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Quasilinear elliptic equations involving critical Sobolev exponents. (English) Zbl 0714.35032

Let \(G\) be a bounded open subset of \(\mathbb R^ N\) with \(C^ 2\) boundary, \(1<p<N\), \(p^*=Np/(N-p)\) and \(a\in L^{\infty}(G)\). The aim of this paper is to study existence of solutions \(u\) for the quasilinear Dirichlet problem: \[ -\mathop{div}(| Du|^{p-2} Du)=a(x)u^{p-1}+u^{p^*-1}\text{ in } G,\quad u>0\text{ in } G;\quad u=0\text{ on } \partial G. \tag{1} \] As \(p^*\) is the critical Sobolev exponent corresponding to the noncompact embedding of \(W_ 0^{1,p}(G)\) into \(L^{p^*}(G)\) it is not possible to solve (1) via simple variational arguments. Instead, techniques initiated by Trudinger, Aubin, Brezis-Nirenberg, Tolksdorf and others are used. The results include the following when \(a\equiv \lambda\) and where \(\lambda_ 1\) is the best Poincaré constant for \(G\). If \(1<p^ 2\leq N\) and \(0<\lambda <\lambda_ 1\) then (1) admits at least one solution \(u\) in \(W_ 0^{1,p}(G)\). If \(1<p<N\), \(\lambda\leq 0\) and \(G\) is star-shaped then (1) admits no solution in \(W_ 0^{1,p}(G)\). If \(1<p\) and \(\lambda \geq \lambda_ 1\) then (1) admits no solution in \(W_ 0^{1,p}(G)\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J62 Quasilinear elliptic equations
35J70 Degenerate elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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