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On a class of nonlocal problem involving a critical exponent. (English) Zbl 1355.35063

Consider the following nonlocal \(p\)-Laplacian with critical exponent \[ \begin{cases} -M\left(\int_{\Omega}|\nabla u|^{p}\right)\Delta_{p} u =\beta h(x)|u|^{q-2}u+|u|^{p^{*}-2}u+f(x) & \text{ in }\Omega, \\ u=0 &\text{ on }\partial\Omega, \end{cases}\tag{1.1} \] where \(\Omega\subset\mathbb R^N\) is a bounded domain with smooth boundary, \(1<p<N,p^{*}=\frac{Np}{N-p},p<q<p^{*},\beta>0,\) \(M:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}\) is a continuous function with \(\inf_{s>0}M(s)>0,\) and \(h\in L^{\frac{p^{*}}{p^{*}-q}}(\Omega),f\in L^{\frac{p}{p-1}}(\Omega).\)
By a version of the Mountain Pass Theorem without Palais-Smale condition, under the following condition \[ \hat{M(t)}\geq M(t)t\text{ for }t>0,\hat{M(t)}=\int_{0}^{t}M(s)ds, \] when \(\beta>0\) large enough and \(|f|_{\frac{p}{p-1}}\) small enough, the existence of nontrivial solutions for problem (1.1) is obtained.

MSC:

35J35 Variational methods for higher-order elliptic equations
35J60 Nonlinear elliptic equations
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B33 Critical exponents in context of PDEs
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