Existence and Nonexistence of a Solution for a Nonlinear -Elliptic Problem with Right-Hand Side Measure

We discuss the existence and nonexistence of solution of a nonlinear problem -elliptic- , where is a Radon measure with bounded total variation, by considering the Sobolev spaces with variable exponents. This study is done in two cases: (i) is absolutely continuous with respect to -capacity. and (ii) is concentrated on a Borel set of null -capacity. 1. Introduction Let be a bounded domain of , let , and let (see definition below). Consider the following nonlinear -elliptic problem: where is a Leray-Lions operator from into its dual . The nonlinear term satisfies a growth condition of the following form: where is a continuous increasing function and and assuming also that , a.e. , , and for , where and are two positive real constants. The second term is a Radon measure on . We are interested in the existence and nonexistence of solution of the problem . We prove the existence of solution if does not charge the sets of null -capacity, in other words, if and only if is absolutely continuous with respect to -capacity. We give a necessary and sufficient condition that allows us the existence of solution . Thus, if is concentrated on a Borel set of of null -capacity, we will show that the problem admits no solution. Boccardo et al. [1] treated for constant. We can also see other variations of this problem; for example, if the nonlinear term is independent of and behaves as , with , the results of the existence and nonexistence of solution depend on the measure ; see Baras and Pierre [2], Brezis [3], and Gallou？t and Morel [4]. 2. Preliminaries Let be a bounded open subset of . The function satisfies the log-H？lder continuity on if with being a positive constant. Let be log-H？lder continuous such that , where and . For , we define the variable exponent Lebesgue space normed by . The space is a separable and reflexive Banach space, and its dual space is isomorphic to , where (see [5, 6]). Proposition 1 (see [5, 6]). (i) For any and , one has the following: (ii) For all such that a.e. in ; then and the embedding is continuous. Proposition 2 (see [5, 6]). If one denotes , , then Define now the Sobolev space with variable exponent by normed by Let be the closure of in and let for . Proposition 3 (see [5, 7]). (i) Assuming that , the spaces and are separable and reflexive Banach spaces. (ii) If and , for any , then embedding is continuous and compact. (iii) The Poincaré inequality: there exists a constant , such that (iv) The Sobolev inequality: there exists another constant , such that Remark 4. By (iii) of the Proposition 3, we deduce that and are equivalent