Positive solutions for generalized nonlinear logistic equations of superdiffusive type
Consider the following generalized nonlinear logistic equations of superdiffusive type \[ \begin{cases} -\Delta_p u = \lambda f(z,u) -g(z,u) & \text{in }\Omega,\\ u>0 & \text{in }\Omega,\\ u=0 & \text{on }\partial \Omega. \end{cases} \tag{\(P_\lambda\)} \] where \(\Omega \subset \mathbb{R}^N\) (\(N \geq 2\)) is a bounded domain with a \(C^2\) boundary \(\partial \Omega\), \(\Delta_p\) denotes the \(p\)-Laplace differential operator and \(\lambda >0\) is a real parameter. Using the mountain pass theorem together with suitable truncation techniques, the authors prove a bifurcation-type theorem for large values of the parameter \(\lambda > 0\). Namely, under the following assumptions on the nonlinearities \(f\) and \(g\) {\parindent=10mm \begin{itemize}\item[(H] \(f: \Omega \times \mathbb{R} \to \mathbb{R}\) is a Carathéodory function (i.e. measurable in \(z \in \Omega\) and continuous in \(x \in \mathbb{R}\)) such that \(f(z, 0) = 0\) for almost all \(z \in \Omega\) and \(f(z, x) > 0\) for almost all \(z \in \Omega\) and all \(x > 0\), and \end{itemize}} {\parindent=12mm \begin{itemize} \item[(i)] \(|f(z, x)| \leq a(z) + c|x|^{r-1}\) for almost all \(z \in \Omega\) and all \(x \in \mathbb{R}\) (\(a \in L^\infty(\Omega)_+, c > 0, p < r < p^*\)); \item [(ii)] \(\lim_{x \to \infty} f(z, x)/x^{p-1} = \infty\), \(\lim_{x \to \infty} f(z, x)/x^{\theta -1} = \infty\) uniformly for almost all \(z \in \Omega\) and \(x \mapsto f(z, x)/x^{\theta -1}\) is nonincreasing in \(]0,\infty[\) for almost all \(z \in \Omega\) (\(\theta > p\)); \item [(iii)] \(\lim_{x \to 0^+} f(z, x)/x^{p-1} = 0\) uniformly for almost all \(z \in \Omega\); \item [(iv)] for every \(\rho > 0\), there exists \(\gamma_\rho > 0\) such that \(f(z, x) \geq \gamma_\rho\) for almost all \(z \in \Omega\) and all \(x \geq \rho\). \end{itemize}} {\parindent=10mm \begin{itemize}\item[(H] \(g: \Omega \times \mathbb{R} \to \mathbb{R}\) is a Carathéodory function (i.e. measurable in \(z \in \Omega\) and continuous in \(x \in \mathbb{R}\)) such that \(g(z, 0) = 0\) for almost all \(z \in \Omega\) and \(g(z, x) > 0\) for almost all \(z \in \Omega\) and all \(x > 0\), and \end{itemize}} {\parindent=12mm \begin{itemize} \item[(i)] \(|f(z, x)| \leq a(z) + c|x|^{r-1}\) for almost all \(z \in \Omega\) and all \(x \in \mathbb{R}\) (\(a, c, r\) as in (H\(_f\))(i)); \item[(ii)] \(\lim_{x \to \infty} g(z, x)/x^{\theta-1} > 0\) uniformly for almost all \(z \in \Omega\) and \(x \mapsto g(z, x)/x^{p -1}\) is nonincreasing for almost all \(z \in \Omega\) (\(\theta\) as in (H\(_f\))(ii)); \item [(iii)] \(\lim_{x \to 0^+} g(z, x)/x^{p-1} \leq \hat{\eta}\) uniformly for almost all \(z \in \Omega\) (\(\hat{\eta} > 0\)); \end{itemize}} {\parindent=10mm \begin{itemize}\item[(H] For every \(\xi > 0\) and every bounded interval \(I \subset \mathbb{R}^+\) there exists \(\sigma_{\xi,I} > 0\) such that \(x \mapsto \lambda f(z, x)-g(z, x)+ \sigma_{\xi,I} x^{\theta -1}\) is nondecreasing in \([0,\xi]\) for all \(\lambda \in I\) and almost all \(z \in \Omega\) (\(\theta\) as in (H\(_f\))(ii)), \end{itemize}} they show that there is a critical value \(\lambda_* > 0\) such that the following holds: if \(\lambda > \lambda_*\), then the problem (\(P_\lambda\)) has two positive solutions; if \(\lambda = \lambda_*\), then there is a positive solution; and finally, if \(0 < \lambda < \lambda_*\), then there are no positive solutions. Moreover, to produce a pair of positive solutions of (\(P_\lambda\)), they prove a strong comparison result which extends a similar result of \textit{M. Guedda} and \textit{L. Veron} [Nonlinear Anal., Theory Methods Appl. 13, No. 8, 879--902 (1989; Zbl 0714.35032)].
scientific article; zbMATH DE number 5995411