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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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