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Existence results for Schrödinger \(p(\cdot)\)-Laplace equations involving critical growth in \(\mathbb{R}^N$. (English) Zbl 1421.35132
Summary: We establish some existence results for Schrödinger \(p(\cdot)\)-Laplace equations in \(\mathbb{R}^N\) with various potentials and critical growth of nonlinearity that may occur on some nonempty set, although not necessarily the whole space \(\mathbb{R}^N\). The proofs are mainly based on concentration-compactness principles in a suitable weighted variable exponent Sobolev space and its imbeddings.

MSC:
35J62 Quasilinear elliptic equations
35J10 Schrödinger operator, Schrödinger equation
35B33 Critical exponents in context of PDEs
35J35 Variational methods for higher-order elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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