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Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents. (English) Zbl 1030.35081

Via the variational method, the authors study the quasilinear elliptic problem \[ -\Delta_pu=\mu|u|^{p^*-2}u+f(x,u), \quad x\in\Omega,\qquad u=0 \quad \text{on} \;\partial\Omega , \] where \(\Delta_pu=\text{ div}\left(|\nabla u|^{p-2}\nabla u\right)\) is the \(p\)-Laplacian, \(\Omega\) is bounded smooth domain in \(\pmb{\mathbb{R}}^N\), \(n\geq 3\), \(1<p<N\), \(p^*=Np/(N-p)\) is the critical Sobolev exponent. The Carathéodory function \(f\), which is odd in \(s\), satisfies a subcritical growth condition: \[ \lim_{|s|\rightarrow\infty}f(x,s)/{|s|^{p^*-1}}=0 \quad \text{uniformly\;a.c.\;in\;} \Omega. \] Under some additional restrictions characterizing the combined behaviour at infinity of the function \(f(x,s)\) and \(F(x,s)=\int_0^sf(x,t) dt\), they prove that for arbitrary \(k \in \pmb{\mathbb{ N}}\), there exists \(\mu_k \in (o,\infty]\) such that the problem under consideration possesses at least \(k\) pairs of nontrivial solutions for all \(\mu \in (0,\mu_k)\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B33 Critical exponents in context of PDEs
35J20 Variational methods for second-order elliptic equations
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References:

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