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Some results on \(p\)-Laplace equations with a critical growth term. (English) Zbl 1131.35325
The paper focuses on sufficient conditions that provide the existence of nontrivial weak solution to the Dirichlet boundary value problem \(\Delta _p u = g(x,u) + | u| ^{p^*-2}\) considered in a bounded \(n\)-dimensional domain; \(p^* = \frac {np}{n-p}\). The function \(g\) is subcritical and parameter \(p\) in definition of the \(p\)-Laplace operator ranges strictly between 1 and \(n\). The problem is treated by the methods of variational calculus (mountain pass theorem, the linking method). The novelty of the results is twofold. First of all, a more general class of lower order nonlinearity \(g\) is considered. (For example \(g\) is not required to be positive or homogeneous with respect to \(u\).) Secondly, this seems to be (one of the) first result(s) where the linking method is used to handle problems involving the \(p\)-Laplacian.

35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)