On a class of singular biharmonic problems involving critical exponents. (English) Zbl 1091.35023
Summary: This paper deals with singular biharmonic problems of the form
\[
\begin{cases} \Delta^2u+V(x)|u|^{q-1}u=|u|^{2^*-2}u,&\text{in}\;\Omega\subset\mathbb R^N,\\u\in D_o^{2,2}(\Omega),&N\geq5,\end{cases}\tag{P}
\]
where \(1\leq q<2^*-1, 2^*=2N/(N-4)\) is the critical Sobolev exponent, \(\Delta^2\) denotes the biharmonic operator, \(\Omega\) is an open domain (not necessarily bounded, it may be equal to \(\mathbb R^N\)) and \(V\) is a potential that changes sign in \(\Omega\) with some points of singularities in \(\Omega\). Some results on the existence of solutions are obtained by combining a mountain pass theorem and the Hardy inequality with some arguments used by H. Brézis and L. Nirenberg [Commun. Pure Appl. Math. 36, No. 4, 437–477 (1983; Zbl 0541.35029)]
MSC:
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 35B33 | Critical exponents in context of PDEs |
| 35J60 | Nonlinear elliptic equations |
| 47J30 | Variational methods involving nonlinear operators |
Keywords:
Biharmonic operator; critical Sobolev exponent; Hardy inequality; Singular; and indefinite potentialCitations:
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