Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

In this work, we study the of positive ground state solution for the semilinear elliptic problem $$ \left\{ \begin{array} [c]{ll}% -\Delta u=u^{p(x)-1},\quad u>0&\mathrm{in}\,G\subseteq\mathbb{R}^{N}% ,\,N\geq3\\ u\in D_{0}^{1,2}(G),&\end{array} \right. $$ where $G$ is either $\mathbb{R}^{N}$ or a bounded domain, and $p:G\rightarrow \mathbb{R}$ is a continuous function assuming critical and subcritical values.


Introduction
In this paper we deal with the existence of a ground state solution for the semilinear elliptic problem . In both cases, N ≥ 3 and p : G → ℝ is a continuous function satisfying the following condition: (H 1 ) There exist a bounded set Ω ⊂ G, with positive N-dimensional Lebesgue measure, and positive constants p − , p + and δ such that The energy functional associated with (P) is given by Our goal in this paper is to obtain a critical point u of I satisfying u > 0 in G and We refer to such a critical point as a ground state solution of (P).
There are several works in the literature dealing with semilinear problems in the particular case p(x) ≡ 2 * . Let us mention some of them.
On the other hand, the subcritical problem has an unbounded set of solutions in H 1 0 (G) (see [13]). Problem (P 2 ) with q = 2 * − ϵ (ϵ > 0) was studied in the papers [3] and [14]. In the former, Atkinson and Peletier considered G a ball and determined the exact asymptotic behavior of the corresponding (radial) solutions u ϵ , as ϵ → 0. In [14], where a general bounded domain G was considered, Garcia Azorero and Peral Alonso provided an alternative for the asymptotic behavior of J ϵ (u ϵ ), as ϵ → 0, where J ϵ denotes the energy functional associated with problem (P 2 ) and q = 2 * − ϵ. More precisely, they showed that if where S denotes the Sobolev constant, then u ϵ converges to either a Dirac mass or a solution of the critical problem We recall that the Sobolev constant is defined by and given explicitly by the expression where Γ(t) = ∫ ∞ 0 s t−1 e −s ds is the Gamma function (see the papers [4] and [20] by Aubin and Talenti, respectively). Furthermore, S is achieved in (1.1) by the Aubin-Talenti function (and also by translations, rescalings and scalar multiples of it). This function also satisfies and is a ground state solution of problem (P) with G = ℝ N and p(x) ≡ 2 ⋆ . In [16], Kurata and Shioji studied the compactness of the embedding H 1 0 (G) → L p(x) (G) for a bounded domain G and a variable exponent 1 ≤ p(x) ≤ 2 * . (For the definition and properties of L p(x) (G) see [12]). They showed the existence of a positive solution of (P) under the hypothesis of existence of a point x 0 ∈ G, a small η > 0, 0 < l < 1 and c 0 > 0 such that p(x 0 ) = 2 * and In [1], Alves and Souto studied the existence of nonnegative solutions for the equation where the variable exponents p(x) and q(x) are radially symmetric functions satisfying 1 < ess inf for constants 0 < δ < R. Finally, in [18], Liu, Liao and Tang proved the existence of a ground state solution for (P) with G = ℝ N and where the constant p belongs to (2, 2 * ) and Ω ⊂ ℝ N has nonempty interior. In Section 2, motivated by the results of [18], we use the concentration-compactness lemma by P. L. Lions and properties of the Nehari manifold N to prove the existence of at least one ground state for problem (P) when G = ℝ N and p ∈ C(ℝ N , ℝ) is a function satisfying condition (H 1 ). A key point in the proof of our existence result is the achievement of the strict inequality and we get this by exploring the "projection" on the Nehari manifold of the sequence (w k ), where w k (x) = w(x + ke N ) and e N = (0, 0, . . . , 1) is the Nth coordinate vector.
In Section 3, we study the case where G is a bounded domain in ℝ N . In this case, the argument based on the sequences of translations of the Aubin-Talenti function is not applicable. Thus, in order to achieve the inequality (1.3) we assume an additional hypothesis (H 2 ) that is stated in terms of a subdomain U of Ω and the valueq := min{q ∈ (2, 2 * ] : where the function g : (2, 2 * ] → (0, ∞) is given by and S q (U) denotes the best constant of the embedding H 1 0 (U) → L q (U), that is, More precisely, we assume that the function p ∈ C(G, ℝ), satisfying (H 1 ), also verifies the following hypothesis, where Ω, p − and p + are defined in (H 1 ) and q is defined by (1.4): (H 2 ) There exists a subdomain U of Ω such that Under the hypotheses (H 1 ) and (H 2 ), we show that problem (P) has at least one ground state solution. We remark that the constant q in the statement of (H 2 ) can be any lower bound for q in the interval (p − , p + ). Considering that the particular value S 2 (U) is available for several domains (especially those with some kind of symmetry), we derive two lower bounds q 1 > q 2 for q in terms of S 2 (U), S and |U| and a third, q 3 , depending only on S and |U|. Moreover, we present sufficient conditions for S 2 (U) ≤ 1 to hold, when the subdomain U is either a ball B R or an annular-shaped domain B R \ B r , with B r ⊂ B R . We also show that if R and R − r are sufficiently large, then S 2 (U) < 1 for U = B R and U = B R \ B r , respectively.

