## 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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### References:

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