Multiplicity of solutions for a quasilinear elliptic equation with ( p , q ) $(p,q)$ -Laplacian and critical exponent on R N $\mathbb{R}^{N}$

Abstract

The multiplicity of solutions for a (p, q)-Laplacian equation involving critical exponent p uq u = λV(x)|u| k-2 u + K(x)|u| p *-2 u, x ∈ R N is considered. By variational methods and the concentration-compactness principle, we prove that the problem possesses infinitely many weak solutions with negative energy for λ ∈ (0, λ *). Moreover, the existence of infinitely many solutions with positive energy is also given for all λ > 0 under suitable conditions.

Similarity Solution of Non-Ideal Fluids Laminar Boundary Layer with Convective Boundary Condition

“…Hence even in the class of convex domains, no maximizer exists for κ 1 (•) prescribing volume. Nevertheless, it is shown in [12] that ball is the unique maximizer of κ 2 (•). This makes the study of (1.11) and its more general form (1.1) more interesting.…”

A Novel Learning Approach to Remove Oscillations in First‐Order Takagi–Sugeno Fuzzy System: Gradient Descent‐Based Neuro‐Fuzzy Algorithm Using Smoothing Group Lasso Regularization

“…$$L_n(n=0)$$ regularization finds broad utilization in variable selection and feature extraction, which generates the sparsest solutions, yet these sparse solutions are often challenging to compute. [ 30,31 ] To overcome this challenge, $$L_n(n=1)$$ regularization is introduced, but it does not exhibit sparsity as strong as L 0 regularization. [ 32 ] It is indicated that $$L_n(0<n<1)$$ regularization assures the generation of sparser solutions in comparison to L 1 regularization, where the index 1/2 assumes a representative role.…”

Singular elliptic problems with unbalanced growth and critical exponent

“…Using critical point theory with truncation arguments and comparison principle authors also proved bifurcation type result. For (p, q)-Laplacian problem with concave-convex nonlinearities in R n we refer to [31].…”