A priori bounds and multiple solutions for superlinear indefinite elliptic problems
Let \[ A:=-\sum_{i,j=1}^na_{ij}\partial_i\partial_j+ \sum_{i=1}^na_i\partial_i+a_0 \] be uniformly elliptic with \(a_{ij}=a_{ji}\in C(\overline \Omega)\) and \(a_i,\;a_0\in L_\infty(\Omega)\), where \(\Omega\subset {\mathbb{R}}^n\) is a bounded smooth domain. If \(\Gamma_0\) and \(\Gamma_1\) are two disjoint open and closed sets of \(\partial \Omega\) with \(\Gamma_0\cup\Gamma_1= \partial \Omega\), let \(Bu:=u\) on \(\Gamma_0\) and \(Bu:=\partial_\beta u+b_0 u\) on \(\Gamma_1\), where \(\beta\) is a smooth outward pointing vector field and \(b_0\) is continuous and non-negative. The authors investigate existence and multiplicity problems for positive solutions of the problem \[ Au=\lambda u+a(x)f(x,u)u \quad\text{in }\Omega, \qquad Bu=0 \quad\text{on }\partial \Omega, \] where \(\lambda\in {\mathbb{R}}\), \(a(x)\) is bounded and measurable, and \(f(x,t)\) is continuous, positive for \(t>0\), vanishing for \(t=0\) and satisfying a superlinear growth condition for \(t\to \infty\). The weight \(a(x)\) is allowed to change sign and to vanish on sets of positive measure. Let \(\Omega_+:=\{x\in \Omega: a(x)>0\}\) and \(\Omega_-:=\{x\in \Omega: a(x)<0\}\). In case of \(| \Omega_+| >0\), it is proved that positive solutions are bounded in \(C(\overline \Omega)\) if they are bounded in \(C(\overline\Omega_+)\). To prove boundedness in \(C(\overline\Omega_+)\), several cases are taken into account. As an example, if \(a^+(x)\) behaves like \(\text{dist}(x,\partial\Omega_+)^\gamma\) with \(\gamma \geq 0\) and if \(f(x,t)\) behaves like \(t^{r-1}\) with \(1<r<(n+1+\gamma)/(n-1)\) and \(r<(n+2)/(n-2)\) when \(n\geq 3\), then it is proved that positive solutions are bounded if \(\lambda\) stays bounded. Other a priori bounds are derived by using a weak Harnack inequality on \(\Omega_+\). In the final part of the paper, \(L_p\) estimates for general elliptic systems of arbitrary order are proved.
scientific article; zbMATH DE number 1212365
scientific article