| Summary: | This article studies the Schrödinger equation with an inhomogeneous combined term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><msub><mo>∂</mo><mi>t</mi></msub><mi>u</mi><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mo>Δ</mo><mo stretchy="false">)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><msub><mi>λ</mi><mn>1</mn></msub><msup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><mrow><mo>−</mo><mi>b</mi></mrow></msup><msup><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow><mi>p</mi></msup><mi>u</mi><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><msup><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow><mi>q</mi></msup><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>,</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>λ</mi><mn>2</mn></msub><mo>=</mo><mo>±</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo><</mo><mi>b</mi><mo><</mo><mrow><mo>{</mo><mn>2</mn><mi>s</mi><mo>,</mo><mi>N</mi><mo stretchy="false">}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. We study the limit behaviour of the infinite blow-up solution at the blow-up time. When the parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><msub><mi>λ</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>λ</mi><mn>2</mn></msub></semantics></math></inline-formula> have different values, we obtain the nonexistence of a strong limit for the non-radial solution and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> concentration for the radial solution. Interestingly, we find that the mass of the finite time blow-up solutions are concentrated in different ways for different parameters.
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