Abstract
In this paper, we study a class of (p, q)-Schrödinger–Kirchhoff type equations involving a continuous positive potential satisfying del Pino–Felmer type conditions and a continuous nonlinearity with subcritical growth at infinity. By applying variational methods, penalization techniques and Lusternik–Schnirelman category theory, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values.
Highlights
1.1 Background and motivationsIn this paper, we consider the following class of (p, q)-Schrödinger–Kirchhoff type problems: ⎧ ⎪ ⎨ −1 + a ∫RN ∇u p dx Δpu − b ∇u q dx Δqu
The authors in [39] combined refined variational methods based on critical point theory with Morse theory and truncation techniques to obtain a multiplicity result for a (p, q)-Laplacian problem in bounded domains
Due to the interest shared by the mathematical community toward quasilinear problems and Kirchhoff type equations, in [12, 31], the authors studied Kirchhoff type equations involving the (p, q)-Laplacian operator with p ≠ q, in a bounded domain and in the whole of R3, respectively
Summary
We consider the following class of (p, q)-Schrödinger–Kirchhoff type problems:. Whose study is motivated by the general reaction diffusion system ut = div [D(u)∇u] + c(x, u) where D(u) = |∇u|p−2 + |∇u|q−2 This system has a wide range of applications in physics and related sciences such as biophysics, plasma physics and chemical reaction design. The authors in [39] combined refined variational methods based on critical point theory with Morse theory and truncation techniques to obtain a multiplicity result for a (p, q)-Laplacian problem in bounded domains. Due to the interest shared by the mathematical community toward quasilinear problems and Kirchhoff type equations, in [12, 31], the authors studied Kirchhoff type equations involving the (p, q)-Laplacian operator with p ≠ q , in a bounded domain and in the whole of R3 , respectively. Motivated by the above works, the purpose of this paper is to study the multiplicity and the concentration of solutions for (1)
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