Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity
The author studies the existence, nonexistence and multiplicity of solutions of the problem \[ -\Delta u = a(x)u^q + \lambda b(x) u^p \quad\text{in } \Omega, \qquad u \geq 0 \quad\text{in } \Omega, \qquad u=0 \quad\text{on } \partial\Omega, \tag{\(P_\lambda\)} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\geq 3\), \(\lambda\in \mathbb{R}\), \(0<q<1<p\leq 2^*-1\) with \(2^*=2N/(N-2)\), and \(a,b\) are bounded functions with \(b\geq 0\). Furthermore, \(\Omega_a=\{x\in\Omega:a(x)\geq 0 \}\), \(\Omega_a^+=\{x\in\Omega:a(x)> 0 \}\), \( \Omega_a^-=\{x\in\Omega:a(x) < 0 \}\) and \(\Omega_b^+ = \{x\in\Omega: b(x)>0 \}\) are assumed to be nonempty, and to satisfy some additional hypotheses. In this situation, solutions can vanish on parts of \(\Omega\). A function \(u\in H_0^1(\Omega)\) is said to be a solution of \((P_\lambda)\) if it is a critical point of the functional \[ F_\lambda(u) = \frac{1}{2} \int_\Omega |\nabla u|^2 - \frac{1}{q+1} \int_\Omega a(x) (u^+)^{q+1} - \frac{\lambda}{p+1} \int_\Omega b(x) (u^+)^{p+1}, \] and in addition \(u(x) > 0\) for a.e. \(x\in \Omega_a^+\). The main results are the following: Theorem 1. Suppose \(0<q<1<p\leq 2^*-1\). There is a \(\lambda^*\in (0,\infty)\) such that \((P_\lambda)\) has at least one solution for \(0<\lambda < \lambda^*\) and this solution is a local minimum for \(F_\lambda\). Furthermore, \((P_\lambda)\) does not have a solution if \(\lambda>\lambda^*\). If \(\partial\Omega\) is smooth, \((P_{\lambda^*})\) has at least one solution. Theorem 2. In the subcritical case \(0<q<1<p<2^*-1\), \((P_\lambda)\) has at least two solutions for \(0<\lambda<\lambda^*\), where \(\lambda^*\) is as in Theorem 1. Theorem 3. Under a further technical assumption on \(b\), in the critical case \(0<q<1 < p=2^*-1\), \((P_\lambda)\) has at least two solutions if \(0<\lambda<\lambda^*\), where \(\lambda^*\) is as in Theorem 1. Similar multiplicity results were obtained by \textit{A.~Ambrosetti, H.~Brezis} and \textit{G.~Cerami} [J. Funct. Anal. 122, No. 2, 519--543 (1994; Zbl 0805.35028)] for \(a \equiv b \equiv 1\) and by \textit{D. G.~de Figueiredo, J.-P.~Gossez} and \textit{P.~Ubilla} [J. Eur. Math. Soc. (JEMS) 8, No. 2, 269--286 (2006; Zbl 1245.35048)] for \(a\geq 0\) and \(b\) indefinite.
scientific article; zbMATH DE number 5978461