Existence of Multiple Solutions for a Class Non-uniformly Elliptic Equations with Critical Exponential Growth

This paper deals with the existence and multiplicity of weak solutions to non-uniformly elliptic problem $$\begin{aligned} -\text {div}(a(x,\nabla u))+V(x)\vert u\vert ^{N-2} u=\lambda \Bigg (\exp \Bigg (\alpha \vert u\vert ^{\frac{N}{N-1}}\Bigg )+f(x,u)\Bigg ), \end{aligned}$$-div(a(x,∇u))+V(x)|u|N-2u=λ(exp(α|u|NN-1)+f(x,u)),in $$\mathbb {R}^N$$RN, where $$N\ge 2$$N≥2, $$\alpha $$α is some positive constant and $$\lambda $$λ is some positive parameter. Our method is mainly based on variational arguments.

where N ≥ 2, α is some positive constant and λ is some positive parameter, V : R N → ]0, +∞[ is a continuous function such that V (x) ≥ V 0 for some positive constant V 0 and V −1 ∈ L 1 N −1 (R N ) or lim V (x) → ∞ as |x| → ∞, i.e., meas{x ∈ R N : V (x) ≤ M} < ∞ for every M > 0. Motivated by [6], we assume that A be a measurable function on R N × R such that A(x, 0) = 0 and a(x, τ ) = ∂ A(x,τ ) ∂τ is a Carathéodory function on R N ×R, and there are positive real numbers c 0 , c 1 , k 1 and two nonnegative measurable functions h 0 , h 1 on R N such that h 0 ∈ L N /(N −1) (R N ), h 1 ∈ L ∞ loc (R N ) and h 1 (x) ≥ 1, satisfying the properties: Then A verifies the growth condition: Note that in the case of N -Laplacian, i.e., A(x, τ ) = 1 N |τ | N , we choose which has been studied extensively, both in the case N = 2 (i.e., Laplacian equation in R 2 ) and in the case N > 3 (i.e., N -Laplacian equation in R N ). The problems of this type are important in many fields of sciences, for example, in electromagnetism, astronomy and fluid dynamics. In fact, these models describe potentials of electric, gravitation and fluid, respectively.
In the case p < N , by the Sobolev embedding, the critical exponent is p * = pN N − p . When p = N , one has another maximal growth for any bounded domain , i.e., N −1 and W N −1 is the surface area of the unit sphere in R N . In this situation, the constant α N is sharp.( [1,4,7]) In the case of unbounded domain, if we replace ||∇u|| L N in the supremum by the standard Sobolev norm, then the following expression (Trudinger-Moser inequality) can be derived [5,9]). Motivated by the Trudinger-Moser inequality, we consider the following assumptions on f which allows us to treat the problem (1.1) variationally in a subspace of Next, we introduce some notations which since V is positive and bounded from below, E is a reflexive Banach space when endowed with the norm with continuous embedding. Thus, there exists a positive constant c 2 such that Using the radial lemma which asserts for all u ∈ W 1,N (R N ) radially nonincreasing symmetric ( [2,5]), there exists a constant c 3 > 0 such that provided that ∇u N ≤ 1, u N ≤ M < ∞ and α < α N . Furthermore, there exists a constant c 4 > 0 such that (1.5) imply that there exists a constant c 5 > 0 such that N . Note that according to [9,Lemma 2.4], the assumptions V −1 ∈ L 1 (R N ) or lim V (x) → ∞ as |x| → ∞, on the potential V (x) lead us to have the compactness of the imbedding E → L q (R N ), for all 1 ≤ q < +∞. Now from (1.2), the imbedding E → L q (R N ) for q = β 1 + 1 and (1.5) and (1.6) Consequently, the critical points of the functional I λ are precisely nontrivial weak solutions of the problem (1.1).
In the present paper, first we obtain the existence of a nontrivial weak solution for the problem (1.1), in a certain range of λ (in Theorem 3.1). Then we prove the existence of the second weak solution for the problem (1.1) distinct from the first one (in Theorem 4.1).

Preliminaries
Our main tool is Theorem 2.2, consequence of a local minimum theorem [3, Theorem 5.1] which is inspired by Ricceri's variational principle.
For a given non-empty set X , and two functionals , : X → R, we define the following functions: Definition 2.1 Let φ 0 and ψ 0 be two continuously Gâteaux differentiable functionals defined on a real Banach space X and fix r 1 , r 2 ∈ [−∞, +∞], with r 1 < r 2 . We say that the functional I 0 = φ 0 − ψ 0 verifies the Palais-Smale condition cut off lower at r 1 and upper at r 2 (in short [
Therefore, there exists u ∈ E such that u n u weakly in E. Thanks to Lemma 2.1. of [9] and using the assumption (1.2), one has This fact together with the compactness of the imbedding On the other hand, using the definition of S N −2 , one has Using the compact imbedding E → L 1 (R N ) together with (1.4) for the first part of the above inequality and E → L 2 (R N ) together with the fact that u n is converges and so bounded in L q for any q ≥ 1, arguing as before, one has Now using (A 1 ), it follows that This inequality together with h 0 ∈ L N /(N −1) (R N ) and h 1 ∈ L ∞ loc (R N ) (u n is converges to u in L 1 and L N ) and using the fact I λ (u n ), u n − u → 0 as n → +∞, it can be derived that (u n ) is strongly convergent to u in E.

Existence of the First Solution
Chosen large enough R > 0 such that for each r > 0 with the property Proof For λ > 0 and R > 0 as in (3.1), define the function with η λ is a real number satisfying First we note that θ λ ∈ E.

Existence of the Second Solution
Now, in this section, we prove the existence of the second local minimum distinct from the first one. For this purpose, we verify the hypotheses of Theorem 2.2. for another range of r .

Remark 4.2
To prove the existence of another local minimum distinct from the first and second ones, we can assume an algebraic condition on f with the classical Ambrosetti-Rabinowitz condition: there exist ν > N and R > 0 such that Clearly, the functional − λ is of class C 1 and ( − λ )(0) = 0. The first part of proof guarantees that u 2 ∈ E is a local nontrivial local minimum for − λ in E. Therefore, there is s > 0 such that inf ||u−u 2 ||=s ( − λ )(u) > ( − λ )(u 2 ).
So the condition [8, Theorem 2.2, (I 1 )] is verified. Now choosing u = 0, from Ambrosetti-Rabinowitz condition, one has Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.