The semilinear elliptic problem in ℝ N
In this section, we consider the semilinear elliptic problem with variable exponent where N ≥ 3 and p : ℝ N → ℝ is a continuous function verifying hypothesis (H 1 ). We recall that the space D 1,2 (ℝ N ) is the completion of C ∞ 0 (ℝ N ) with respect to the norm The dual space of D 1,2 (ℝ N ) will be denoted by D −1 . The energy functional I : D 1,2 (ℝ N ) → ℝ associated with (2.1) is given by where u + (x) = max{u(x), 0}. Hence, under hypothesis (H 1 ), we can write For a posterior use, let us estimate the second term in the above expression. For this, let u ∈ D 1,2 (ℝ N ) and consider the set E = {x ∈ Ω δ : |u(x)| < 1}. Then where we have used (H 1 ) and Hölder's inequality. Hence, it follows from (1.1) and (H 1 c) that We observe from (2.2) that the functional I is well defined. The next lemma establishes that I is of class C 1 . Since its proof is standard, it will be omitted.

Remark 2.2.
The previous lemma ensures that u ∈ D 1,2 (ℝ N ) is a weak solution of (2.1) if, and only if, u is a critical point of I (i.e. I (u) = 0). We remark that a critical point u of I is nonnegative, since where u − (x) = min{u(x), 0}. Consequently, according to the Strong Maximum Principle, if u ̸ ≡ 0 is a critical point of I, then u > 0 in ℝ N .

The Nehari manifold
In this subsection we prove some properties of the Nehari manifold associated with (2.1), which is defined by Of course, critical points of I belong to N. In the sequel we show important properties involving the Nehari manifold, which are crucial in our approach.

Proposition 2.4. Assume that (H 1 ) holds. Then m > 0.
Proof. For an arbitrary u ∈ N we have Thus, it follows from (2.2) that where C 1 and C 2 denote positive constants that do not depend on u. Consequently, from which we conclude that there exists η > 0 such that Therefore, Proof. Let We note that for all t ∈ (0, +∞).
Thus, we can see that f (t) > 0 for all t > 0 sufficiently small and also that f (t) < 0 for all t ≥ 1 sufficiently large. Therefore, there exists t u > 0 such that showing that t u u ∈ N.
In order to prove the uniqueness of t u , let us assume that 0 < t 1 < t 2 satisfy f (t 1 ) = f (t 2 ) = 0. Then Hence, for all x ∈ ℝ N , the above equality leads to the contradiction u + ≡ 0. Proposition 2.6. Assume that (H 1 ) holds. Then where η was given in (2.3). Hence, J (u) ̸ = 0 for all u ∈ N. Proof. Since m is the minimum of I on N, Lagrange multiplier theorem implies that there exists λ ∈ ℝ such that According to the previous proposition, λ = 0, and so, The next proposition shows that, under (H 1 ), there exists a Palais-Smale sequence for I associated with the minimum m.
Proof. According to the Ekeland variational principle (see [ It follows from (2.4) that This implies that (v n ) is bounded in D 1,2 (ℝ N ). Hence, taking into account that Using the fact that I (v n )(v n ) = 0, we conclude from Proposition 2.6 that λ n → 0. Consequently, Moreover, Now, let us fix t n > 0 such that u n := t n v + n ∈ N. Using the fact that I (u n )u n = 0 and I (v + n )v + n = o n (1), a simple computation gives t n → 1, so that I(u n ) − I(v + n ) = o n (1) and I (u n ) − I (v + n ) = o n (1). Hence, u n ∈ N, u n ≥ 0, I(u n ) → m and I (u n ) → 0.
This proves the proposition.
The next proposition provides a special upper bound for m. A direct computation shows that ‖w k ‖ 2 * = ‖w‖ 2 * and ‖w k ‖ 1,2 = ‖w‖ 1,2 . Moreover, exploring the expression of w, we can easily check that w k → 0 uniformly in bounded sets and, therefore, for any α > 0. By Proposition 2.5, there exists t k > 0 such that t k w k ∈ N, which means that Hence, and then, by using (2.5) for α = 2 * , we can verify that the sequence (t k ) is bounded, where we have used that the maximum value of the function t ∈ [0, ∞) → t 2 2 − t 2 * 2 * is 1 N .
Combining the boundedness of the sequence (t k ) with the fact that w k → 0 uniformly in Ω δ , we can select k sufficiently large, such that t k w k ≤ 1 in Ω δ . Therefore, for this k, since the latter integrand is strictly negative in Ω, which has positive N-dimensional Lebesgue measure.

