Si prova la maggior sommabilità del gradiente dei minimi locali di funzionali integrali della forma $${\int }_{\mathrm{\Omega }}f\left(Du\right)dx,$$ dove $f$ soddisfa l'ipotesi di crescita $${\left|\xi \right|}^p-c_1\le f\left(\xi \right)\le c\left(1+{\left|\xi \right|}^q\right),$$ con $1<p<q\le 2$. L'integrando $f$ è $C^2$ e $DDf$ ha crescita $p-2$ dal basso e $q-2$ dall'alto.

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 note, we verify the conjecture of Barron, Evans and Jensen [3] regarding the characterization of viscosity solutions of general Aronsson equations in terms of the properties of associated forward and backwards Hamilton-Jacobi flows. A special case of this result is analogous to the characterization of infinity harmonic functions in terms of convexity and concavity of the functions $r\mapsto {max}_{y\in B_r\left(x\right)}u\left(y\right)$ and $r\mapsto {min}_{y\in B_r\left(x\right)}u\left(y\right)$, respectively.

We apply general Hardy type inequalities, recently obtained by the author. As a consequence we obtain a family of Hardy-Poincaré inequalities with certain constants, contributing to the question about precise constants in such inequalities posed in [3]. We confirm optimality of some constants obtained in [3] and [8]. Furthermore, we give constants for generalized inequalities with the proof of their optimality.