Existence Results for a Class of the Quasilinear Elliptic Equations with the Logarithmic Nonlinearity

Abstract

In this paper, the nonlinear quasilinear elliptic problem with the logarithmic nonlinearity <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"><mo>−</mo><mi mathvariant="normal">div</mi><mfenced open="(" close=")"><mrow><msup><mrow><mfenced open="|" close="|"><mrow><mo>∇</mo><mi>u</mi></mrow></mfenced></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mrow></mfenced><mo>=</mo><mi>a</mi><mfenced open="(" close=")"><mrow><mi>x</mi></mrow></mfenced><msub><mrow><mi>φ</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced open="(" close=")"><mrow><mi>u</mi></mrow></mfenced><mi mathvariant="normal">log</mi><mfenced open="|" close="|"><mrow><mi>u</mi></mrow></mfenced><mo>+</mo><mi>h</mi><mfenced open="(" close=")"><mrow><mi>x</mi></mrow></mfenced><msub><mrow><mi>ψ</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced open="(" close=")"><mrow><mi>u</mi></mrow></mfenced></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> was studied. By means of a double perturbation argument and Nehari manifold, the authors obtain the existence results.

Locations

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