| Abstract: |
| During the last decade the so-called double phase operator has drawn attention from researchers. Originally it was introduced by Zhikov in the context of homogenization and elasticity theory and as an example for the Lavrentiev phenomenon. It regained popularity after some novel regularity results for local minimizers of the corresponding functional.
In the first part of this talk we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We discuss useful properties of the corresponding Musielak-Orlicz Sobolev spaces and of this new double phase operator. In contrast to the previously known constant exponent case we are able to weaken the assumptions on the data.
After that we consider a problem with superlinear right-hand side and we show, under very general assumptions, a multiplicity result for such problems, whereby we show the existence of a positive solution, a negative one and a solution with changing sign. The sign-changing solution is obtained via the Nehari manifold approach and, in addition, we can also give information on its nodal domains. Furthermore, we derive a priori estimates on the solutions in the L$^\infty$-norm under the very general setting used above. |
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