Quasilinear Scalar Field Equations Involving Critical Sobolev Exponents and Potentials Vanishing at Infinity

Abstract

We are concerned with the existence of positive weak solutions, as well as the existence of bound states (i.e. solutions inW1,p(ℝN)), for quasilinear scalar field equations of the form$$ - \Delta _pu + V(x) \vert u \vert ^{p - 2}u = K(x) \vert u \vert ^{q - 2}u + \vert u \vert ^{p^ * - 2}u,\qquad x \in {\open R}^N,$$where Δpu: =div(|∇u|p−2∇u), 1 <p<N,p*: =Np/(N−p) is the critical Sobolev exponent,q∈ (p, p*), whileV(·) andK(·) are non-negative continuous potentials that may decay to zero as |x| → ∞ but are free from any integrability or symmetry assumptions.

Existence of bound state solutions for p-Laplacian equation with critical growth

“…In [31], under without assuming the Ambrosetti-Rabinowitz type condition, Liu obtained the existence of ground state solutions for problem (1.2), where f (x, u) is subcritical and p-superlinear and V is periodic or has a bounded potential well. In [34], Lyberopoulos obtained a bound state solution of problem (1.1), when N > p 2 , p < q < p * , V and a satisfied the following assumptions: (V ′ ) V : R N → R is continuous and non-negative. Furthermore, there exist α > 0 , Λ > 0 and r 0 > 0 such that inf…”

Fractional Schrödinger–Poisson system with critical growth and potentials vanishing at infinity

“…Recently, Deng, Li, and Shuai [20] generalized Alves and Souto's results to the p ‐Laplacian case and applied it to study the following p ‐Laplacian equation: $$$$\begin{equation} -\Delta _p u+V(x)|u|^{p-2}u=K(x)f(u)+P(x)|u|^{p^*-2}u,\nobreakspace \nobreakspace x\in \mathbb {R}^N,\end{equation}$$$$to get a positive solution by means of variational methods. For more existence results of Equation (1.3), see, for instance, [11, 20, 22, 34] and references therein.…”

“…to get a positive solution by means of variational methods. For more existence results of Equation (1.3), see, for instance, [11,20,22,34] and references therein. which has been introduced in [12] as a model describing a quantum particle interacting with an electromagnetic field.…”

“…Highlighted by [1], in the first step, we establish the variational framework such that positive solutions can be obtained for a auxiliary equation instead, which is obtained by properly penalizing the right-hand side of (1.1). This idea has originated from del Pino and Felmer [19], who used it to distinguish the local mountain pass for the classical Schrödinger equation; since then, it has been widely used at various situations for the classical elliptic problems, see [3,13,15,22,34,37,38] and references therein. We remark that, while dealing with (1.1), we shall face two major difficulties: One is the severe lack of compactness due to the critical nonlinearity.…”