On the existence of two positive solutions for spatially nonhomogeneous elliptic problems

Abstract We treat the semilinear elliptic Dirichlet problem Δ u + λu ( x ) q ( x )−1 = 0 with a convex and concave nonlinear reaction <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mspace width="0.17em"/> <m:mo>⋅</m:mo> <m:mspace width="0.17em"/> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:mi>s</m:mi> <m:mo>⟼</m:mo> <m:mo stretchy="false">|</m:mo> <m:mi>s</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>s</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msub> <m:mo>⊂</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>→</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:math> $f\left(x, \cdot \right) : s{\longmapsto}\vert s{\vert }^{q\left(x\right)-2}s : {\mathbb{R}}_{+}\subset \mathbb{R}\to \mathbb{R}$ which (for s ≥ 0) is convex in an nonempty open subset <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msub> <m:mrow> <m:mspace width="0.17em"/> <m:mover> <m:mrow> <m:mo>=</m:mo> </m:mrow> <m:mrow> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi mathvariant="normal">e</m:mi> <m:mi mathvariant="normal">f</m:mi> </m:mrow> </m:mrow> </m:mover> <m:mspace width="0.17em"/> </m:mrow> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:mi>q</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>&gt;</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> ${{\Omega}}_{+} \mathrm{d}\mathrm{e}\mathrm{f}{=} \left\{x\in {\Omega} : q\left(x\right){ &gt;}2\right\}$ and concave in another nonempty open subset <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> </m:mrow> </m:msub> <m:mrow> <m:mspace width="0.17em"/> <m:mover> <m:mrow> <m:mo>=</m:mo> </m:mrow> <m:mrow> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi mathvariant="normal">e</m:mi> <m:mi mathvariant="normal">f</m:mi> </m:mrow> </m:mrow> </m:mover> <m:mspace width="0.17em"/> </m:mrow> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:mi>q</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>&lt;</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> ${{\Omega}}_{-} \mathrm{d}\mathrm{e}\mathrm{f}{=} \left\{x\in {\Omega} : q\left(x\right){&lt; }2\right\}$ of a bounded domain <m:math xmlns:m="http://www.w3.org/