Orlicz–Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations

In this work, we study the following boundary value problem ( P ) { − div ( a ( | ∇ u | ) ∇ u ) = f ( x , u ) , in  Ω , u = 0 , on  ∂ Ω , with nonhomogeneous principal part. By assuming the nonlinearity f ( x , t ) corresponds to subcritical growth, we prove a regularity result for weak solutions. Using the regularity result we show that C 1 -local minimizers are also local minimizers in the Orlicz–Sobolev space. So, similar to the approach for the p -Laplacian equation, the sub–supersolution method for this problem is developed. Applying these results and critical point theory, we prove the existence of multiple solutions of problem (P) in the Orlicz–Sobolev space. The result for the sign-changing solution is new for the p -Laplacian equation.