| Summary: | We study the following nonlinear eigenvalue problem <inline-formula><math display="inline"><semantics><mrow><mo>−</mo><msup><mi>u</mi><mrow><msup><mrow></mrow><mo>″</mo></msup></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>λ</mi><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mspace width="5.0pt"></mspace><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn><mo>,</mo><mspace width="5.0pt"></mspace><mi>t</mi><mo>∈</mo><mi>I</mi><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mspace width="5.0pt"></mspace><mi>u</mi><mrow><mo>(</mo><mo>±</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mo form="prefix">log</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is a parameter. Then <inline-formula><math display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> is a continuous function of <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is the maximum norm <inline-formula><math display="inline"><semantics><mrow><mrow><mi>α</mi><mo>=</mo><mo>∥</mo></mrow><msub><mi>u</mi><mi>λ</mi></msub><msub><mrow><mo>∥</mo></mrow><mo>∞</mo></msub></mrow></semantics></math></inline-formula> of the solution <inline-formula><math display="inline"><semantics><msub><mi>u</mi><mi>λ</mi></msub></semantics></math></inline-formula> associated with <inline-formula><math display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>. We establish the precise asymptotic formula for <inline-formula><math display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mi>λ</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> as <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula> up to the third term of <inline-formula><math display="inline"><semantics><mrow><mi>λ</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>.
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