Abstract
We consider the following elliptic problem: \t\t\t{−div(|∇u|p−2∇u|y|ap)=|u|q−2u|y|bq+f(x)in Ω,u=0on ∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\operatorname{div} ( \\frac{ \\vert \\nabla u \\vert ^{p-2} \\nabla u}{ \\vert y \\vert ^{ap}} ) = \\frac { \\vert u \\vert ^{q-2} u}{ \\vert y \\vert ^{bq}} + f(x) & \\mbox{in } \\Omega,\\\\ u = 0 & \\mbox{on } \\partial\\Omega, \\end{cases} $$\\end{document} in an unbounded cylindrical domain \t\t\tΩ:={(y,z)∈Rm+1×RN−m−1;0<A<|y|<B<∞},\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\Omega:= \\bigl\\{ (y,z)\\in \\mathbb{R}^{m+1}\\times\\mathbb{R}^{N-m-1} ; 0< A< \\vert y \\vert < B < \\infty \\bigr\\} , $$\\end{document} where 1leq m< N-p, q=q(a,b):=frac{Np}{N-p(a+1-b)}, p>1 and A,Binmathbb{R}_{+}. Let p^{*}_{N,m}:=frac {p(N-m)}{N-m-p}. We show that p^{*}_{N,m} is the true critical exponent for this problem. The starting point for a variational approach to this problem is the known Maz’ja’s inequality (Sobolev Spaces, 1980) which guarantees, for the q previously defined, that the energy functional associated with this problem is well defined. This inequality generalizes the inequalities of Sobolev (p=2, a=0 mbox{ and } b=0) and Hardy (p=2, a=0 mbox{ and } b=1). Under certain conditions on the parameters a and b, using the principle of symmetric criticality and variational methods, we prove that the problem has at least one solution in the case fequiv0 and at least two solutions in the case f notequiv0, if p< q< p^{*}_{N,m}.
Highlights
Consider the class of degenerate singular quasilinear elliptic equations in RN– div A(x, ∇u)∇u = g(x, u) ∀x ∈ RN, ( )where A is a nonnegative unbounded function that vanishes at some points of RN
We show that p∗N,m is the true critical exponent for this problem
Under certain conditions on the parameters a and b, using the principle of symmetric criticality and variational methods, we prove that the problem has at least one solution in the case f ≡ 0 and at least two solutions in the case f ≡ 0, if p < q < p∗N,m
Summary
Where A is a nonnegative unbounded function that vanishes at some points of RN. we consider variants of this class of equations of the type. Note that ∗N,m is the critical Sobolev exponent in dimension N – m, which is greater than the usual critical To prove these results, we study an auxiliary problem and show that its solutions are axially symmetric and belong to the space W ,p(S) ⊂ W ,p( ), where S := (A, B) × RN–m–. We study an auxiliary problem and show that its solutions are axially symmetric and belong to the space W ,p(S) ⊂ W ,p( ), where S := (A, B) × RN–m– As usual, this is done by defining an energy functional I : W ,p(S) → R and by showing the existence of critical points for I in the space W ,p(S). To ensure the existence of solutions to the auxiliary problem, we use the results of the previous section as well as Ekeland’s variational principle
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