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A Dirichlet problem involving critical exponents. (English) Zbl 0828.35042

The goal of this work is to find a nontrivial solution of the equation: \[ - \Delta_p (u) \equiv - \text{div} \bigl( |Du |^{p - 2} Du \bigr) = |u |^{r - 2} u + \lambda g(u), \quad u > 0 \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega, \tag{1} \] where \(\Omega\) is a bounded and smooth domain of \(\mathbb{R}^N\). \(\lambda_1\) is the first eigenvalue of \(- \Delta_p\) on \(\Omega\) with Dirichlet boundary condition. The authors obtain the following main result:
Assume: \(1 < p\), \(r > p\), \(g : \mathbb{R} \to \mathbb{R}\) is a Carathéodory function and \(\exists c_1 > 0\), \(\exists q \in (1,p)\): \(\forall s \geq 0\), \(g(s) \leq c_1 s^{q - 1}\); the function \(u \mapsto |u |^{r - 2} u + \lambda g(u)\) is nondecreasing; \(c_0 = \varliminf_{z \to 0^+} g(z)/z^{p - 1}\); \(\forall \tau > 0\) \(C_3 (\tau) = \inf_{0 \leq z \leq (\tau \lambda_1)^{1/(r - p)}} g(z)/z^{p - 1}\).
Then there exists a positive constant \(\lambda_0\) such that, if \(\lambda_1 < c_0 \lambda_0\) then for all \(\lambda \in (\lambda_1/c_0, \lambda_0]\) there is a solution \(u \in W_0^{1,p}\) \((\Omega) \cap L^\infty (\Omega)\) of (1). If for some \(\tau > 1\), \(C_3 (\tau) > 0\) and \(\lambda > \lambda_1/C_3 (\tau)\) then problem (1) has no solution in \(W^{1,p}_0 (\Omega)\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
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