Corrigendum to “Existence of solutions for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-Laplacian problems on a bounded domain” [J. Math. Anal. Appl. 306 (2005) 604–618]

Journal of Mathematical Analysis and Applications

2007

We study the boundary value problem − div(log(1 + |∇u| q )|∇u| p−2 ∇u) = f (u) in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in R N with smooth boundary. We distinguish the cases where either f (u) = −λ|u| p−2 u + |u| r−2 u or f (u) = λ|u| p−2 u − |u| r−2 u, with p, q > 1, p + q < min{N, r}, and r < (Np − N + p)/(N − p). In the first case we show the existence of infinitely many weak solutions for any λ > 0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz–Sobolev space setting

Mihăilescu

Journal of Mathematical Analysis and Applications

2007

“…The experimental research has been done mainly in the USA, for instance in NASA laboratories. For more information on properties, modelling and the application of variable exponent spaces to these fluids we refer to Acerbi and Mingione [1], Alves and Souto [3], Chabrowski and Fu [6], Diening [8], Fan et al [14,15], Mihȃilescu and Rȃdulescu [22][23][24], Rajagopal and Ruzicka [32], and Ruzicka [33].…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent

Mihăilescu

Pucci

Journal of Mathematical Analysis and Applications

2008

177

In this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term. One of the main features of our work is that the differential operator involves partial derivatives with different variable exponents, so that the functional-analytic framework relies upon anisotropic Sobolev and Lebesgue spaces. Existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients. Our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations.

“…A second major application of non-homogeneous differential operators with variable exponent is the modeling of some materials with inhomogeneities, for instance, electrorheological (non-Newtonian) fluids (sometimes referred to as 'smart fluids'), cf. [2][3][4][5][6][7]. Materials requiring such more advanced theory have been studied experimentally since the middle of the last century.…”

Section: Introductionmentioning

confidence: 99%

Multiple Solutions of Neumann Problems: An Orlicz–Sobolev Space Setting

Afrouzi

Bull. Malays. Math. Sci. Soc.

Shokooh

2015

In the present paper, we establish the range of two parameters for which a non-homogeneous boundary value problem admits at least three weak solutions. The proof of the main results relies on recent variational principles due to Ricceri.Keywords Three solutions · Non-homogeneous differential operator · Orlicz-Sobolev space Mathematics Subject Classification Primary 34B37 · Secondary 35J60 · 35J70 · 46N20 · 58E05 Communicated by