On hardy-sobolev inequality and its application to certain singular problems

Abstract

Inequalities involving integrals of a function and its derivatives appear frequently in various branches of mathematics and represent a useful tool, e.g., in the theory and practice of differential equations, in the theory of approximation, etc. A general question in this direction can be stated as: Let ? be an open subset of ?? and w, w?, …, w? be given weight functions (measurable and positive almost everywhere). Now the question is: do there exist positive real numbers p and q such that the following inequality (???u(x)?pw(x)?dx)1/p?C?i=1n(????iu(x)?qwi(x)?dx)1/q(0.1)\left( \int_\Omega |u(x)|^p w(x) \, dx \right)^{1/p} \leq C \sum_{i=1}^n \left( \int_\Omega |\partial_i u(x)|^q w_i(x) \, dx \right)^{1/q} \tag{0.1}(????u(x)?pw(x)dx)1/p?Ci=1?n?(?????i?u(x)?qwi?(x)dx)1/q(0.1) holds for all u?Cc?(?)u \in C_c^\infty(\Omega)u?Cc??(?)? If w, w?, …, w? are bounded weights, the study of inequality (0.1) is quite well understood. Here, we are mainly interested in the case when w is a singular weight and w? = 1 for all i = 1, …, n. In 1920, G. H. Hardy proved the following inequality: Let 1<p<?1 < p < \infty1<p<?, then ?0??u(x)x?pdx?(pp?1)p?0??u?(x)?pdx,(0.2)\int_0^\infty \left| \frac{u(x)}{x} \right|^p dx \leq \left( \frac{p}{p-1} \right)^p \int_0^\infty |u'(x)|^p dx, \tag{0.2}?0???xu(x)??pdx?(p?1p?)p?0???u?(x)?pdx,(0.2) for all u?Cc?(0,?)u \in C_c^\infty(0,\infty)u?Cc??(0,?). Moreover, (pp?1)p\left( \frac{p}{p-1} \right)^p(p?1p?)p is the best constant and the inequality is strict unless u=0u = 0u=0. Analogues of inequality (0.2) exist in higher dimensions. More precisely, for any p,1<p<np, 1 < p < np,1<p<n, the following Hardy–Sobolev inequality holds: ?Rn?u(x)?p?x?pdx?(pn?p)p?Rn??u(x)?pdx,(0.3)\int_{\mathbb{R}^n} \frac{|u(x)|^p}{|x|^p} dx \leq \left( \frac{p}{n-p} \right)^p \int_{\mathbb{R}^n} |\nabla u(x)|^p dx, \tag{0.3}?Rn??x?p?u(x)?p?dx?(n?pp?)p?Rn???u(x)?pdx,(0.3) for all u?D1,p(Rn)u \in D^{1,p}(\mathbb{R}^n)u?D1,p(Rn), where D1,p(Rn)D^{1,p}(\mathbb{R}^n)D1,p(Rn) is the completion of Cc?(Rn)C_c^\infty(\mathbb{R}^n)Cc??(Rn) in the norm ?u?1,p=??u?p\|u\|_{1,p} = \|\nabla u\|_p?u?1,p?=??u?p?. The constant An,p:=(pn?p)pA_{n,p} := \left( \frac{p}{n-p} \right)^pAn,p?:=(n?pp?)p is the best in inequality (0.3). Moreover, for any bounded domain ? containing 0, inequality (0.3) holds for any uuu in the Sobolev space W01,p(?)W_0^{1,p}(\Omega)W01,p?(?), which is the completion of Cc?(?)C_c^\infty(\Omega)Cc??(?) in the norm ?u?1,p,?:=(???u(x)?pdx+????u?pdx)1/p,\|u\|_{1,p,\Omega} := \left( \int_\Omega |u(x)|^p dx + \int_\Omega |\nabla u|^p dx \right)^{1/p},?u?1,p,??:=(????u(x)?pdx+?????u?pdx)1/p, and An,pA_{n,p}An,p? is independent of the domain. The most interesting fact about inequality (0.3) is that the constant An,pA_{n,p}An,p? is never achieved (in the sense that equality holds for some u?0?W01,p(?)u \neq 0 \in W_0^{1,p}(\Omega)u?=0?W01,p?(?)) in any domain, not even in the whole space, and this will be the source of many problems which we deal with in this thesis. Some questions arise: Is it possible to add a second term (at least when ? is a bounded domain) on the right-hand side of inequality (0.3)? If yes, what would be the ‘best’ term? Does this process stop? In Chapter 2, we answer these questions and present the following theorem: Theorem. Let R>sup???x?R > \sup_\Omega |x|R>sup???x? and 1<p<n1 < p < n1<p<n. Then there exists C>0C > 0C>0, depending on n, p, and R, such that ???u(x)?p?x?p(log?(R/?x?))?dx?C????u?pdx,\int_\Omega \frac{|u(x)|^p}{|x|^p (\log(R/|x|))^\gamma} dx \leq C \int_\Omega |\nabla u|^p dx,????x?p(log(R/?x?))??u(x)?p?dx?C?????u?pdx, for any u?W01,p(?)u \in W_0^{1,p}(\Omega)u?W01,p?(?) if and only if (i) ?>2\gamma > 2?>2 when 1<p<n1 < p < n1<p<n, (ii) ?>n\gamma > n?>n when p=np = np=n. More generally, for 2<p<n2 < p < n2<p<n and for any 1<q<p?:=npn?p1 < q < p^* := \frac{np}{n-p}1<q<p?:=n?pnp?, there exists C1>0C_1 > 0C1?>0 depending on n, p, q, R, and ? such that ???u(x)?q?x?qdx+?\int_\Omega \frac{|u(x)|^q}{|x|^q} dx + \cdots????x?q?u(x)?q?dx+? (Corollary and further results follow similarly.)

Later chapters deal with:

The differential operator related to inequality (0.3) in the case p=2p = 2p=2, i.e., study of the operator L?:=?????x?2L_\mu := -\Delta - \frac{\mu}{|x|^2}L??:=????x?2??. Pohozaev-type identities showing non-existence of positive solutions for certain eigenvalue problems. Existence and non-existence results for nonlinear weighted problems. Perturbation of operators and improved Hardy–Sobolev inequalities applied to parabolic problems.