Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent

Abstract

Abstract. We establish the existence of an entire solution for a class of stationary Schrödinger equations with subcritical discontinuous nonlinearity and lower bounded potential that blows-up at infinity. The abstract framework is related to Lebesgue-Sobolev spaces with variable exponent. The proof is based on the critical point theory in the sense of Clarke and we apply Chang's version of the Mountain Pass Lemma without the PalaisSmale condition for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz [35] on the existence of ground-state solutions of the nonlinear Schrödinger equation.

The Liouville theorems for elliptic equations with nonstandard growth

“…In the case of p = const operators A as in Definition 2 have been investigated by e.g. Mitidieri-Pokhozaev, see condition (11) in [20]. Results of our Theorems 4, 5 and Corollary 1 partially generalize Theorem 2 and Corollary 2 in [20].…”

“…We would like to emphasize that, according to our best knowledge, the Liouville-type theorems in the setting of equations with nonstandard growth have not yet been studied systematically in the literature. In fact, it appears that there exists only one paper about the Liouville theorem in the context of variable exponent, see Wang [25]; see also Pucci-Zhang [22] and Dinu [11] for related topics. We hope that our results will attract wider audience and lead to deeper studies of Liouville type theorems and nonexistence results for PDEs with nonstandard growth.…”

Compact embedding from W01,2(Ω) to LqKurata1, Shioji2 2008Journal of Mathematical Analysis and Applications17000View full textAdd to dashboardCite

“…Recently, under the uniform subcritical condition, i.e. ess inf x∈Ω (Np(x)/(N − p(x)) − q(x)) > 0, nonlinear elliptic boundary value problems of the type −div(|∇u| p(x)−2 ∇u) = |u| q(x)−2 u with variable exponents has been studied by using the critical point theory (see [4,6,[8][9][10]12] and references therein). In [2], Alves and Souto studied the existence of nonnegative solutions of −∇(|∇u| p(x)−2 ∇u) = u q(x)−1 in R N under the following conditions on p(x) and q(x): p(x) and q(x) are radially symmetric,…”

A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces

“…Typical examples of equations involving the variable exponent setting include models of electrorheological fluids [3,175,177], image restoration processing [43], elasticity equations [201], and thermistor model [202]. Since the setting is carefully examined and broad range of problems is already addressed, we mention only a few attempts to basic properties of PDEs such as existence [75,93,148,174], regularity results [1,2], maximal principle [118], and nonexistence [4,83]. Existence to L 1 -data problems within isotropic approach were studied in [24,195] and anisotropic case in [22,23].…”