)dx,u\in H_0^1\left(B_1\right)$ and its restrictions ${E|}_{M_{\Lambda }}$, where $M_{\Lambda }:=\{u\in H_0^1\left(B_1\right){:\parallel u\parallel }_{H_0^1}^2=\Lambda \}$ for $\Lambda >0$. We prove that if a sequence $u_k$ of positive critical points of ${E|}_{M_{{\Lambda }_k}}$ (for some ${\Lambda }_k>0$) blows up as $k\to \infty$, then ${\Lambda }_k\to 4\pi$, and $u_k\to 0$ weakly in $H_0^1\left(B_1\right)$ and strongly in $C_{\mathrm{loc}}^1({\overline{B}}_1\setminus \left\{0\right\})$. Using this fact we also prove that when $\Lambda$ is large enough, then ${E|}_{M_{\Lambda }}$ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

In this preliminary Note we outline the results of the forthcoming paper [2] dealing with a class on nonlinear Schrödinger equations with potentials vanishing at infinity. Working in weighted Sobolev spaces, the existence of a ground state is proved. Furthermore, the behaviour of such a solution, as the Planck constant tends to zero (semiclassical limit), is studied proving that it concentrates at a point.

In this paper we consider Riemannian manifolds $(M^n,g)$ of dimension $n\ge 5$, with semi-positive $Q$-curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive $Q$-curvature. Modifying the test function construction of Esposito-Robert, we show...

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V\left(x\right)\sim {\left|x\right|}^{-\alpha }$, $0<\alpha <2$, and $K\left(x\right)\sim {\left|x\right|}^{-\beta }$, $\beta >0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\epsilon }$ belonging to $W^{1,2}\left({ℝ}^N\right)$ is proved under the assumption that $\sigma <p<(N+2)/\left(N-2\right)$ for some $\sigma ={\sigma }_{N,\alpha ,\beta }$. Furthermore, it is shown that $v_{\epsilon }$ are spikes concentrating at a minimum point of $𝒜=V^{\theta }{K}^{-2/\left(p-1\right)}$, where $\theta =(p+1)/\left(p-1\right)-1/2$.

In this paper we perform a fine blow up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere $({𝕊}^n,h)$. We derive from this analysis some a priori estimates in dimension $5$ and $6$. On ${𝕊}^5$ these a priori estimates, combined with the perturbation result in the first part of the present work, allow us to obtain some existence result using a continuity method. On ${𝕊}^6$ we prove the existence of at least one...

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation...