Volume 3, Issue 1

Abstract. This article completes the picture in the study of positive radial solutions in the function space 𝒟1,2(ℝN)∩L2(ℝN,|x|-αdx)∩Lp(ℝN)${{\mathcal {D}^{1,2}({\mathbb {R}^N}) \cap L^2({{\mathbb {R}^N}, | x |^{-\alpha } dx})\cap L^p({\mathbb {R}^N})}}$ for the equation -Δu+A|x|αu=up-1inℝN∖{0}withN≥3,A>0,α>0,p>2.$- \Delta u + \frac{A}{| x |^\alpha } u = u^{p-1} \quad \mbox{in } {\mathbb {R}^N}\setminus \lbrace 0\rbrace \mbox{ with } N\ge 3, A> 0, \alpha > 0, p>2. $ An energy balance identity is employed to prove nonexistence of such solutions in the last remaining open region in the (α,p)${{(\alpha , p)}}$ plane.

Abstract. We establish local Calderón–Zygmund estimates for solutions to certain parabolic problems with irregular obstacles and nonstandard p(x,t)${p(x,t)}$-growth. More precisely, we will show that the spatial gradient Du${Du}$ of the solution to the obstacle problem is as integrable as the obstacle ψ${\psi }$, i.e. |Dψ|p(·),|∂tψ|γ1'∈L loc q⇒|Du|p(·)∈L loc q,foranyq>1,$ |D\psi |^{p(\,\cdot \,)},|\partial _t\psi |^{\gamma _1^{\prime }}\in L^q_\mathrm {loc}\Rightarrow |Du|^{p(\,\cdot \,)}\in L^q_\mathrm {loc},\quad \text{for any}~q>1, $ where γ1'=γ1γ1-1${\gamma _1^{\prime }=\frac{\gamma _1}{\gamma _1-1}}$ and γ1${\gamma _1}$ is the lower bound for p(·)${p(\,\cdot \,)}$.

Abstract. We turn back to some pioneering results concerning, in particular, nonlinear potential theory and non-homogeneous boundary value problems for the so-called p -Laplace operator. Unfortunately these results, obtained at the very beginning of the seventies, were kept in the shade. We believe that our proofs are still of interest, in particular due to their extreme simplicity. Moreover, some contributions seem to improve the results quoted in the current literature.