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Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents

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Abstract

We study the following elliptic equations with variable exponents

$$-\text{div}(\phi(x,|\nabla u|)\nabla u)=\lambda f(x,u)\quad\text{in}\Omega$$

which is subject to Dirichlet boundary condition. Under suitable conditions on \({\phi}\) and f, employing the variational methods, we show the existence of nontrivial solutions of a class of quasilinear elliptic problems with variable exponents. Also we show the existence of positivity of the infimum of all eigenvalues for the above problem and then give some examples to demonstrate our main result.

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Kim, I.H., Kim, YH. Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents. manuscripta math. 147, 169–191 (2015). https://doi.org/10.1007/s00229-014-0718-2

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