Existence of Solutions to a Semilinear Elliptic System through Orlicz-Sobolev Spaces

Abstract

Using Orlicz-Sobolev spaces and a variant of the Mountain-Pass Lemma of Ambrosetti-Rabinowitz we obtain existence of a (positive) solution to a semilinear system of elliptic equations. The admissible nonlinearities are such that the system is superlinear and subcritical. The Orlicz setting used here allows us to consider nonlinearities which are not (asymptotically) pure powers. Moreover, by an interpolation theorem of Boyd we find an elliptic regularity result in Orlicz-Sobolev spaces. A bootstrapping argument implies that the above mentioned solutions are classical. Mathematics Subject Classification (2000). Primary 35J50; Secondary 46N20.

Mountain pass type periodic solutions for Euler–Lagrange equations in anisotropic Orlicz–Sobolev space

“…To show that the action functional satisfies the Palais-Smale condition we need index q ∞ G . Similar observation can be found in [12,13,14,15] where the existence of elliptic systems via the Mountain Pass Theorem is considered. In [13] authors deal with an anisotropic problem.…”

“…Without loss of generality we can assume, that u n W 1 L G > 0. Substituting u n into (11) and by (12) we obtain…”

Classification of isolated singularities for nonhomogeneous operators in divergence form

“…For more details we refer to the book by Adams [1], and to the papers by Clément et al [5], Lieberman [11,12] and Martínez and Wolanski [13].…”

Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth

“…Now, using the results discussed in Section 2 for Orlicz and Orlicz-Sobolev spaces, we mention that the following embedding are continuous W 1,m 0 (Ω) ֒→ W 1,Φǫ 0 (Ω) ֒→ W 1,ℓǫ 0 (Ω) (cf. [7]) and W 1,Φǫ 0 (Ω) ֒→ W 1,1 0 (Ω) (cf. [1]).…”