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A variational approach to superlinear semipositone elliptic problems. (English) Zbl 1367.35070

The authors prove the existence of positive solutions to a class of elliptic problems with superlinear semipositone nonlinearities via mountain-pass variational methods.
More precisely, let \(\Omega\) be a bounded smooth domain in \(\mathbb{R}^N\) with \(N\geq 3\). Consider the problem \[ \begin{cases} -\Delta u=\lambda a(x)(f(u)-\ell) &\text{ in } \Omega,\\ u=0 &\text{ on } \partial\Omega, \end{cases} \] where the parameters \(\lambda, \ell>0\), \(a\in C(\overline{\Omega})\) is a continuous function, and \(f:[0,\infty)\rightarrow \mathbb{R}\) is a continuous function satisfying \(f(0)=0\) and of regular variation of index \(p\) at infinity with \(1<p<\frac{N+2}{N-2}\): \[ \lim_{s\rightarrow \infty}\frac{f(\mu s)}{f(s)}=\mu^p \text{ for every } \mu>0. \] Note that such a function \(f\) does not need to be asymptotic to the power \(s^p\) at infinity. The authors prove that if \(a\geq 0\), or \(a\) changes sign but has a thick zero set, then there exists \(\lambda_0>0\) such that the problem has a positive solution \(u_\lambda\) for every \(0<\lambda<\lambda_0\).
For the proof the authors consider suitable rescaled problems adapted to the nonlinearities, and apply a combination of variational and continuity arguments.

MSC:

35J15 Second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35J61 Semilinear elliptic equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
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