Existence of a ground state solution
Our main result in this section is the following. We prove this theorem throughout this subsection by using the following well-known result. [17]). Let (u n ) be a sequence in D 1,2

Lemma 2.11 (Lions' lemma
Then there exist an at most countable set of indices I, points (x i ) i∈I and positive numbers (ν i ) i∈I such that

S for any i ∈ I, where δ x i denotes the Dirac measure supported at x i .
We know from Proposition 2.8 that there exists a sequence (u n ) ⊂ N satisfying u n ≥ 0 in ℝ N , I(u n ) → m and I (u n ) → 0 in D −1 . Since (u n ) is bounded in D 1,2 (ℝ N ), we can assume (by passing to a subsequence) that there exists u ∈ D 1,2 (ℝ N ) such that u n ⇀ u in D 1,2 (ℝ N ), u n → u in L s loc (ℝ N ) for 1 ≤ s < 2 * and u n (x) → u(x) a.e. in ℝ N . Moreover, |∇u n | 2 ⇀ μ and |u n | 2 * ⇀ ν in M(ℝ N ).
We claim that u ̸ ≡ 0. Indeed, let us suppose, by contradiction, that u ≡ 0. We affirm that this assumption implies that the set I given by Lions' lemma is empty. Otherwise, let us fix i ∈ I, x i ∈ ℝ N and ν i > 0 as in Lions' lemma. Let ϕ ∈ C ∞ c (ℝ N ) such that and 0 ≤ ϕ(x) ≤ 1 for all x ∈ ℝ N , where B 1 and B 2 denotes the balls centered at the origin, with radius 1 and 2, respectively. For ϵ > 0 fixed, define Since (u n ) is bounded in D 1,2 (ℝ N ), the same holds for the sequence (ϕ ϵ u n ). Thus,

Consequently,
According to Lions' lemma, Combining this inequality with part (ii) of Lions' lemma, we obtain ν i ≥ S N 2 . It follows that we have Since p : ℝ → ℝ is continuous, for each ϵ > 0, there exists Ω δ,ϵ ⊂ Ω δ such that Then, since u n → 0 in L s loc (ℝ N ), for s ∈ [1, 2 * ), and ϵ is arbitrary, we conclude that Therefore, by making n → ∞ in (2.7), we obtain which contradicts Proposition 2.9, showing that I = 0. Hence, it follows from Lions' lemma that In particular, u n → 0 in L 2 * (Ω δ ), so that Thus, by making n → ∞ in the equality we obtain m = L N ≥ 1 N S N 2 , which contradicts Proposition 2.9 and proves that u ̸ ≡ 0. Now, combining the weak convergence with the fact that I (u n ) → 0 in D −1 , we conclude that meaning that u is a nontrivial critical point of I.
Thus, taking into account Proposition 2.7, in order to complete the proof that u is a ground state solution for (2.1) we need to verify that I(u) = m. Indeed, since the weak convergence u n ⇀ u in D 1,2 (ℝ N ) and Fatou's lemma imply that showing that I(u) = m.

