Local superlinearity and sublinearity for indefinite semilinear elliptic problems.
The authors study the semilinear elliptic boundary value problem (*) \(-\Delta u = f(x,u)\), \(u\geq 0\) and \(u\not\equiv 0\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \({\mathbb R}^{N}\) and \(f\) is sublinear at 0 and superlinear at infinity (in a weaker sense than usual; in particular a superlinearity condition is somewhat weaker than that of Ambrosetti and Rabinowitz). A typical example is the concave-convex nonlinearity \[ f(x,u) = \lambda a(x)u^q+b(x)u^p, \] where \(\lambda>0\), \(0\leq q<1<p\) and \(a\), \(b\) may be \(\leq 0\) on some subsets of \(\Omega\). The first result asserts that if \(f\) is of subcritical growth (in the example above this means \(p<2^{*}-1=(N+2)/(N-2)\)), then, under some additional hypotheses, (*) possesses at least 2 solutions. These solutions are respectively found as a mountain pass point and a local minimum of the associated energy functional. In the second result it is shown that under somewhat different conditions (*) has at least 1 solution. The proof uses the method of upper and lower solutions, and this time \(f\) is allowed to grow supercritically (\(p<+\infty\) in the example). Finally, a nonexistence result for (*) is proved. The results of the paper are related to and extend those of \textit{A. Ambrosetti, H. Brézis} and \textit{G. Cerami} [J. Func. Anal. 122, 519--543 (1994; Zbl 0805.35028)] and [\textit{H. Tehrani}, On elliptic equations with nonlinearities that are sum of a sublinear and superlinear term, volume dedicated to Dr. S. Shahshahani, Iran Dept. Math. Sci., Univ. Nevada, Las Vegas, Nevada 89154, 123--131 (2002)].
scientific article; zbMATH DE number 1915836
scientific article