Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity
Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity
Abstract This paper is concerned with the following Kirchhoff-type problem with convolution nonlinearity: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>-</m:mo><m:mrow><m:mo maxsize="210%" minsize="210%">(</m:mo><m:mi>a</m:mi><m:mo>+</m:mo><m:mi>b</m:mi><m:msub><m:mo largeop="true" symmetric="true">∫</m:mo><m:msup><m:mi>ℝ</m:mi><m:mn>3</m:mn></m:msup></m:msub><m:mrow><m:mo fence="true" stretchy="false">|</m:mo><m:mo>∇</m:mo><m:mi>u</m:mi><m:mpadded width="+1.7pt"><m:msup><m:mo stretchy="false">|</m:mo><m:mn>2</m:mn></m:msup></m:mpadded><m:mi mathvariant="normal">d</m:mi><m:mi>x</m:mi><m:mo maxsize="210%" minsize="210%">)</m:mo></m:mrow><m:mi mathvariant="normal">Δ</m:mi><m:mi>u</m:mi><m:mo>+</m:mo><m:mi>V</m:mi><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow><m:mi>u</m:mi><m:mo>=</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msub><m:mi>I</m:mi><m:mi>α</m:mi></m:msub><m:mo>*</m:mo><m:mi>F</m:mi><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>u</m:mi><m:mo stretchy="false">)</m:mo></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow><m:mi>f</m:mi><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>u</m:mi><m:mo stretchy="false">)</m:mo></m:mrow><m:mo rspace="12.5pt">,</m:mo><m:mi>x</m:mi><m:mo>∈</m:mo><m:msup><m:mi>ℝ</m:mi><m:mn>3</m:mn></m:msup><m:mo rspace="4.2pt">,</m:mo><m:mi>u</m:mi><m:mo>∈</m:mo><m:msup><m:mi>H</m:mi><m:mn>1</m:mn></m:msup><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi>ℝ</m:mi><m:mn>3</m:mn></m:msup><m:mo stretchy="false">)</m:mo></m:mrow><m:mo>,</m:mo></m:mrow></m:mrow></m:math> -\bigg{(}a+b\int_{\mathbb{R}^{3}}\lvert\nabla u|^{2}\,\mathrm{d}x\bigg{)}% \Delta u+V(x)u=(I_{\alpha}*F(u))f(u),\quad x\in{\mathbb{R}}^{3},\,u\in H^{1}(% \mathbb{R}^{3}), where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo><m:mi>b</m:mi></m:mrow><m:mo>></m:mo><m:mn>0</m:mn></m:mrow></m:math> {a,b>0} …