Multiplicity of solutions for a nonlinear degenerate problem in anisotropic variable exponent spaces
The author of this paper studies the following anisotropic problem \[ -\sum_{i=1}^N\partial_{x_i}(|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u)=\lambda (u^{\alpha(x)-1}-u^{\beta(x)-1})\,\,\,\, \text{in}\,\,\, \Omega \] \[ u=0 \,\,\,\, \text{on}\,\,\, \partial\Omega, \] \[ u\geq 0 \,\,\,\, \text{in}\,\,\, \Omega\,, \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\), \(N\geq 3\), with smooth boundary \(\partial \Omega\), \(p_i\) are continuous functions on \(\overline \Omega\) such that \(p_i(x)\geq 2\) for any \(x\in \overline \Omega\) and \(i=1, \dots, N\), \(\lambda\) is a real parameter, while \(\alpha, \beta: \overline \Omega \to \mathbb R\) are continuous functions such that \[ 1<\beta^-\leq \beta(x)\leq \alpha(x)\leq \alpha^+<P^-_-\leq P^+_+<P_{-, \infty}\,, \qquad x\in \Omega \] and there exists \(x_0\in \Omega\) such that \[ \beta(x_0) < \alpha(x_0)\,. \] Here \(P_+^+=\max\{p_1^+, \dots p_N^+\}\), \(P_-^+=\max\{p_1^-, \dots p_N^-\}\), \(P_-^-=\min\{p_1^-, \dots p_N^-\}\) and \(P_{-, \infty}=\max\{P_-^+, P_-^*\}\), where \[ P_-^*=\frac{N}{\sum_{i=1}^N 1/p_i^- -1} \] with \(\sum_{i=1}^N 1/p_i^- >1\). In the main result of the paper, the author proves that the problem admits at least two distinct nontrivial nonnegative weak solutions, provided the parameter \(\lambda\) is sufficiently large. The proof of this result is based on variational methods. Precisely, the first solution is found using the direct methods in the calculus of variations, that is as a minimum of the energy functional naturally associated with the problem while the second solution comes from the mountain pass theorem.
direct methods in the calculus of variations
scientific article; zbMATH DE number 6143143