The semilinear elliptic problem in a bounded domain
In this section we consider the elliptic problem where G is a smooth bounded domain of ℝ N , N ≥ 3, and p : G → ℝ is a continuous function verifying (H 1 ) and an additional hypothesis (H 2 ), which is stated in the sequel. We recall that the usual norm in H 1 0 (G) is given by We denote the dual space of H 1 0 (G) by H −1 . The energy functional I : H 1 0 (G) → ℝ associated with problem (3.1) is defined by It belongs to C 1 (H 1 0 (G), ℝ) and its derivative is given by Thus, a function u ∈ H 1 0 (G) is a weak solution of (3.1) if, and only if, u is a critical point of I. Moreover, as in Section 2, the nontrivial critical points of I are positive in G (a consequence of the Strong Maximum Principle).
We maintain the notation of Section 2. Thus, We gather in the next lemma some results that can be proved as in Section 2. Unfortunately, hypothesis (H 1 ) by itself is not sufficient to guarantee that m < S N 2 N as in Proposition 2.9. The reason is that the translation argument used in the proof of that Proposition does not apply to a bounded domain. So we assume an additional assumption (H 2 ). In order to properly state such an assumption, we need some background information.
Let U ⊂ ℝ N be a bounded domain and define S q (U) := inf{ ‖∇v‖ 2 It is well known that if 1 ≤ q < 2 * , then the infimum in (3.2) is attained by a positive function ϕ q in H 1 0 (U). Actually, this follows from the compactness of the embedding H 1 0 (U) → L q (U). Another well-known fact is that in the case q = 2 * the infimum in (3.2) coincides with the best Sobolev constant, i.e.
Moreover, in this case the infimum (3.2) is not attained if U is a proper subset of ℝ N .
In the sequence we make use of the function for all q ∈ (2,q ).
Taking into account that S 2 * (U) = S, we can easily check that g(2 * ) = 1 N S N 2 . Thus, defininḡ Proof. Let ϕ q ∈ H 1 0 (U) denote a positive extremal function of S q (U). Thus, ϕ q > 0 in U and . Let us define the functionφ q ∈ H 1 0 (G) bỹ For each t > 0 we have Taking it is easy to see that t qφ ∈ N and Since S 2 (U) ≤ 1 and p − ≤ q < min{p + ,q }, it follows from Lemma 3.3 that This implies that m < 1 N S N 2 .
The main result in this section is the following. Proof. According to item (iv) of Lemma 3.2, there exists a sequence (u n ) ⊂ N satisfying I(u n ) → m and I (u n ) → 0 in H −1 . Since (u n ) is bounded in H 1 0 (G), there exist u ∈ H 1 0 (G) and a subsequence, still denoted by (u n ), such that u n ⇀ u in H 1 0 (G), u n → u in L p (G), for 1 ≤ p < 2 * , and u n (x) → u(x) a.e. in G. Arguing as in Section 2, we can combine Lions' lemma and Lemma 3.4 to prove that u ̸ ≡ 0, I (u) = 0 and I(u) = m, showing thus that u is a ground state solution of (3.1).

On hypothesis (H 2 )
In this subsection we present some lower bounds for the value of q, defined by (3.6), which can be used as the value constant for p(x) in hypothesis (H 2 ). Moreover, we give some examples of simple bounded domains U such that S 2 (U) ≤ 1.
The value of q depends on the function q → g(q), which in turn depends on the function q → S q (U). It is well known that S q (U) is the least value of λ for which the Dirichlet problem has a nontrivial weak solution. When p = 2, this is the well-studied eigenvalue problem for (−∆, H 1 0 (U)) and S 2 (U) is its first eigenvalue. It follows that S 2 (U) can be found analytically for some simple domains as balls, rectangles and other domains enjoying some kind of symmetry. For instance, if U is a ball of radius R, then where j α,1 denotes the first positive root of the first kind Bessel function of order α.
When q ̸ = 2 the above problem is no longer linear and, consequently, it is more difficult to be solved analytically, even for simple domains. For this reason, determining an analytical expression for the function g on the interval (2, 2 * ) is a hard task and we do not know the exact value ofq given by (3.6). However, the inequality (3.5) allows us to derive lower bounds forq , in terms of S 2 (U), |U| and S, which can be used as the value constant for p(x) in hypothesis (H 2 ). In fact, assuming S 2 (U) ≤ 1, we can easily verify that g 1 (q) > 0 for all q ∈ (2, 2 * ], where the function q → g 1 (q) is defined in (3.5). Therefore, taking into account that lim q→2 + g 1 (2) = 0 and g 1 (q ) > g(q ) = 1 N S N 2 , there exists a unique value q 1 ∈ (2,q ) such that an equation that can be solved at least numerically. A rougher but explicit lower bound q 2 forq follows from the inequality g 1 (q) < g 2 (q) := |U| N S 2 (U) q q−2 , q ∈ (2, 2 * ), which is obtained from (3.5) by observing that 1 2 − 1 q ≤ 1 2 − 1 2 * = 1 N . Indeed, since the function g 2 enjoys the same properties as g 1 , there exists a unique point q 2 ∈ (2,q ) satisfying g 2 (q 2 ) = 1 N S N 2 . A simple calculation yields Of course, 2 < q 2 < q 1 < q. A third lower bound q 3 forq also follows from (3.5). Indeed, by using that S 2 (U) ≤ 1 in (3.5), we obtain g(q) < g 3 (q) := |U|( Hence, since g 3 > 0 and g 3 (2) = 0, there exists a unique point q 3 ∈ (2,q ) satisfying g 3 (q 3 ) = S N 2 N . Such a point is given explicitly by Another conclusion that follows easily from the monotonicity of the function q → |U| In the sequel, we present sufficient conditions for the inequality S 2 (U) ≤ 1 to hold when U is either a ball or an annulus. We will denote by B R (y) the ball centered at y with radius R > 0. When y = 0, we will write simply B R . Moreover, a simple scaling argument (or (3.7)) yields So, if R ≥ S 2 (B 1 ) 1 2 , then S 2 (U) = S 2 (B R (y)) ≤ 1.
Thus, we can replace the condition S(U) ≤ 1 in (H 2 ) by either R ≤ S 2 (B 1 )